# On expansions for nonlinear systems, error estimates and convergence issues

Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach

► To cite this version:

Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach. On expansions for nonlinear systems, error estimates and convergence issues. Comptes Rendus. Mathématique, 2023, 361, pp.97-189. 10.5802/crmath.395. hal-03204519v3

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https://hal.science/hal-03204519v3

Submitted on 10 Mar 2023

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Distributed under a Creative Commons Attribution 4.0 International License
On expansions for nonlinear systems,
error estimates and convergence issues

Karine Beauchard\*, Jérémy Le Borgne\*, Frédéric Marbach\*

March 10, 2023

Abstract

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.

First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.

Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.

Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion.

Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

Contents

<table><tr><td><b>1 Introduction</b></td><td><b>2</b></td></tr><tr><td>1.1 Motivations</td><td>2</td></tr><tr><td>1.2 Short historical survey</td><td>3</td></tr><tr><td>1.3 Main goals and organization of this article</td><td>5</td></tr><tr><td><b>2 Formal expansions for linear dynamics</b></td><td><b>6</b></td></tr><tr><td>2.1 Notations</td><td>6</td></tr><tr><td>2.2 Formal differential equations and iterated integrals</td><td>9</td></tr><tr><td>2.3 Logarithm of flows, coordinates of the first kind</td><td>11</td></tr><tr><td>2.4 Interaction picture, coordinates of the pseudo-first kind</td><td>20</td></tr><tr><td>2.5 Infinite product, coordinates of the second kind</td><td>24</td></tr></table>

<footnote>*Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France</footnote>
<table>
  <tbody>
    <tr>
      <td>3</td>
      <td>Technical tools about functions and vector fields</td>
      <td>28</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Functional spaces for finite or analytic regularity</td>
      <td>28</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Estimates for differential operators and Lie brackets</td>
      <td>30</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Well-posedness of ordinary differential equations</td>
      <td>32</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Flows, compositions and pushforwards</td>
      <td>33</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Error estimates in time for nonlinear vector fields</td>
      <td>37</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Linearization principle for nonlinear vector fields</td>
      <td>37</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Chen-Fliess expansion</td>
      <td>39</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Magnus expansion in the usual setting</td>
      <td>39</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Magnus expansion in the interaction picture</td>
      <td>44</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Sussmann's infinite product expansion</td>
      <td>46</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Convergence results and issues</td>
      <td>48</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Chen-Fliess expansion</td>
      <td>49</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Magnus expansion in the usual setting</td>
      <td>51</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Magnus expansion in the interaction picture</td>
      <td>56</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Sussmann's infinite product expansion</td>
      <td>59</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Error estimates for control systems</td>
      <td>67</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Chen-Fliess expansion</td>
      <td>67</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Magnus expansion in the usual setting: a counter-example</td>
      <td>69</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Magnus expansion in the interaction picture</td>
      <td>70</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Sussmann's infinite product expansion</td>
      <td>70</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Refined error estimates for scalar-input affine systems</td>
      <td>76</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Auxiliary system trick</td>
      <td>76</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>A new formulation of the Chen-Fliess expansion</td>
      <td>77</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Magnus expansion in the interaction picture</td>
      <td>78</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>Sussmann's infinite product expansion</td>
      <td>83</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>Failure of the primitive estimate for multiple inputs</td>
      <td>85</td>
    </tr>
    <tr>
      <td>8</td>
      <td>On direct intrinsic representations of the state</td>
      <td>86</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Approximate direct intrinsic representations</td>
      <td>86</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Diffeomorphisms and Lie brackets</td>
      <td>88</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Replacing the Magnus flow by a diffeomorphism</td>
      <td>89</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Proving that the logarithm of the flow is a Lie series</td>
      <td>92</td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>Using shuffle relations and Ree's theorem</td>
      <td>92</td>
    </tr>
    <tr>
      <td>A.2</td>
      <td>Using Friedrich's criterion</td>
      <td>93</td>
    </tr>
    <tr>
      <td>B</td>
      <td>Elementary numerical identities</td>
      <td>94</td>
    </tr>
    <tr>
      <td>B.1</td>
      <td>Bernoulli numbers</td>
      <td>94</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

## 1.1 Motivations

There are multiple situations in which one desires to compute the solution to a differential equation whose dynamics depend on time. One often looks for explicit formulas, depending preferentially on intrinsic quantities, which describe the composition of flows, or even the continuous composition of flows. Some important applications are listed below.
* **Control theory.** Here, the dynamics depend on time mostly through the choice of time-varying controls. One looks for explicit formulas of the continuous product of flows in order to be able to construct controls for which this resulting flow drives a given initial state to a desired target state. In order to establish necessary and sufficient conditions for controllability, one is interested in intrinsic formulas. It is our main motivation.

* **Numerical splitting methods.** Here, the splitting algorithm applies sequentially a succession of basic flows, composed with appropriate time steps. One is interested in choosing correctly the base flows and the time steps in order to approximate the most precisely possible the solution to the true complex flow. Formulas concerning the composition of flows are essential to compute the order of the resulting numerical scheme. We refer to the survey [20] and the introduction books [19, 58]. Composition of flows formulas are also very useful in particular settings like Hamiltonian systems [22] or in the presence of a small perturbation of a reference flow [81]. Concerning numerical methods, more generally, we refer to [82] (respectively [37]) for a survey on Butcher series (resp. post-Lie algebras), algebraic tools related to the algebras manipulated in the sequel.

* **Stochastic differential equations.** Here, the dynamics depend on time through the sources of randomness, say Brownian motions. One wishes to investigate the influence of the randomness on the final state and thus looks for explicit formulas involving iterated Stratanovich integrals to construct a representation of the flow, see e.g. [12, 15, 28, 32].

* **Differential equations on Lie groups.** Sometimes, the state itself of the differential equation belongs to a Lie group, as in [64]. Then, looking for an intrinsic approximation of the state helps to preserve structure which would be lost otherwise. In particular, writing the product of multiple flows as a single flow is important. There are also control problems for differential equations set on Lie groups, as in [67]. Some works, e.g. [30], also tackle the hard question of obtaining Magnus-type expansions, which are intrinsically linear, for nonlinear equations within matrix Lie groups.

* **Analysis of time-periodic systems.** When investigating the behavior of time-periodic systems, some authors borrow tools from “chronological calculus” or expressions of the “logarithm of the flow” (described below). For example, such techniques are used to study stability and asymptotic stability of time-periodic systems of ODEs; see the non linear Floquet Theorem 3.2 and the high-order averaging procedure Theorem 7.1 in [93], or the recent higher-order averaging results of [79].

## 1.2 Short historical survey

We start with a short survey of some of the many approaches related with the computation of solutions to formal linear differential equations, say

$$ \dot{x}(t) = X(t)x(t), \qquad (1) $$

together with some initial condition $x(0)$. We recall in Section 1.2.4 the consequences of such results for nonlinear ordinary differential equations.

### 1.2.1 Iterated integration and Chen-Fliess expansion

A straightforward approach to solving (1) consists in what can be seen as a Picard iteration. For small times, starting from the initial approximation $x(t) \approx x(0)$, one then enhances the approximation iteratively by substituting it in the equation and obtains successively $x(t) \approx x(0) + \int_0^t X(s)x(0) ds$, then $x(t) \approx x(0) + \int_0^t X(s)x(0) ds + \int_0^t X(s) \int_0^s X(s')x(0) ds' ds$ and so on.

In the context of control theory, this expansion is known as the Chen-Fliess expansion, after being popularized by the works [33, 45]. Its main advantages are its simplicity and nice convergence
properties (see Section 5.1). However, it also has some strong drawbacks, which we detail in Remark 17 and Remark 167 and motivate the investigation of other expansions.

### 1.2.2 Magnus expansion

When $X(t)$ is piecewise constant, for example with values $X_1$ for $t \in [0, 1]$ and $X_2$ for $t \in [1, 2]$, one has formally, $x(2) = e^{X_2}e^{X_1}x(0)$. Hence, the computation of solutions to (1) has a deep link with the famous Campbell [29], Baker [11], Hausdorff [61], Dynkin [40] formula ("CBHD formula" in the sequel).

This formula has a long and rich history which involves forgotten contributions of other authors such as Schur, Poincaré, Pascal or Yosida. As noted by Bourbaki in [25], "chacun considère que les démonstrations de ses prédécesseurs ne sont pas convaincantes" (each one considers that the proofs of his predecessors are not convincing). We therefore encourage the reader to dive into the fascinating retrospectives [2] and [23] to understand the progressive construction of its proof throughout the decades. This formula is a formal identity expressing the product of the exponentials of two (non-commutative) indeterminates $X_1$ and $X_2$ as the single exponential of a series of Lie brackets (i.e. nested commutators) of these indeterminates, of which the first terms are well-known:

$$e^{X_2}e^{X_1} = \exp \left( X_1 + X_2 + \frac{1}{2}[X_2, X_1] + \dots \right). \qquad (2)$$

When more than two exponentials are multiplied, say $e^{X_1}$ through $e^{X_n}$, one can of course iterate the formula (2) with itself to formally express the product of $n$ exponentials as the single exponential of a complicated series. Letting $n \to +\infty$, one is lead to computing a continuous product of exponentials, which corresponds, heuristically, to solving (1).

Magnus performed a breakthrough by deriving in [80] the first formal representation of the solution to (1) as the exponential of a series, of which the first terms are

$$x(t) = \exp \left( \int_{0}^{t} X(\tau_1) d\tau_1 + \frac{1}{2} \int_{0}^{t} \int_{0}^{\tau_1} [X(\tau_1), X(\tau_2)] d\tau_2 d\tau_1 + \dots \right) x(0). \qquad (3)$$

This formula can be seen as the continuous counterpart of the CBHD formula and highlights important structural properties of the solutions to (1) (see Section 2.3).

### 1.2.3 Infinite products

The CBHD formula and the Magnus formula share the goal of expressing the desired quantity as the exponential of a single, although complicated, object. Other approaches go the other way around and try to express the desired quantity as a long (infinite) product of exponentials of very simple objects.

A well-known example is the Lie-Trotter product formula (see e.g. [101]), often used for numerical splitting methods which attempts to give a meaning to the equality

$$e^{X_1+X_2} = \lim_{n \to +\infty} \left( e^{\frac{X_1}{n}} e^{\frac{X_2}{n}} \right)^n, \qquad (4)$$

the interest relying on the fact that the exponentials of $X_1$ and $X_2$ are assumed to be easier to compute in some sense than the direct exponential of $X_1 + X_2$.

Another related formula is the Zassenhaus expansion, described by Magnus in [80], which allows to decompose the same quantity $e^{X_1+X_2}$ as an infinite product of exponentials of linear combinations of nested commutators of strictly increasing lengths, whose first terms are

$$e^{X_1+X_2} = e^{X_1} e^{X_2} \exp \left( -\frac{1}{2}[X_1, X_2] \right) \exp \left( \frac{1}{3}[X_2, [X_1, X_2]] + \frac{1}{6}[X_1, [X_1, X_2]] \right) \cdots \qquad (5)$$
In the context of differential equations such as (1), a nice formula is Sussmann’s infinite product expansion, introduced in [100]. When $X(t)$ is given as a linear combination of elementary generators, e.g. $X(t) = a_1(t)X_1 + a_2(t)X_2$, Sussmann’s infinite product expansion is given by a product of exponentials of Lie monomials, such as

$$x(t) = e^{\xi_1 X_1} e^{\xi_2 X_2} e^{\xi_{12}[X_1, X_2]} e^{\xi_{112}[X_1, [X_1, X_2]]} e^{\xi_{212}[X_2, [X_1, X_2]]} \dots x(0), \quad (6)$$

where the $\xi_i$ are scalar functions of time given by explicit formulas from the functions $a_1$ and $a_2$. Compared to other expansions, this formula is both intrinsic (such as the Magnus expansion) and involves coefficients which are easily computed by induction (such as the Chen-Fliess expansion).

### 1.2.4 Consequences for nonlinear ordinary differential equations

Although the expansions mentioned above concern linear formal differential equations, they can be adapted to ordinary nonlinear differential equations on smooth manifolds governed by smooth vector fields. Indeed, one can identify vector fields with linear operators acting on smooth functions, and points of the manifold with the linear operator on smooth functions corresponding to evaluation at this point. This method allows to recast the nonlinear equation into a linear equation set on a larger space, for which the formal linear expansions can be used (see Section 4.1).

This linearization technique has been used by Sussmann in [99, Proposition 4.3] to prove the convergence of the Chen-Fliess expansion for nonlinear ordinary differential equations driven by analytic vector fields, by Agrachev and Gamkrelidze in the context of control theory (see [3, 4, 49] in which they derive an exponential representation of flows, very similar to Magnus’ expansion, using the chronological calculus framework) and by Strichartz (see [97] and his derivation of the generalized CBHD formula, with applications related to sub-Riemannian geometry).

At a formal level, all identities mentioned above (almost) always make sense. However, if the indeterminates are replaced by objects coming from analysis (say vector fields, matrices or differential operators), convergence issues arise. Generally speaking, convergence often requires that one either assumes that the objects are small enough or that the generated Lie algebra has additional structure, like nilpotence.

## 1.3 Main goals and organization of this article

This article is both a survey on some classical expansions for nonlinear systems, a research article containing new results and counter-examples and a toolbox for future works. In particular, we aim at the following goals.

* In Section 2 we give a **didactic review of classical expansions for formal linear differential equations**. Our introduction to this algebraic topic is written with a view to making it understandable by readers with minimal algebraic background. We review the following classical expansions:

1. the Chen-Fliess formula,

2. the Magnus or generalized CBHD formula (associated with coordinates of the first kind),

3. Sussmann’s infinite product formula (associated with coordinates of the second kind).

* We introduce a **new formal mixed expansion**, inspired by quantum mechanics, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation (see Theorem 41), which we name Magnus expansion in the interaction picture and for which we define coordinates of the pseudo-first kind by analogy with first and second kind coordinates.

* We recall in Section 3 **classical well-posedness results and estimates** for products and Lie brackets of analytic vector fields, which are used throughout the article.
* In the context of nonlinear ordinary differential equations driven by regular vector fields, we give in Section 4 **rigorous proofs of error estimates** between the exact solution and finite approximations of each of these four formal expansions. These estimates are part of the mathematical folklore for the Chen-Fliess and Magnus expansions, but are new for our mixed expansion (see Proposition 99) and for Sussmann's infinite product expansion (see Proposition 104). We strive towards providing estimates with similar structures for the four expansions and which are valid under parsimonious regularity assumptions.

* We investigate the convergence of these expansions in Section 5. We recall **known positive convergence results** for smooth vector fields generating nilpotent Lie algebras and for small linear dynamics (matrices). For our new expansion, we investigate the subtle convergence under a natural partial nilpotency assumption (see Corollary 122). In this case, convergence requires analyticity, contrary to the proofs we give for the other expansions under a full nilpotency assumption.

* For analytic vector fields, only the Chen-Fliess expansion is known to converge. We give in Section 5.2 **new strong counter examples** to the convergence of CBHD and Magnus expansions, which **disprove the convergence** of these expansions even for analytic vector fields and in very weak senses (see Proposition 114). We state an open problem concerning the convergence of Sussmann's infinite product for analytic vector fields (see Open problem 135).

* When the system involves a time-invariant drift and a time-varying perturbation, we show in Section 6 that only the Magnus expansion fails to provide **well-behaved estimates with respect to the perturbation size**. For the three other expansions, it turns out to be possible to obtain such estimates by summing well-defined infinite partial series which converge for analytical vector fields (see mostly Propositions 139 and 143).

* In the particular case of scalar-input systems, we prove in Section 7 **new errors estimates involving a negative Sobolev norm of the time-varying input** (see mostly Propositions 149 and 157). Such estimates are the best compatible with the regularity of the input-to-state map and can be helpful for specific applications.

* Eventually, we derive in Section 8 **approximate direct intrinsic representations of the state** for nonlinear systems, which don't require the computation of flows (see Proposition 161). Our formulas can be viewed as almost-diffeomorphisms and might be useful for applications in control theory. Unfortunately, we also study a counter-example which demonstrates that one cannot obtain an exact representation through a diffeomorphism.

# 2 Formal expansions for linear dynamics

In this section, we consider formal linear differential equations, recall classical expansions valid in this formal setting (for which there is no convergence issue; see nevertheless Remark 12) and introduce a new mixed expansion which isolates the role of a perturbation in the dynamics. Here and in the sequel, the adjective *formal* denotes situations in which we work within the realm of formal power series (see Definition 11).

## 2.1 Notations

We recall classical definitions and notations for usual algebraic objects. In the sequel, $\mathbb{K}$ denotes the field $\mathbb{R}$ or $\mathbb{C}$. All statements and proofs hold for both base fields. It will be implicit that all vector spaces and algebras are constructed from the base field $\mathbb{K}$.
### 2.1.1 Free algebras

We refer to the books [65, 90] for thorough introductions to Lie algebras and free Lie algebras.

**Definition 1** (Indeterminates). Let $I$ be a finite set. At the formal level, we consider a set $X := \{X_i; i \in I\}$ of indeterminates, indexed by $I$. For applications, we will substitute in their place matrices or vector fields. Most often, we will write $I = [[1, q]]$ for some $q \in \mathbb{N}^*$, or $I = [[0, q]]$ when we want to isolate the role of the indeterminate $X_0$.

**Definition 2** (Free monoid). For $I$ as above, we denote by $I^*$ the free monoid over $I$, i.e. the set of finite sequences of elements of $I$ endowed with the concatenation operation. More precisely, if $\sigma = (\sigma_i)_{1 \le i \le \ell}$ and $\sigma' = (\sigma'_i)_{1 \le i \le \ell'}$ are elements of $I^*$, then the concatenation of $\sigma$ and $\sigma'$ is the sequence $\sigma \cdot \sigma' = (\sigma''_i)_{1 \le i \le \ell+\ell'}$ where $\sigma''_i = \sigma_i$ if $1 \le i \le \ell$ and $\sigma''_i = \sigma'_{i-\ell}$ if $\ell+1 \le i \le \ell+\ell'$. It is common to write the elements of $I^*$ as words whose letters are elements of $I$, by juxtaposition of the elements of the sequence. With this point of view, the concatenation operation is the juxtaposition of words. For a more detailed exposition, see [26, §7.2].

For $\sigma = (\sigma_1, \dots, \sigma_k) \in I^*$, where $k$ is the length of $\sigma$ also denoted by $|\sigma|$, we let $X_\sigma := X_{\sigma_1} \cdots X_{\sigma_k}$. This operation defines an homomorphism from $I^*$ to $X^*$, the free monoid over $X$ (monomials over $X$).

**Definition 3** (Free algebra). For $X$ as above, we consider $\mathcal{A}(X)$ the free associative algebra generated by $X$ over the field $\mathbb{K}$, i.e. the unital associative algebra of polynomials of the non commutative indeterminates $X$ (see also [26, Chapter 3, Section 2.7, Definition 2]). $\mathcal{A}(X)$ can be seen as a graded algebra:

$$ \mathcal{A}(X) = \bigoplus_{n \in \mathbb{N}} \mathcal{A}_n(X), \qquad (7) $$

where $\mathcal{A}_n(X)$ is the finite-dimensional $\mathbb{K}$-vector space spanned by monomials of degree $n$ over $X$. In particular $\mathcal{A}_0(X) = \mathbb{K}$ and $\mathcal{A}_1(X) = \text{span}_{\mathbb{K}}(X)$.

**Definition 4** (Free Lie algebra). For $X$ as above, $\mathcal{A}(X)$ is endowed with a natural structure of Lie algebra, the Lie bracket operation being defined by $[a, b] = ab - ba$. This operation satisfies $[a, a] = 0$ and the Jacobi identity $[a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0$. We also write $[a, b]$ as $\text{ad}_a(b)$ (respectively $\text{ad}_b(a)$) which allows for iterated left (resp. right) bracketing. We consider $\mathcal{L}(X)$, the free Lie algebra generated by $X$ over the field $\mathbb{K}$, which is defined as the Lie subalgebra generated by $X$ in $\mathcal{A}(X)$. It can be seen as the smallest linear subspace of $\mathcal{A}(X)$ containing all elements of $X$ and stable by the Lie bracket (see also [90, Theorem 0.4]). $\mathcal{L}(X)$ is a graded Lie algebra:

$$ \mathcal{L}(X) = \bigoplus_{n \in \mathbb{N}} \mathcal{L}_n(X), \quad [\mathcal{L}_m(X), \mathcal{L}_n(X)] \subset \mathcal{L}_{m+n}(X) \qquad (8) $$

where, for each $n \in \mathbb{N}$, we define $\mathcal{L}_n(X) := \mathcal{L}(X) \cap \mathcal{A}_n(X)$.

**Definition 5** (Nilpotent Lie algebra). Let $L$ be a Lie algebra. We define recursively the following two-sided Lie ideals: $L^1 := L$ and, for $k \ge 1$, $L^{k+1} := [L, L^k]$ i.e. $L^{k+1}$ is the linear subspace of $L$ generated by brackets of the form $[a, b]$ with $a \in L$ and $b \in L^k$. Let $m \in \mathbb{N}^*$. We say that $L$ is a nilpotent Lie algebra of index $m$ when $L^m = \{0\}$ and $m$ is the smallest integer for which this property holds.

**Definition 6** (Free nilpotent Lie algebra). Let $m \in \mathbb{N}^*$. The free $m$-nilpotent Lie algebra over $X$ is the quotient $\mathcal{N}_m(X) := \mathcal{L}(X)/\mathcal{L}(X)^m$ (with the notation of Definition 5). Then the canonical surjection $\sigma_m : \mathcal{L}(X) \to \mathcal{N}_m(X)$ is an homomorphism of Lie algebras.

The universal properties of the various free algebras constructed above allow to transport on algebras relations proved at the free level.

**Lemma 7.** The following universal properties hold.
* For each unital associative algebra $A$ and map $\Lambda : X \to A$, there exists a unique homomorphism of algebras $\mathcal{A}(X) \to A$ that extends $\Lambda$.

* For each Lie algebra $L$ and map $\Lambda : X \to L$, there exists a unique homomorphism of Lie algebras $\mathcal{L}(X) \to L$ that extends $\Lambda$.

* Let $m \in \mathbb{N}^*$. For each nilpotent Lie algebra $L$ of index $m$ and map $\Lambda : X \to L$, there exists a unique homomorphism of Lie algebras $\mathcal{N}_m(X) \to L$ that extends $\Lambda$.

### 2.1.2 Iterated brackets and evaluation

**Definition 8 (Iterated brackets).** For $X$ as above, we consider $\mathrm{Br}(X)$ the set of iterated brackets of elements of $X$. This set can be defined by induction: for $X_i \in X$, $X_i \in \mathrm{Br}(X)$ and if $b_1, b_2 \in \mathrm{Br}(X)$, then the ordered pair $(b_1, b_2)$ belongs to $\mathrm{Br}(X)$. More rigorously, one can define $\mathrm{Br}(X)$ as the free magma over $X$ or as the set of rooted full binary trees, with leaves labeled by $X$.

For $b \in \mathrm{Br}(X)$, we will use the following notations:

* $|b|$ will denote the length of $b$ (i.e. the number of leaves of the tree).

* If $|b| > 1$, there exists a unique pair $b_1 \in \mathrm{Br}(X)$ and $b_2 \in \mathrm{Br}(X)$ such that $b = (b_1, b_2)$ (left and right factors) which are denoted as $\lambda(b) = b_1$ and $\mu(b) = b_2$. We also write $(b_1, b_2)$ as $\mathrm{ad}_{b_1}(b_2)$ (respectively $\mathrm{ad}_{b_2}(b_1)$) which allows iterated left (resp. right) bracketing.

* For $i \in I$, $n_i(b)$ denotes the number of occurrences of the indeterminate $X_i$ in $b$. When $I = [0, q]$ we will also write $n(b) = n_1(b) + \dots + n_q(b) = |b| - n_0(b)$.

**Remark 9.** There is a natural evaluation mapping $\mathrm{E}$ from $\mathrm{Br}(X)$ to $\mathcal{L}(X)$ defined by induction by $\mathrm{E}(X_i) := X_i$ for $X_i \in X$ and $\mathrm{E}((b_1, b_2)) := [\mathrm{E}(b_1), \mathrm{E}(b_2)]$. Through this mapping, $\mathrm{Br}(X)$ spans $\mathcal{L}(X)$ over $\mathbb{K}$, i.e. $\mathcal{L}(X) = \mathrm{span}_{\mathbb{K}} \mathrm{E}(\mathrm{Br}(X))$. This mapping is however not injective: for example, $(X_1, X_1)$ and $(X_2, (X_1, X_1))$ are two different elements of $\mathrm{Br}(X)$, both evaluated to zero in $\mathcal{L}(X)$. More precisely, the $\mathrm{E}$ map extends to a surjective homomorphism of algebras from the nonassociative free algebra over $X$ (which is the free vector space over $\mathrm{Br}(X)$, whose elements are (finite) linear combinations of elements of $\mathrm{Br}(X)$, endowed with the natural product map induced by the product in $\mathrm{Br}(X)$). Moreover the kernel of the extended $\mathrm{E}$ is precisely the ideal generated by the relations that define anticommutativity and the Jacobi identity in $\mathcal{L}(X)$. This gives an alternative description of $\mathcal{L}(X)$ as a quotient of the free vector space over $\mathrm{Br}(X)$.

**Definition 10 (Subspaces of brackets).** When $I = [0, q]$ and $M \in \mathbb{N}$, $S_M$ denotes the vector subspace $\mathcal{L}(X)$ defined by

$$S_M := \mathrm{span}_{\mathbb{K}} \{\mathrm{E}(b); b \in \mathrm{Br}(X), n(b) \le M\}. \qquad (9)$$

### 2.1.3 Formal power series, exponential and logarithms

**Definition 11 (Formal power series).** We consider the (unital associative) algebra $\hat{\mathcal{A}}(X)$ of formal power series generated by $\mathcal{A}(X)$. An element $a \in \hat{\mathcal{A}}(X)$ is a sequence $a = (a_n)_{n \in \mathbb{N}}$ written $a = \sum_{n \in \mathbb{N}} a_n$, where $a_n \in \mathcal{A}_n(X)$ with, in particular, $a_0 \in \mathbb{K}$ being its constant term. We also define the Lie algebra of formal Lie series $\hat{\mathcal{L}}(X)$ as the Lie algebra of formal power series $a \in \hat{\mathcal{A}}(X)$ for which $a_n \in \mathcal{L}(X)$ for each $n \in \mathbb{N}$. For $a \in \hat{\mathcal{A}}(X)$ and $\sigma \in I^*$, $\langle a, X_\sigma \rangle$ denotes the coefficient of $X_\sigma$ in $a$: $a = \sum_{\sigma \in I^*} \langle a, X_\sigma \rangle X_\sigma$.

**Remark 12.** The definition of $\hat{\mathcal{A}}(X)$ can be made more rigorous by considering $\mathrm{val} : \mathcal{A}(X) \to \mathbb{N} \cup \{\infty\}$ defined by $\mathrm{val}(a) := \inf\{n \in \mathbb{N}; a \in \bigoplus_{k \ge n} \mathcal{A}_k(X)\}$. Then $\delta(a, b) := e^{-\mathrm{val}(b-a)}$ is a distance on $\mathcal{A}(X)$, that induces the discrete topology on each $\mathcal{A}_n(X)$, and $\hat{\mathcal{A}}(X)$ is defined as
the completion of the metric space $\mathcal{A}(X)$, to which the operations on $\mathcal{A}(X)$ naturally extend as
continuous operations, endowing it with a structure of topological algebra. This distance verifies a
stronger triangular inequality: $\delta(a, b) \le \max\{\delta(a, c), \delta(b, c)\}$ (usually referred to as the ultrametric
inequality). This construction allows to write, for $a \in \hat{\mathcal{A}}(X)$ with components $a_n \in \mathcal{A}_n(X)$,

$$
a = \lim_{N \to +\infty} \sum_{n \le N} a_n, \qquad (10)
$$

where the convergence holds with respect to the topology described above. This justifies the notation
$a = \sum_{n \in \mathbb{N}} a_n$ used in Definition 11. To avoid confusion with convergence issues associated with
the evaluation of formal power series when substituting the indeterminates by objects coming from
analysis we shall however not use the term convergence in this context.

If $a \in \hat{\mathcal{A}}(X)$ has zero constant term, we define $\exp(a) \in \hat{\mathcal{A}}(X)$ and $\log(1+a) \in \hat{\mathcal{A}}(X)$ as

$$
\exp(a) := \sum_{m \ge 0} \frac{a^m}{m!}, \qquad (11)
$$

$$
\log(1+a) := \sum_{m \ge 1} \frac{(-1)^{m-1}}{m} a^m. \qquad (12)
$$

Since $a$ has zero constant term, one checks that the right-hand sides of (11) and (12) indeed define
formal power series of $\hat{\mathcal{A}}(X)$ (and the sums converge in the sense of the topology constructed in
Remark 12). In particular, $\log(\exp(a)) = a$ and $\exp(\log(1+a)) = 1+a$.

**Lemma 13.** Let $a, b \in \hat{\mathcal{A}}(X)$ with zero constant term. Then $a = b$ if and only if $\exp(a) = \exp(b)$.

*Proof.* The forward implication is obvious. Conversely, if $\exp(a) = \exp(b)$ in $\hat{\mathcal{A}}(X)$, then, for every $r \ge 1$, their components in $\mathcal{A}_r$ are equal. Moreover, from (11), one has:

$$
(\exp(a))_r = \sum_{k=1}^{r} \sum_{r_1 + \dots + r_k = r} \frac{a_{r_1} \dots a_{r_k}}{k!} = a_r + \Theta_r (a_1, \dots a_{r-1}), \quad (13)
$$

for some function $\Theta_r$ depending only on the $a_{r'}$ for $r' < r$. Hence, we obtain by induction on $r \ge 1$
that $a_r = b_r$ from the equalities $(\exp(a))_r = (\exp(b))_r$. $\square$

2.2 Formal differential equations and iterated integrals

Using the notations of Section 2.1, for $i \in I$, let $a_i \in L^1(\mathbb{R}_+; \mathbb{K})$ and define $a$ by

$$
a(t) := \sum_{i \in I} a_i(t) X_i. \qquad (14)
$$

In this section, we consider the following formal ordinary differential equation set on $\hat{\mathcal{A}}(X)$, driven
by $a$ and associated with some initial data $x^*$,

$$
\begin{cases}
\dot{x}(t) = x(t)a(t), \\
x(0) = x^*,
\end{cases}
\qquad (15)
$$

whose solutions are precisely defined in the following way.

**Definition 14 (Solution to a formal differential equation).** Let $a_i \in L^1(\mathbb{R}_+; \mathbb{K})$ for $i \in I$ and define $a$ by (14). Let $x^* \in \hat{\mathcal{A}}(X)$ with homogeneous components $x_n^* \in \mathcal{A}_n(X)$. The solution to the formal differential equation (15) is the formal-series valued function $x : \mathbb{R}_+ \to \hat{\mathcal{A}}(X)$, whose homogeneous
components $x_n : \mathbb{R}_+ \to \mathcal{A}_n(X)$ are the unique continuous functions that satisfy, for every $t \ge 0$, $x_0(t) = x_0^*$ and, for every $n \in \mathbb{N}^*$,

$$x_n(t) = x_n^* + \int_0^t x_{n-1}(\tau)a(\tau)d\tau. \qquad (16)$$

**Definition 15** (Ordered simplex). For $r \in \mathbb{N}^*$ and $t > 0$, we introduce

$$\Delta^r(t) := \{(\tau_1, \dots, \tau_r) \in (0, t)^r; 0 < \tau_1 < \dots < \tau_r < t\}. \qquad (17)$$

Iterating this integral formula yields the following power series expansion, which is the most direct way to compute the solution to (15) and was introduced in [33, 34] and popularized in control theory by [45]. In the “chronological calculus” terminology (not used in the present article), it is called “(right) formal Volterra chronological series” [3, Section 1.5].

**Lemma 16** (Chen series). In the context of Definition 14, the solution to (15) with initial data $x^* = 1$ can be expanded as

$$x(t) = \sum_{\sigma \in I^*} \left( \int_{0}^{t} a_{\sigma} \right) X_{\sigma}, \qquad (18)$$

where $\int_0^t a_\emptyset = 1$ by convention and, for $\sigma \in I^*$ with $|\sigma| \ge 1$, we introduce the notation

$$\int_{0}^{t} a_{\sigma} := \int_{\Delta^{n}(t)} a_{\sigma_{1}}(\tau_{1}) \cdots a_{\sigma_{n}}(\tau_{n}) d\tau. \qquad (19)$$

*Proof.* Expansion (18) is a direct consequence of the iterated application of (16) and of the definition of $X_\sigma$ in Definition 2 and can be proved by induction on the length of $\sigma$. $\square$

**Remark 17.** Despite its simplicity, the Chen series suffers from a major drawback: it involves non intrinsic quantities and is redundant. As an illustration, this has the following consequences:

* The functionals $\int_0^t a_\sigma$ for $\sigma \in I^*$ are not algebraically independent. For example, for every solution to (15) and every $t \ge 0$, one has the identity

$$\langle x(t), X_1 X_2 \rangle + \langle x(t), X_2 X_1 \rangle - \langle x(t), X_1 \rangle \langle x(t), X_2 \rangle = 0 \qquad (20)$$

* In the context of nonlinear ordinary differential equations, the representation (18) can fail to converge for smooth vector fields despite strong structural assumptions (see Section 5.1.1).

* In the context of nonlinear ordinary differential equations, the representation (18) will not be invariant by diffeomorphism (see Remark 167), which would be a desirable invariance.

This drawback motivates the search for more intrinsic representations of the solutions, which will turn out to involve Lie algebras.

The Chen series give rise to Fliess operators (stemming from [45, 46]) which can be defined, given some $c \in \hat{\mathcal{A}}(X)$, as $a \mapsto \sum_{\sigma \in I^*} \langle c, X_\sigma \rangle \int_0^t a_\sigma$. Such operators are well-defined (converge) provided that the coefficients $\langle c, X_\sigma \rangle$ satisfy an appropriate asymptotic behavior. Fliess operators can be used to model input-output systems and feedback groups. For manipulations of such operators thanks to an underlying Hopf algebra structure, we refer to [53, 54, 55, 56], which investigate the question of whether an interconnection of such operators remains a Fliess operator, and its convergence, both in scalar and multivariate settings. See also [57] for the investigation of global convergence issues, and realization of such formal operators on concrete systems.
## 2.3 Logarithm of flows, coordinates of the first kind

In the particular case where $a(t)$ is a constant element $a \in A_1(X)$, evaluating the iterated integrals in (19) yields the elegant formula $x(t) = x^* \exp(ta)$, with the notation of (11). Of course, it is no longer valid for a time-varying dynamic (because the indeterminates do not commute), but one can wish to find an object of which the flow is the exponential, the logarithm of the flow.

In this section, we recall and prove Theorem 27, which states that the logarithm of flows of formal linear differential equations is given by explicit Lie brackets. The key argument is the structure result Theorem 20, which states that the logarithm of the flow is a Lie series, and of which we give an elementary proof based on the differential equation satisfied by the logarithm of the flow. We rely on well-known algebraic results, which we recall, for the sake of giving a self-contained presentation.

### 2.3.1 A differential equation for the logarithm of the flow

We start by deriving the formal differential equation (23) satisfied by the logarithm of the flow. This equation is well-known (see e.g. [80, Theorem III], [34, Theorem 4.1] or [3, formula (5.2)]). We provide an elementary derivation (see Remark 19).

**Proposition 18.** *The following statements hold.*

1. Let $T > 0$ and $z \in C^1([0, T]; \hat{A}(X))$. Then, for every $t \in [0, T]$,

$$ \frac{d}{dt} \exp(z(t)) = \exp(z(t)) \sum_{n=0}^{+\infty} \frac{(-1)^n}{(n+1)!} \mathrm{ad}_{z(t)}^n (\dot{z}(t)). \quad (21) $$

2. Let a be given by (14) and $x$ denote the solution to (15) with initial data $x^* = 1$. Then $z := \log x$ satisfies, for almost every $t \in \mathbb{R}_+$,

$$ \sum_{n=0}^{+\infty} \frac{(-1)^n}{(n+1)!} \mathrm{ad}_{z(t)}^n (\dot{z}(t)) = a(t), \quad (22) $$

$$ \dot{z}(t) = \sum_{n=0}^{+\infty} \frac{(-1)^n B_n}{n!} \mathrm{ad}_{z(t)}^n (a(t)), \quad (23) $$

where the Bernoulli numbers ($B_n$)$_{n \in \mathbb{N}}$ are defined in (527).

*Proof.* We prove the two claims successively.

1. The regularity assumption $z \in C^1([0, T]; \hat{A}(X))$ is to be understood component by component, i.e. means that for each $\sigma \in I^*$, $t \mapsto \langle z(t), X_\sigma \rangle$ belongs to $C^1([0, T]; \mathbb{K})$. We have

$$ \begin{aligned} \frac{d}{dt} \exp(z(t)) &= \frac{d}{dt} \left( \sum_{k=0}^{+\infty} \frac{z^k(t)}{k!} \right) = \sum_{k=0}^{+\infty} \frac{1}{(k+1)!} \sum_{j=0}^{k} z^j(t) \dot{z}(t) z^{k-j}(t) \\ &= \exp(z(t)) \left( \sum_{l=0}^{+\infty} \frac{(-1)^l}{l!} z^l(t) \right) \left( \sum_{k=0}^{+\infty} \frac{1}{(k+1)!} \sum_{j=0}^{k} z^j(t) \dot{z}(t) z^{k-j}(t) \right). \end{aligned} \quad (24) $$

Letting $n := k + l$ and $i := l + j$, we obtain that

$$ \frac{d}{dt} \exp(z(t)) = \exp(z(t)) \sum_{n=0}^{+\infty} \frac{1}{(n+1)!} \sum_{i=0}^{n} z^i(t) \dot{z}(t) z^{n-i}(t) \sum_{l=0}^{i} (-1)^l \binom{n+1}{l} \quad (25) $$
The following formulas, which can be proved by induction using Pascal's rule,

$$ \sum_{l=0}^{i} (-1)^l \binom{n+1}{l} = (-1)^i \binom{n}{i}, \qquad (26) $$

$$ \sum_{i=0}^{n} (-1)^i \binom{n}{i} z^i y^{n-i} = (-1)^n \mathrm{ad}_z^n(y) \qquad (27) $$

give the conclusion. Of course, if $z \in W^{1,1}((0,T); \hat{\mathcal{A}}(X))$ (i.e. absolutely continuous), equations (21) remains true as an equality in $L^1((0,T); \hat{\mathcal{A}}(X))$, i.e. holding for almost every $t \in (0,T)$.

2. Since $z = \log x$ and $\dot{x} = xa$, (22) is an immediate consequence of (21), using the preceding comment since both $x$ and $z$ have $W^{1,1}$ regularity in time when $a$ has $L^1$ regularity in time. Starting from (22) and applying $\sum_k (-1)^k B_k/k! \mathrm{ad}_{z(t)}^k$ to both sides yields (23) because

$$ \sum_{k=0}^{+\infty} \frac{(-1)^k B_k}{k!} \sum_{\ell=0}^{+\infty} \frac{(-1)^\ell}{(\ell+1)!} \mathrm{ad}_{z(t)}^{k+\ell}(\dot{z}(t)) = \dot{z}(t). \qquad (28) $$

This follows from the change of index $n := k + \ell$ and the combinatorial relation (529). $\square$

**Remark 19.** The historical proofs of Proposition 18 are written using the Poisson bracket notation $\{\cdot, z^k\} := (-1)^k \mathrm{ad}_z^k(\cdot)$ which allows to write (23) as the nice equality

$$ \dot{z} = \left\{ a, \frac{z}{e^z - 1} \right\}, \qquad (29) $$

using the generating series (527) of the Bernoulli numbers. This approach allows elegant computations, but requires some setup (see [80, Section III] or [34, Section 1]), which is why we prefer here the elementary computations used in the preceding proof.

### 2.3.2 The logarithm of the flow is a Lie series

The fundamental result concerning the logarithm of the flow is that it is a Lie series. We repeat here the proof given in [34, Theorem 4.2] for the sake of completeness. At least two other approaches can be used: one relying on shuffle relations and Ree's theorem (see Appendix A.1) and another one relying on Friedrich's criterion (see Appendix A.2).

**Theorem 20.** Let $a$ be given by (14) and $x$ be the solution to (15) with initial data $x^* = 1$. Then, for every $t > 0$, $\log x(t) \in \hat{\mathcal{L}}(X)$.

*Proof.* The proof relies on an iterated integration of (23), where $z = \log x$. More precisely, writing $z = \sum z_n$ where $z_n \in \mathcal{A}_n(X)$, we prove by induction on $n$ that, for every $t$, $z_n(t) \in \mathcal{L}(X)$. First, for every $t > 0$, since $x_0(t) = 1$, one has $z_0(t) = 0$ so $z_0(t) \in \mathcal{L}(X)$. Then, for every $n \ge 1$, by (23),

$$ \dot{z}_n(t) = \sum_{k=0}^{n-1} \frac{(-1)^k B_k}{k!} \sum_{n_1+\cdots+n_k=n-1} [z_{n_1}(t), [z_{n_2}(t), \cdots [z_{n_k}(t), a(t)] \cdots] \qquad (30) $$

where the sum ranges over indexes $n_i \ge 1$. Moreover, for every $T > 0$ and $y \in L^1((0,T); \mathcal{L}(X))$ one checks that, for every $t \in [0,T]$, $\int_0^t y \in \mathcal{L}(X)$. By the induction assumption, $z_{n_i}(t) \in \mathcal{L}(X)$ for each $n_i \le n-1$ and every $t$. By the previous comment, this property is preserved by the time-integration of (30), so $z_n(t) \in \mathcal{L}(X)$ for every $t$. $\square$
### 2.3.3 Notations

We start with an abstract definition of the truncated logarithm of a time-dependent dynamic.

**Definition 21.** For $m, r \in \mathbb{N}^*$, we define the set of ordered positive partitions of size $m$ of $r$,

$$ \mathbb{N}_r^m := \{ \mathbf{r} = (\mathbf{r}_1, \dots, \mathbf{r}_m) \in (\mathbb{N}^*)^m; \mathbf{r}_1 + \cdots + \mathbf{r}_m = r \}, \quad (31) $$

where $\mathbb{N}_r^m = \emptyset$ when $r < m$. For each $\mathbf{r} \in \mathbb{N}_r^m$ and $t > 0$, we also define the product of simplexes

$$ \Delta^{\mathbf{r}}(t) := \Delta^{\mathbf{r}_1}(t) \times \cdots \times \Delta^{\mathbf{r}_m}(t). \quad (32) $$

**Example 22.** The sets $\Delta^{\mathbf{r}}(t)$ will be used as integration domains. As easy examples, one has

$$ \Delta^{(3)}(t) = \{ \tau = (\tau_1, \tau_2, \tau_3) \in (0, t)^3; 0 < \tau_1 < \tau_2 < \tau_3 < t \}, \quad (33) $$

$$ \Delta^{(1,1,1)}(t) = \{ \tau = (\tau_1, \tau_2, \tau_3) \in (0, t)^3 \}. \quad (34) $$

A more complex example for $r = 4$, $m = 2$ and $\mathbf{r} = (2, 2) \in \mathbb{N}_4^2$ is

$$ \Delta^{(2,2)}(t) = \{ \tau = (\tau_1, \tau_2, \tau_3, \tau_4) \in (0, t)^4; 0 < \tau_1 < \tau_2 < t \text{ and } 0 < \tau_3 < \tau_4 < t \}. \quad (35) $$

We now give a notation for the (truncated or complete) logarithm of a time-dependent dynamic. We will see in the sequel why this quantity indeed corresponds to a logarithm.

**Definition 23** (Abstract logarithm of a time-varying field). Let $M \in \mathbb{N}$ or $M = +\infty$, $t > 0$ and $F$ be a map from $[0, t]$ with values in some algebra. We introduce the notation

$$ \mathrm{Log}_M\{F\}(t) := \sum_{r=1}^{M} \frac{1}{r} \sum_{m=1}^{r} \frac{(-1)^{m-1}}{m} \sum_{\mathbf{r} \in \mathbb{N}_r^m} \int_{\Delta^{\mathbf{r}}(t)} [\cdots [F(\tau_1), F(\tau_2)], \cdots F(\tau_r)] d\tau. \quad (36) $$

**Remark 24.** In such an abstract setting, the right-hand side of (36) does not make sense since we are not able to define an integral over an abstract algebra (without topology on the algebra and without time-regularity on $F$). At this stage, we see (36) as an abstract formula or notation. We will check, each time we use it, that we can give a meaning to the integrals.

### 2.3.4 A preliminary algebraic result

Since the monomials form a basis of $\mathcal{A}(X)$, one can define the following linear map $\beta$ from $\mathcal{A}(X)$ to $\mathcal{L}(X)$ by setting its values on the monomials by $\beta(1) := 0$, $\beta(X_i) := X_i$ for $1 \le i \le q$, and, for $1 \le i_1, \dots, i_k \le q$ with $k \in \mathbb{N}^*$,

$$ \beta(X_{i_1} X_{i_2} \cdots X_{i_k}) := [\cdots [X_{i_1}, X_{i_2}], \cdots, X_{i_k}]. \quad (37) $$

This process defines a standard way, the “left to right” or “left normed” bracketing, to associate a Lie bracket to each monomial. The following important result, proved successively by Dynkin [39], Specht [95] and Wever [104] states that, if a polynomial is a Lie element, then it is equal to its left normed bracketing.

**Lemma 25** (Dynkin's theorem). For $a \in \mathcal{A}_n(X)$, $a \in \mathcal{L}(X)$ if and only if $\beta(a) = na$.

*Proof.* This statement is contained in the equivalence between (i) and (v) of [90, Theorem 1.4]. $\square$

**Example 26.** The element $X_1X_2$ does not belong to $\mathcal{L}(X)$. And indeed, $\beta(X_1X_2) = X_1X_2 - X_2X_1 \ne 2X_1X_2$. On the contrary, the element $[X_1, X_2] = X_1X_2 - X_2X_1$ belongs to $\mathcal{L}(X)$. And indeed, $\beta([X_1, X_2]) = (X_1X_2 - X_2X_1) - (X_2X_1 - X_1X_2) = 2[X_1, X_2]$.
### 2.3.5 An explicit formula for the logarithm of the flow

We now state an explicit expansion of the state as the exponential of the logarithm of the flow. It is the continuous analogue of the well-known CBHD formula (which we recall in Section 2.3.7 as a corollary). It was originally derived by Magnus in [80, Theorem III] and is thus often referred to as the “Magnus expansion”.

**Theorem 27.** For $t \in \mathbb{R}_+$ and $x^* \in \hat{\mathcal{A}}(X)$, the solution $x$ to (15) satisfies

$$x(t) = x^* \exp(\mathrm{Log}_{\infty}\{a(t)\}), \qquad (38)$$

with the notation of Definition 23.

*Proof.* First, by linearity, it suffices to prove (38) for $x^* = 1$. Repeated integration of (16) yields the following formula (which is a slightly different form of the Chen series of Lemma 16),

$$x(t) = 1 + \sum_{r \ge 1} \int_{\Delta^r(t)} a(\tau_1) \cdots a(\tau_r) d\tau. \qquad (39)$$

Hence, recalling the definitions (31) of $\mathbb{N}_r^m$ and (32) of $\Delta^r(t)$, one has

$$\log(x(t)) = \sum_{r=1}^{+\infty} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_r^m} \frac{(-1)^{m-1}}{m} \int_{\Delta^r(t)} a(\tau_1)a(\tau_2)\cdots a(\tau_r) d\tau. \qquad (40)$$

By Theorem 20, for each $t \ge 0$, $\log(x(t)) \in \hat{\mathcal{L}}(X)$. Hence, applying Lemma 25 to each of its homogeneous components in $\mathcal{A}_r$ proves that

$$\log(x(t)) = \sum_{r=1}^{+\infty} \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_r^m} \frac{(-1)^{m-1}}{m} \int_{\Delta^r(t)} [\cdots[a(\tau_1), a(\tau_2)], \dots, a(\tau_r)] d\tau. \qquad (41)$$

Recalling the notation (23) and taking the exponential concludes the proof of (38). $\square$

Magnus expansions (also called BCH expansions) have been extended to more general structures than Lie algebras, for instance to pre-Lie (another name for “chronological algebras”) and post-Lie algebras [47], Rota-Baxter algebras [6, 43] and dendriform algebras [41, 42].

### 2.3.6 Coordinates of the first kind

Although the expansion (41) already has some interest by itself, it is not written on a basis of $\mathcal{L}(X)$, which has some drawbacks. In this paragraph, we define canonical representations for this expansion, in appropriate bases of $\mathcal{L}(X)$.

**Definition 28 (Monomial basis).** Let $\mathcal{B} \subset \mathcal{L}(X)$. We say that $\mathcal{B}$ is a basis of $\mathcal{L}(X)$ when each element $a \in \mathcal{L}(X)$ can be written as a unique finite linear combination of elements of $\mathcal{B}$. We say that $\mathcal{B}$ is a monomial basis of $\mathcal{L}(X)$ when moreover $\mathcal{B} \subset \mathcal{E}(\mathrm{Br}(X))$. In particular, for such bases, if $b \in \mathcal{B}$, one can define $|b|$, $n_i(b)$ for $i \in I$ and $n(b)$ as in Section 2.1.2 by importing these notions from $\mathrm{Br}(X)$. In particular, for $n \in \mathbb{N}^*$, we use the notations $\mathcal{B}_n := \{b \in \mathcal{B}; |b| = n\}$ and $\mathcal{B}_{[1,n]} := \{b \in \mathcal{B}; |b| \le n\}$.

**Proposition 29.** Let $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$. There exists a unique set of functionals $(\zeta_b)_{b \in \mathcal{B}}$, with $\zeta_b \in \mathcal{C}^0(\mathbb{R}_+ \times L^1(\mathbb{R}_+; \mathbb{K})^{|I|}; \mathbb{K})$, such that, for every $a_i \in L^1(\mathbb{R}_+; \mathbb{K})$, $x^* \in \hat{\mathcal{A}}(X)$ and $t \ge 0$, the solution to (15) satisfies

$$x(t) = x^* \exp \left( \sum_{b \in \mathcal{B}} \zeta_b(t, a)b \right). \qquad (42)$$

Moreover, the functionals $\zeta_b$ are “causal” in the sense that, for every $t \ge 0$, $\zeta_b(t, a)$ only depends on the restrictions of the functions $a_i$ to $[0, t]$.
**Proof.** For each $b \in \mathcal{B}$, since $\mathcal{B}$ is monomial, only a finite number of summands of the right-hand side of (41) have a non vanishing component along $b$ (indeed, only terms sharing the same homogeneity can be involved). Hence, it is clear that the functionals thereby defined are continuous on $\mathbb{R}_+ \times L^1(\mathbb{R}_+; \mathbb{K})^q$, due to their explicit expression. The sum in (42) is understood in the sense of a well-defined formal power series. Indeed, for each word $\sigma \in I^*$, only a finite number of elements $b \in \mathcal{B}$ have a non-vanishing component $\langle b, X_\sigma \rangle$. $\square$

**Definition 30** (Coordinates of the first kind). The functionals $\zeta_b$ are usually called coordinates of the first kind associated to the (monomial) basis $\mathcal{B}$ of $\mathcal{L}(X)$.

The terminology coordinates of the first kind or first species and the opposition with coordinates of the second kind (see Section 2.5.3) is classical, see e.g. [24, III.4.3]. See also Section 2.3.8 for references concerning the computation of such coordinates in the context of control theory.

**Remark 31.** Thanks to the monomial nature of the basis, one does not need to specify the full basis in order to define a given functional. For example, if $\lambda \in \mathbb{N}^I$ is a given homogeneity, let

$$ \mathrm{Br}_{\lambda}(X) := \{b \in \mathrm{Br}(X); \forall i \in I, n_i(b) = \lambda_i\}. \qquad (43) $$

Then the coordinates of the first kind $\zeta_b$ for $b \in \mathcal{B} \cap \mathrm{Br}_{\lambda}(X)$ only depend on $\mathcal{B} \cap \mathrm{Br}_{\lambda}(X)$.

**Remark 32.** An important particular case for applications to control theory is the case $X = \{X_0, X_1\}$, with $a_0(t) = 1$ and $a_1(t) = u(t)$. This corresponds to formal scalar-input control-affine systems $\dot{x}(t) = x(t)(X_0 + u(t)X_1)$. One often writes $\zeta_b(t, u)$ (omitting the dependency on $a_0 \equiv 1$) to denote the coordinates of the first kind in this particular context.

### 2.3.7 Campbell Baker Hausdorff Dynkin formula

As a corollary, we obtain the classical finite CBHD formula.

**Corollary 33.** Let $X$ be a finite set, $n \in \mathbb{N}^*$ and $y_1, \dots, y_n \in \hat{\mathcal{L}}(X)$ without constant term. There exists a unique $w \in \hat{\mathcal{L}}(X)$ such that

$$ e^{y_1} \dots e^{y_n} = e^w. \qquad (44) $$

We will use the notation $w = \mathrm{CBHD}_{\infty}(y_1, \dots, y_n)$. Moreover, for each monomial basis $\mathcal{B}$ of $\mathcal{L}(\{Y_1, \dots, Y_n\})$, there exists a unique sequence $(\alpha_b)_{b \in \mathcal{B}} \subset \mathbb{K}^{\mathcal{B}}$ such that, for every finite set $X$ and $y_1, \dots, y_n \in \hat{\mathcal{L}}(X)$

$$ \mathrm{CBHD}_{\infty}(y_1, \dots, y_n) = \sum_{b \in \mathcal{B}} \alpha_b y_b \qquad (45) $$

where $y_b := \Lambda(b)$ and $\Lambda: \mathcal{L}(\{Y_1, \dots, Y_n\}) \to \hat{\mathcal{L}}(X)$ is the homomorphism of Lie algebras such that $\Lambda(Y_j) = y_j$ for $j \in \llbracket 1, n \rrbracket$.

**Proof.** We prove that (44) holds with

$$ w := \mathrm{Log}_{\infty} \left\{ \sum_{j=1}^{n} y_j 1_{[j-1, j]} \right\} (n) \qquad (46) $$

in the sense of Definition 23.

**Step 1:** Proof when $X = \{X_1, \dots, X_n\}$ and $y_j = X_j$ for $j \in \llbracket 1, n \rrbracket$. The solution to (15) with $a(t) = \sum_{j=1}^n X_j 1_{[j-1, j]}(t)$ is $x(t) = x^* e^{X_1} \dots e^{X_n}$. By Theorem 27, $w$ satisfies (44). By injectivity of the exponential (see Lemma 13), it is the unique solution. By Proposition 29, the equality (45) holds with $\alpha_b := \zeta_b(n, 1_{[0,1]}, \dots, 1_{[n-1,n]})$.

**Step 2:** Proof in the general case. Let $X$ be a finite set, $n \in \mathbb{N}^*$, $y_1, \dots, y_n \in \hat{\mathcal{L}}(X)$. Let $Y := \{Y_1, \dots, Y_n\}$ be another set of indeterminates.
The map $\Lambda: Y \to \hat{\mathcal{L}}(X)$ defined by $\Lambda(Y_j) = y_j$ for $j \in [1, n]$ extends into an homomorphism of algebras $\hat{\mathcal{A}}(Y) \to \hat{\mathcal{A}}(X)$, which is also an homomorphism of Lie algebras $\hat{\mathcal{L}}(Y) \to \hat{\mathcal{L}}(X)$, that we still denote $\Lambda$. Indeed Lemma 7 ensures the extension as an homomorphism of algebras $\mathcal{A}(Y) \to \hat{\mathcal{A}}(X)$ (resp. an homomorphism of Lie algebras $\mathcal{L}(Y) \to \hat{\mathcal{L}}(X)$). The extension can be done on $\hat{\mathcal{A}}(Y)$ (resp. $\hat{\mathcal{L}}(Y)$) because $y_1, \dots, y_n$ do not have constant terms and the target space $\hat{\mathcal{A}}(X)$ (resp. $\hat{\mathcal{L}}(X)$) is a space of formal power series.

Let $W := \text{Log}_{\infty}\{\sum_{j=1}^n Y_j 1_{[j-1,j]}\}(n) \in \hat{\mathcal{L}}(Y)$. Then $\Lambda(W) = w$. By applying the homomorphism of algebras $\Lambda$ to the relation $e^{Y_1} \cdots e^{Y_n} = e^W$ we get (44). By applying the homomorphism of Lie algebras $\Lambda$ to the relation $W = \sum_{b \in B} \alpha_b b$ we get (45). $\square$

Despite the fact that the product $e^{y_1} \cdots e^{y_n}$ is of course non-commutative, there is some structure and symmetry inside its logarithm, which we highlight for future use in the following result.

**Proposition 34.** There exists a family of elements $F_{q,h}(Y_1, \dots, Y_q) \in \mathcal{L}(\{Y_1, \dots, Y_q\})$ for $q \in \mathbb{N}^*$ and $h = (h_1, \dots, h_q) \in (\mathbb{N}^*)^q$, such that

* for each $i \in [1, q]$, $F_{q,h}(Y_1, \dots, Y_q)$ is of homogeneity $h_i$ with respect to $Y_i$,

* for every $n \ge 2$, $y_1, \dots, y_n \in \hat{\mathcal{L}}(X)$ with zero constant term,

$$ \mathrm{CBHD}_{\infty}(y_1, \dots, y_n) = \sum_{\substack{q \in [1, n], h \in (\mathbb{N}^*)^q \\ j_1 < \cdots < j_q \in [1, n]}} F_{q,h}(y_{j_1}, \dots, y_{j_q}), \quad (47) $$

where $F_{q,h}(y_{j_1}, \dots, y_{j_q})$ denotes the image of $F_{q,h}(Y_1, \dots, Y_q)$ by the homomorphism of algebras from $\mathcal{L}(\{Y_1, \dots, Y_q\})$ to $\hat{\mathcal{L}}(X)$ which sends $Y_i$ to $y_{j_i}$ for each $i \in [1, q]$.

For $q = 1$, $F_{1,(1)}(Y) = Y_1$ and $F_{1,(h_1)} = 0$ for $h_1 \ge 2$. For $q = 2$ and $h_1 + h_2 \le 4$,

$$ \begin{align} F_{2,(1,1)}(Y) &= \frac{1}{2}[Y_1, Y_2] & F_{2,(2,2)}(Y) &= -\frac{1}{24}[Y_2, [Y_1, [Y_1, Y_2]]] \nonumber \\ F_{2,(2,1)}(Y) &= \frac{1}{12}[Y_1, [Y_1, Y_2]] & F_{2,(3,1)}(Y) &= 0 \tag{48} \\ F_{2,(1,2)}(Y) &= \frac{1}{12}[Y_2, [Y_2, Y_1]] & F_{2,(1,3)}(Y) &= 0. \nonumber \end{align} $$

For higher order terms, we state below a recursive formula.

*Proof.* Using the same Lie algebra homomorphism arguments as in the proof of Corollary 33, it is sufficient to consider the case where $y_i = Y_i$ is an indeterminate.

For $n = 2$, the statement is merely a rewriting of (45) where the terms are grouped by their homogeneity with respect to $y_1$ and $y_2$. This defines the elements $F_{1,(1)}(Y_1) = Y_1$ and $F_{1,(h)}(Y_1) = 0$ for $h \ge 2$ and $F_{2,h}(Y_1, Y_2)$ for $h \in (\mathbb{N}^*)^2$ according to the usual two-variables formula, of which the well-known low-order terms are recalled in (48).

We define by induction on $n \ge 3$ the functions $F_{n,h}$ by the relations

$$ F_{n,h}(Y_1, \dots, Y_n) := \sum_{m|h_1, \dots, h_{n-1}} F_{2,(m,h_n)} \left( F_{n-1,\left(\frac{h_1}{m}, \dots, \frac{h_{n-1}}{m}\right)}(Y_1, \dots, Y_{n-1}), Y_n \right). \quad (49) $$

We now prove the result by induction on $n$. Let $n \ge 3$. By associativity of the product, the formula
for two indeterminates and the induction hypothesis at step $n-1$, we obtain

$$
\begin{align}
\mathrm{CBHD}_{\infty}(Y_1, \dots, Y_n) & \\
&= \mathrm{CBHD}_{\infty}(\mathrm{CBHD}_{\infty}(Y_1, \dots, Y_{n-1}), Y_n) \nonumber \\
&= \mathrm{CBHD}_{\infty}(Y_1, \dots, Y_{n-1}) + Y_n + \sum_{g \in (\mathbb{N}^*)^2} F_{2,g}(\mathrm{CBHD}_{\infty}(Y_1, \dots, Y_{n-1}), Y_n) \tag{50} \\
&= Y_n + \sum_{\substack{q \in [1, n-1], h' \in (\mathbb{N}^*)^q \\ j_1 < \dots < j_q \in [1, n-1]}} \left( F_{q,h'}(Y_{j_1}, \dots, Y_{j_q}) + \sum_{g \in (\mathbb{N}^*)^2} F_{2,g}(F_{q,h'}(Y_{j_1}, \dots, Y_{j_q}), Y_n) \right) \nonumber
\end{align}
$$

We now check that the right-hand side of (50) is the same as the right-hand side of (47). Since we are working in the free Lie algebra over $Y_1, \dots, Y_n$, we can proceed by homogeneity.

• The terms not involving $Y_n$ are equal, since they have the same expression.

• The term involving only $Y_n$ on both sides is $Y_n$ itself, so they are equal.

• Now, let $q \in [1, n-1]$, $j_1 < \dots < j_q \in [1, n-1]$ and $h \in (\mathbb{N}^*)^{q+1}$. We look for the term involving $h_i$ times $Y_{j_i}$ for $i \in [1, q]$ and $h_{q+1}$ times $Y_n$, which is $F_{q+1,h}(Y_{j_1}, \dots, Y_{j_q}, Y_n)$ in (47). In (50), it is

$$
\sum_{h' \in (\mathbb{N}^*)^q} \sum_{g \in (\mathbb{N}^*)^2} F_{2,g}(F_{q,h'}(Y_{j_1}, \dots, Y_{j_q}), Y_n), \quad (51)
$$

where the sum is restricted to $g_1h'_i = h_i$ and $g_2 = h_{q+1}$. Hence, both terms are equal thanks to the definition (49).

This concludes the proof and gives a way to compute the elements $F_{q,h}$ iteratively. $\square$

<figure><img src="image_2.png" /></figure>

**Remark 35.** In particular, the component of $\mathrm{CBHD}_{\infty}(y_1, \dots, y_n)$ homogeneous with degree $h = (h_1, \dots, h_q)$ with respect to $(y_{j_1}, \dots, y_{j_q})$ is $F_{q,h}(y_{j_1}, \dots, y_{j_q})$. It depends neither on the total number $n$ of arguments in the initial product, nor on the selection of indexes $(j_1, \dots, j_q)$. This is the natural symmetry that we wish to highlight.

Algorithms to compute iteratively the terms in the CBHD formula are investigated for instance in [85, Section 4.a] or in [18, 31] within Hall bases, or in [10] for an expansion on right-nested brackets, which uses fewer terms.

2.3.8 Computation of some coordinates of the first kind

In this paragraph, we focus on the case $X = \{X_0, X_1\}$. Computing the coordinates of the first kind is of paramount interest for applications (see e.g. [70] where the first 14 such coordinates are computed, and [31, 86] for efficient algorithms and explicit formulas obtained by an approach relying on rooted binary labeled trees).

Here, we calculate as an illustration (and because they will be used later) all coordinates of the
first kind on a basis of

$$
\dot{S}_1 := \operatorname{span}_{\mathbb{K}} \{ \mathrm{E}(b); b \in \operatorname{Br}(X), n_1(b) = 1 \} \subset \mathcal{L}(X), \quad (52)
$$

where this notation is chosen so that $S_1 = \mathbb{K}X_0 \oplus \dot{S}_1$ (see Definition 10). We define moreover

$$
S_2^+ := \operatorname{span}_{\mathbb{K}} \{ \mathrm{E}(b); b \in \operatorname{Br}(X), n_1(b) \ge 2 \} \subset \mathcal{L}(X), \quad (53)
$$

thanks to which we can write the direct sum decomposition $\mathcal{L}(X) = \mathbb{K}X_0 \oplus \dot{S}_1 \oplus S_2^+$.
**Lemma 36.** The family $(\mathrm{ad}_{X_0}^k(X_1))_{k \in \mathbb{N}}$ is a (monomial) basis of $\dot{S}_1$.

*Proof.* From (52), $\dot{S}_1$ is spanned by the evaluations in $\mathcal{L}(X)$ of the iterated brackets $b \in \mathrm{Br}(X)$ involving $X_1$ exactly once. Let $b \in \mathrm{Br}(X)$ be such an iterated bracket. We assume $\mathrm{E}(b) \neq 0$ in $\mathcal{L}(X)$ and $b \neq X_1$. Then $\mathrm{E}(b) = [\mathrm{E}(\lambda(b)), \mathrm{E}(\mu(b))]$ thus $\mathrm{E}(\lambda(b))$ and $\mathrm{E}(\mu(b))$ are non null in $\mathcal{L}(X)$. Moreover, either $\lambda(b)$ or $\mu(b)$ does not involve $X_1$ and is thus equal to $X_0$. Therefore $\mathrm{E}(b) = \pm [X_0, \mathrm{E}(\bar{b})]$ where $\bar{b} \in \mathrm{Br}(X)$ involves $X_1$ exactly once and $\mathrm{E}(\bar{b}) \neq 0$. Working by induction on the number $k$ of occurrences of $X_0$ in $b$, we obtain $\mathrm{E}(b) = \pm \mathrm{ad}_{X_0}^k(X_1)$.

The previous argument proves that the given family spans $\dot{S}_1$. Moreover, this family is linearly independent in $\mathcal{L}(X)$ because two different elements have different lengths. $\square$

We now compute the coordinates of the first kind associated with these elements. Up to our knowledge, the following explicit expression is new.

**Proposition 37.** Let $\mathcal{B}$ a monomial basis of $\mathcal{L}(X)$ containing $X_0$ and the family $(\mathrm{ad}_{X_0}^k(X_1))_{k \in \mathbb{N}}$. The associated coordinates of the first kind satisfy, for each $t > 0$, $a_0, a_1 \in L^1((0, t); \mathbb{K})$ and $k \in \mathbb{N}$,

$$ \zeta_{\mathrm{ad}_{X_0}^k(X_1)}(t, a_0, a_1) = (-1)^k \sum_{\ell=0}^{k} A_0(t)^{k-\ell} \frac{B_{k-\ell}}{(k-\ell)!} \int_{\Delta^{\ell+1}(t)} a_1(\tau_1)a_0(\tau_2) \cdots a_0(\tau_{\ell+1}) \mathrm{d}\tau, \quad (54) $$

where $A_0(t) := \int_0^t a_0$ and the Bernoulli numbers $(B_n)_{n \in \mathbb{N}}$ are defined in (527).

*Proof.* First, the considered coordinates are well-defined independently on the exact choice of $\mathcal{B}$ (see Remark 31). Let $x$ be the solution to (15) starting from $x^* = 1$. To simplify the notations in this proof, we write $x(t)$, $\zeta_k(t)$ and $Z(t)$ instead of $x(t, a)$, $\zeta_{\mathrm{ad}_{X_0}^k(X_1)}(t, a_0, a_1)$ and $\mathrm{Log}_{\infty}\{a\}(t)$. From (42),

$$ Z(t) = \sum_{b \in \mathcal{B}} \zeta_b(t, a)b = \zeta_{X_0}(t, a)X_0 + Z_1(t) + Z_2(t), \quad (55) $$

where $Z_2(t) \in S_2^+$ and

$$ Z_1(t) := \sum_{k=0}^{+\infty} \zeta_k(t) \mathrm{ad}_{X_0}^k(X_1). \quad (56) $$

First, a straightforward identification in (36) yields $\zeta_{X_0} = A_0$ and $\zeta_{X_1}(t) = \int_0^t a_1$. Let $k \in \mathbb{N}^*$. The proof consists in computing $\langle x(t), X_1 X_0^k \rangle$ in two ways: first by the differential equation (15), then by the formula $x(t) = e^{Z(t)}$. By definition of the solution to (15), we have, for every word $\sigma \in I^*$ and $t > 0$

$$ \langle x(t), X_{\sigma} X_{0} \rangle = \int_{0}^{t} \langle x(\tau), X_{\sigma} \rangle a_{0}(\tau) \mathrm{d}\tau. \quad (57) $$

Taking into account that $\langle x(t), X_1 \rangle = \int_0^t a_1$, we obtain

$$ \langle x(t), X_1 X_0^k \rangle = \int_{\Delta^{k+1}(t)} a_1(\tau_1) a_0(\tau_2) \cdots a_0(\tau_{k+1}) \mathrm{d}\tau. \quad (58) $$

On the other hand, we deduce from the expansion of $x(t) = e^{Z(t)}$ that

$$ \langle x(t), X_1 X_0^k \rangle = \langle Z(t), X_1 X_0^k \rangle + \sum_{\ell=2}^{k+1} \frac{1}{\ell!} \langle Z(t)^\ell, X_1 X_0^k \rangle \quad (59) $$

because, for $\ell \ge (k+2)$, $Z(t)^\ell$ is a sum of words with length at least $(k+2)$. For $\ell \in \llbracket 2, k+1 \rrbracket$,

$$ Z(t)^{\ell} = \sum_{j=0}^{\ell-1} (A_0(t)X_0)^j Z_1(t) (A_0(t)X_0)^{\ell-1-j} + Z_{2,\ell}(t), \quad \text{where} \quad Z_{2,\ell}(t) \in S_2^+. \quad (60) $$
Thus

$$ \langle Z(t)^{\ell}, X_1 X_0^k \rangle = \langle Z_1(t) (A_0(t) X_0)^{\ell-1}, X_1 X_0^k \rangle = A_0(t)^{\ell-1} (-1)^{k-\ell+1} \zeta_{k-\ell+1}(t), \quad (61) $$

because the word $X_1 X_0^{k-\ell+1}$ appears in the decomposition of $\mathrm{ad}_{X_0}^n(X_1)$ iff $k-\ell+1=n$ and then it appears with coefficient $(-1)^n$. We deduce from (59) and (61) that

$$ \langle x(t), X_1 X_0^k \rangle = (-1)^k \zeta_k(t) + \sum_{\ell=2}^{k+1} \frac{(-1)^{k+1-\ell}}{\ell!} A_0(t)^{\ell-1} \zeta_{k+1-\ell}(t). \quad (62) $$

Using (58) and the index change $j=k+1-\ell \in [0, k-1]$, we obtain

$$ \int_{\Delta^{k+1}(t)} a_1(\tau_1) a_0(\tau_2) \cdots a_0(\tau_{k+1}) \, d\tau = (-1)^k \zeta_k(t) + \sum_{j=0}^{k-1} \frac{(-1)^j A_0(t)^{k-j}}{(k+1-j)!} \zeta_j(t), \quad (63) $$

When $A_0(t) = 0$, this formula yields (54) immediately. When $A_0(t) \neq 0$, let, for $j \in \mathbb{N}$,

$$ \alpha_j := \frac{\langle x(t), X_1 X_0^j \rangle}{A_0(t)^{j+1}} \quad \text{and} \quad \beta_j := \frac{(-1)^j \zeta_j(t)}{A_0(t)^{j+1}} \quad (64) $$

we deduce from (63) that

$$ \alpha_k = \sum_{j=0}^{k} \frac{\beta_j}{(k+1-j)!}. \quad (65) $$

We have

$$ z \left( \sum_{k \ge 0} \alpha_k z^k \right) = \sum_{k \ge 0} \sum_{j=0}^{k} \beta_j z^j \frac{z^{k+1-j}}{(k+1-j)!} = \left( \sum_{j \ge 0} \beta_j z^j \right) (e^z - 1) \quad (66) $$

or equivalently

$$ \sum_{j \ge 0} \beta_j z^j = \frac{z}{e^z - 1} \left( \sum_{k \ge 0} \alpha_k z^k \right) = \sum_{n \ge 0} \sum_{k \ge 0} B_n \frac{z^n}{n!} \alpha_k z^k. \quad (67) $$

Thus, for every $j \in \mathbb{N}^*$

$$ \beta_j = \sum_{k=0}^{j} \frac{B_{j-k}}{(j-k)!} \alpha_k. \quad (68) $$

Finally (64) and (58) give (54). $\square$

**Remark 38.** Formula (54) bears a strong similarity with the differential equation (23) satisfied by $z(t)$, which also involves the Bernoulli numbers. Unfortunately, we have not been able to obtain a shorter proof using this equation.

In particular, using Proposition 37, we recover the following very classical formula for the partial coefficients of the CBHD formula (see e.g. [103, equation (2)] or [90, Corollary 3.24]).

**Corollary 39.** One has $e^{X_1}e^{X_0} = e^Z$ where $Z = X_0 + Z_1 + Z_2$, $Z_2 \in S_2^+$ (see (53)) and

$$ Z_1 := \sum_{n=0}^{+\infty} \frac{B_n}{n!} \mathrm{ad}_{X_0}^n(X_1) = X_1 - \frac{1}{2}[X_0, X_1] + \sum_{n=1}^{+\infty} \frac{B_{2n}}{(2n)!} \mathrm{ad}_{X_0}^{2n}(X_1). \quad (69) $$

*Proof.* We apply the previous result to the controls $a_0(t) = \mathbf{1}_{(1,2)}(t)$ and $a_1(t) = \mathbf{1}_{(0,1)}(t)$, for which the solution to (15) with $x^* = 1$ satisfies $x(2) = e^{X_1}e^{X_0}$. For $\ell \in \mathbb{N}^*$ and $0 < \tau_1 < \cdots < \tau_{\ell+1} < 2$,
the real number $a_1(\tau_1)a_0(\tau_1) \cdots a_0(\tau_{\ell+1})$ does not vanish iff $0 < \tau_1 < 1$ and $1 < \tau_2 < \cdots < \tau_{\ell+1} < 2$ and then it equals 1. Thus, for every $k \ge 2$, using (54) and (528),

$$(-1)^k \zeta_k(2) = \sum_{\ell=0}^{k} \frac{B_{k-\ell}}{(k-\ell)!} \frac{1}{\ell!} = \sum_{j=0}^{k} \frac{B_j}{j!(k-j)!} = \frac{B_k}{k!} \quad (70)$$

We conclude by noticing, thanks to (54), that $\zeta_0(2) = 1 = B_0$ and $\zeta_1(2) = -\frac{1}{2} = B_1$. $\square$

**Example 40.** As an example and for later use in the sequel, we compute the coordinates of the first kind for the particular choice $a_0(t) := 1$ and $a_1(t) := t$. Let $k \in \mathbb{N}$. Using formula (54) of Proposition 37 and the identity (530) we obtain

$$\begin{aligned}
\zeta_{\text{ad}^k_{X_0}(X_1)}(t, a) &= (-1)^k \sum_{\ell=0}^{k} t^{k-\ell} \frac{B_{k-\ell}}{(k-\ell)!} \frac{t^{\ell+2}}{(\ell+2)!} \\
&= (-1)^k t^{k+2} \sum_{\ell=0}^{k} \frac{B_{k-\ell}}{(k-\ell)!(\ell+2)!} \quad (71) \\
&= (-1)^{k+1} t^{k+2} \frac{B_{k+1}}{(k+1)!}.
\end{aligned}$$

## 2.4 Interaction picture, coordinates of the pseudo-first kind

In quantum mechanics, the *interaction picture* is an intermediate representation between the *Schrödinger picture* (in which the state vectors are time-dependent and the operators are time-independent) and the *Heisenberg picture* (in which the state vectors are time-independent and the operators are time-dependent). The interaction picture is particularly useful when the dynamics can be written as the sum of a time-independent part, which can be solved exactly, and a time-dependent perturbation. In this section, we introduce and study a formal counterpart of this situation, that can be useful for applications.

### 2.4.1 A new formal expansion

In this paragraph, we therefore consider $I = [0, q]$ to isolate the role of $X_0$. For some given $a_i \in L^1(\mathbb{R}_+; \mathbb{K})$ for $i \in [1, q]$, we assume that $a$ takes the form

$$a(t) = X_0 + \sum_{i=1}^{q} a_i(t) X_i. \quad (72)$$

**Theorem 41.** For $t \in \mathbb{R}_+$, $x^* \in \hat{\mathcal{A}}(X)$ and $a$ of the form (72), the solution $x$ to (15) satisfies

$$x(t) = x^* \exp(tX_0) \exp(\mathcal{Z}_\infty(t, X, a)), \quad (73)$$

where $\mathcal{Z}_\infty(t, X, a) := \text{Log}_\infty\{b_t(t)\}$ with the notation of Definition 23 and

$$b_t(s) := e^{-(t-s)X_0} \left( \sum_{i=1}^{q} a_i(s) X_i \right) e^{(t-s)X_0} = \sum_{i=1}^{q} \sum_{k=0}^{+\infty} \frac{(-1)^k}{k!} (t-s)^k a_i(s) \text{ad}_{X_0}^k(X_i) \quad (74)$$

i.e.

$$\mathcal{Z}_\infty(t, X, a) = \sum \frac{(-1)^{m-1}}{mr} \int_{\Delta^r(t)} \frac{(\tau_1 - t)^{k_1}}{k_1!} \cdots \frac{(\tau_r - t)^{k_r}}{k_r!} a_{i_1}(\tau_1) \cdots a_{i_r}(\tau_r) \, d\tau \quad (75) \\ [\cdots[\text{ad}_{X_0}^{k_1}(X_{i_1}), \text{ad}_{X_0}^{k_2}(X_{i_2})], \cdots \text{ad}_{X_0}^{k_r}(X_{i_r})],$$

where the sum is taken over $r \in [1, \infty]$, $m \in [1, r]$, $\mathbf{r} \in \mathbb{N}_r^m$, $k_1, \dots, k_r \in \mathbb{N}$ and $i_1, \dots, i_r \in [1, q]$.
Proof. First, note that the second equality in (74) stems from the fact that both functions $g_1(\tau) := e^{-\tau X_0} X_i e^{\tau X_0}$ and $g_2(\tau) := \sum_{k=0}^{+\infty} \frac{(-\tau)^k}{k!} \text{ad}_{X_0}^k(X_i)$ solve the Cauchy problem $\dot{g}(\tau) = [g(\tau), X_0]$ and $g(0) = X_i$, so they are equal.

Let $t > 0$. A key point is to remark that all the definitions and results from the previous paragraphs which are stated for a finite set $I$ of indeterminates are still valid if $I$ is an infinite set. For mathematicians with a background in analysis, all equalities can be understood “in the weak sense” as equalities holding along each monomial. Therefore, for a set of indeterminates $\{Y_{k,i}\}_{k \in \mathbb{N}, i \in [1,q]}$, the solution to

$$ \dot{z}(s) = z(s)\gamma_t(s) \quad \text{where} \quad \gamma_t(s) := \sum_{k,i} \frac{(-1)^k}{k!} (t-s)^k a_i(s) Y_{k,i}, \qquad (76) $$

with initial data $z(0) = 1$ satisfies, thanks to Theorem 27,

$$ z(t) = \exp(\text{Log}_{\infty}\{\gamma_t(t)\}). \qquad (77) $$

Let $\Theta$ be the unique homomorphism of algebras from $\hat{\mathcal{A}}(\{Y_{k,i}\}_{k \in \mathbb{N}, i \in [1,q]})$ to $\hat{\mathcal{A}}(X)$ defined by

$$ \Theta(Y_{k,i}) := \text{ad}_{X_0}^k(X_i). \qquad (78) $$

Then $z_{\Theta}(s) = \Theta(z(s))$ satisfies on the one hand $z_{\Theta}(0) = 1$ and $\dot{z}_{\Theta}(s) = z_{\Theta}(s)b_t(s)$, and on the other hand $z_{\Theta}(t) = \exp(\text{Log}_{\infty}\{b_t(t)\})$.

We introduce the change of variables $y(s) := x(s)e^{(t-s)X_0}$. Then,

$$ \dot{y}(s) = \dot{x}(s)e^{(t-s)X_0} - x(s)X_0 e^{(t-s)X_0} = x(s) \left( \sum_{i=1}^{q} a_i(s) X_i \right) e^{(t-s)X_0} = y(s)b_t(s). \qquad (79) $$

Hence

$$ x(t) = y(t) = y(0)z_{\Theta}(t) = x^* e^{tX_0} \exp(\text{Log}_{\infty}\{b_t(t)\}), \qquad (80) $$

which concludes the proof of (73). $\square$

**Remark 42.** In the above proof, $\mathcal{Z}_{\infty}(t, X, a)$ is defined by the logarithm of the product of two flows: the one associated with $-X_0$ and the one associated with $a(t)$. It is a particular case of the construction of the chronological logarithm of the product of two flows associated with two non-autonomous vector fields, see [4, Section 2.2] or [72, p. 92].

**Remark 43.** In expansion (73), the choice to write $\exp(tX_0)$ to the left of the formal logarithm is arbitrary. One could obtain a similar formula with $\exp(tX_0)$ to the right. Depending on the application one has in mind, both choices can be helpful.

### 2.4.2 Coordinates of the pseudo-first kind

**Proposition 44.** Let $q \in \mathbb{N}^*$, $X = \{X_0, X_1, \dots, X_q\}$ and $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$. There exists a unique set of functionals $(\eta_b)_{b \in \mathcal{B}}$, with $\eta_b \in \mathcal{C}^0(\mathbb{R}_+ \times L^1(\mathbb{R}_+; \mathbb{K})^q; \mathbb{K})$, such that, for every $a_i \in L^1(\mathbb{R}_+; \mathbb{K})$ and $t \ge 0$

$$ \mathcal{Z}_{\infty}(t, X, a) = \sum_{b \in \mathcal{B}} \eta_b(t, a)b \quad \text{in} \quad \hat{\mathcal{L}}(X). \qquad (81) $$

Moreover, $\eta_{X_0} = 0$ and the functionals $\eta_b$ are “causal” in the sense that, for every $t \ge 0$, $\eta_b(t, a)$ only depends on the restrictions of the functions $a_i$ to $[0, t]$.
Proof. For every $r \in \mathbb{N}^*$ and $\nu \in \mathbb{N}$ we introduce the finite sum of brackets

$$ Z_{\infty}^{r,\nu}(t, X, a) = \sum \frac{(-1)^{m-1}}{mr} \int_{\Delta^r(t)} \frac{(\tau_1 - t)^{k_1}}{k_1!} \cdots \frac{(\tau_r - t)^{k_r}}{k_r!} a_{i_1}(\tau_1) \cdots a_{i_r}(\tau_r) \, d\tau \quad (82) \\ [\cdots[ad_{X_0}^{k_1}(X_{i_1}), ad_{X_0}^{k_2}(X_{i_2})], \cdots ad_{X_0}^{k_r}(X_{i_r})], $$

where the sum is taken over $m \in \llbracket 1, r \rrbracket$, $\mathbf{r} \in \mathbb{N}_r^m$, $k_1, \dots, k_r \in \mathbb{N}$ such that $k_1 + \cdots + k_r = \nu$ and $i_1, \dots, i_r \in \llbracket 1, q \rrbracket$. For each term in this sum, the bracket

$$ [\cdots[ad_{X_0}^{k_1}(X_{i_1}), ad_{X_0}^{k_2}(X_{i_2})], \cdots ad_{X_0}^{k_r}(X_{i_r})]] \quad (83) $$

has a unique expansion on the basis $\mathcal{B}_{r,\nu} = \{b \in \mathcal{B}; n(b) = r \text{ and } n_0(b) = \nu\}$. By summing these expansions we obtain causal functions $(\eta_b)_{b \in \mathcal{B}_{r,\nu}}$ in $\mathcal{C}^0(\mathbb{R}_+ \times L^1(\mathbb{R}_+; \mathbb{K})^q; \mathbb{K})$ such that the following equality holds in $\mathcal{L}(X)$

$$ Z_{\infty}^{r,\nu}(t, X, a) = \sum_{b \in \mathcal{B}_{r,\nu}} \eta_b(t, a)b. \quad (84) $$

By summing these relations, we get (81). $\square$

**Definition 45** (Coordinates of the pseudo-first kind). We call the functionals $\eta_b$ coordinates of the pseudo-first kind associated to the (monomial) basis $\mathcal{B}$ of $\mathcal{L}(X)$, by analogy with coordinates of the first kind.

### 2.4.3 Structure constants and estimates for the coordinates

At the formal level, series such as (81) make sense. However, in the sequel, we will need to give a meaning to such series where the indeterminates are replaced by true objects. To make sure that the resulting series converge, it will be necessary to have estimates on the coordinates of the pseudo-first kind. In this paragraph, we suggest a criterion based on the structure constants of $\mathcal{L}(X)$ relative to the underlying monomial basis to obtain such estimates.

**Definition 46** (Structure constants). Let $\mathcal{B}$ be a basis of $\mathcal{L}(X)$. For every $a, b \in \mathcal{B}$, since $[a, b] \in \mathcal{L}(X)$, it can be written as a finite linear combination of basis elements, say

$$ [a, b] = \sum_{c \in \mathcal{B}} \gamma_{a,b}^c c, \quad (85) $$

where the coefficients $\gamma_{a,b}^c \in \mathbb{K}$ and only a finite number of them are non-zero. The set of these coefficients are called the structure constants of $\mathcal{L}(X)$ relative to the basis $\mathcal{B}$.

**Definition 47** (Geometric growth). Let $X$ be a finite set and $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$. We say that $\mathcal{B}$ has geometric growth when there exists $\Gamma \ge 1$ such that, for every $b_1, b_2 \in \mathcal{B}$,

$$ \sum_{c \in \mathcal{B}} |\gamma_{b_1, b_2}^c| \le \Gamma^{|b_1|+|b_2|}. \quad (86) $$

**Definition 48** (Asymmetric geometric growth). Let $q \in \mathbb{N}^*$, $X = \{X_0, X_1, \dots, X_q\}$ and $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$. We say that $\mathcal{B}$ has geometric growth with respect to $X_0$ when, for every $k \in \mathbb{N}$, there exists $\Gamma(k) \ge 1$ such that, for every $b_1, b_2 \in \mathcal{B}$ with $n(b_1) + n(b_2) \le k$,

$$ \sum_{c \in \mathcal{B}} |\gamma_{b_1, b_2}^c| \le \Gamma(k)^{|b_1|+|b_2|}. \quad (87) $$

Asymmetric geometric growth is a weaker notion than geometric growth (which can be seen as asymmetric geometric growth with a constant $\Gamma$ independent of $k$). These definitions therefore lead to the following algebraic open problem:
**Open problem 49.** Which monomial bases $\mathcal{B}$ of $\mathcal{L}(X)$ have (asymmetric) geometric growth?

**Remark 50.** A family of examples of monomial bases of $\mathcal{L}(X)$ is given by Hall bases (see Section 2.5.1, and in particular Definition 56). In the paper [13] dedicated to studying the growth of structure constants for Hall bases of $\mathcal{L}(X)$, we provide examples of Hall bases of $\mathcal{L}(X)$ that have geometric growth (in particular, the classical examples of length-compatible Hall bases and the Lyndon basis have geometric growth, see Theorem 1.5 and Theorem 1.6 in the cited work). More importantly, we show that every Hall basis has asymmetric geometric growth (see Theorem 1.9 in the cited work).

For such bases, we can prove nice estimates for the coordinates of the pseudo-first kind. We start with an estimate concerning the decomposition of the Lie brackets involved in (82).

**Lemma 51.** Let $q \in \mathbb{N}^*$, $X = \{X_0, X_1, \dots, X_q\}$, $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$ with geometric growth with respect to $X_0$. For every $r \ge 1$, there exists $C(r) \ge 1$ such that, for every $\ell \ge 2$, $b_1, \dots, b_\ell \in \mathcal{B} \setminus \{X_0\}$ with $n(b_1) + \dots + n(b_\ell) \le r$ and $b \in \mathcal{B}$,

$$|\langle[\dots[b_1, b_2], \dots b_\ell], b\rangle_{\mathcal{B}}| \le C(r)^{|b|}, \quad (88)$$

where the bra-ket denotes the component of the Lie bracket along $b$ in its decomposition on $\mathcal{B}$.

**Proof.** Any $a \in \mathcal{L}(X)$ can be written as a linear combination of basis elements, say $a = \sum_{c \in \mathcal{B}} \alpha_c^a c$, where the coefficients $\alpha_c^a \in \mathbb{K}$ and only a finite number of them are non-zero. We endow $\mathcal{L}(X)$ with the norm $\|a\|_{\mathcal{B}} := \sum_{c \in \mathcal{B}} |\alpha_c^a|$. Then, Definition 48 gives, for every $b_1, b_2 \in \mathcal{B}$, $\|[b_1, b_2]\|_{\mathcal{B}} \le \Gamma(n(b_1) + n(b_2))^{|b_1|+|b_2|}$. We prove by induction on $\ell \ge 2$ that, for every $b_1, \dots, b_\ell \in \mathcal{B} \setminus \{X_0\}$,

$$\|[\dots[b_1, b_2], \dots b_\ell]\|_{\mathcal{B}} \le \Gamma(n(b_1) + \dots + n(b_\ell))^{(\ell-1)(|b_1|+\dots+|b_\ell|)} \quad (89)$$

which implies Lemma 51 with $C(r) = \Gamma(r)^{r-1}$. The result for $\ell = 2$ is already known. Let $\ell \ge 2$ and $b_1, \dots, b_{\ell+1} \in \mathcal{B} \setminus \{X_0\}$. Then $[\dots[b_1, b_2], \dots b_\ell] = \sum_{d \in \mathcal{B}} \alpha_d d$ where the sum is finite and $\sum_{d \in \mathcal{B}} |\alpha_d| \le \Gamma(n(b_1) + \dots + n(b_\ell))^{(\ell-1)(|b_1|+\dots+|b_\ell|)}$. Then

$$[[\dots[b_1, b_2], \dots b_\ell], b_{\ell+1}] = \sum_{d \in \mathcal{B}} \alpha_d [d, b_{\ell+1}] = \sum_{d \in \mathcal{B}} \alpha_d \sum_{c \in \mathcal{B}} \gamma_{d, b_{\ell+1}}^c c = \sum_{c \in \mathcal{B}} \left( \sum_{d \in \mathcal{B}} \alpha_d \gamma_{d, b_{\ell+1}}^c \right) c, \quad (90)$$

where the sums are finite and indexed by $d \in \mathcal{B}$ such that $n(d) + n(b_{\ell+1}) = n(b_1) + \dots + n(b_{\ell+1})$ and $|d| + |b_{\ell+1}| = |b_1| + \dots + |b_{\ell+1}|$ thus

$$\begin{aligned} \|[[\dots[b_1, b_2], \dots b_\ell], b_{\ell+1}]\|_{\mathcal{B}} &= \sum_{c \in \mathcal{B}} \left| \sum_{d \in \mathcal{B}} \alpha_d \gamma_{d, b_{\ell+1}}^c \right| \le \sum_{d \in \mathcal{B}} |\alpha_d| \sum_{c \in \mathcal{B}} |\gamma_{d, b_{\ell+1}}^c| \quad (91) \\ &\le \Gamma(n(b_1) + \dots + n(b_\ell))^{(\ell-1)(|b_1|+\dots+|b_\ell|)} \Gamma(n(b_1) + \dots + n(b_{\ell+1}))^{|b_1|+\dots+|b_{\ell+1}|} \end{aligned}$$

which gives the conclusion. $\square$

**Proposition 52.** Let $q \in \mathbb{N}^*$, $X = \{X_0, X_1, \dots, X_q\}$, $\mathcal{B}$ be a monomial basis of $\mathcal{L}(X)$ with geometric growth with respect to $X_0$. Then, for every $M \in \mathbb{N}^*$, there exists $C_M > 0$ such that, for every $T \ge 0$, $u \in L^1((0, T); \mathbb{K}^q)$, $b \in \mathcal{B}$ with $n(b) \le M$ and $t \in [0, T]$,

$$|\eta_b(t, u)| \le \frac{C_M^{|b|}}{|b|!} t^{n_0(b)} \|u\|_{L^1(0,t)}^{n(b)} \quad (92)$$

**Proof.** We may assume that $(C(r))_{r \in \mathbb{N}^*}$ given by Lemma 51 is non-decreasing. Then, for every $i_1, \dots, i_r \in [1, q]$ and $k_1, \dots, k_r \in \mathbb{N}$, for every $b \in \mathcal{B}$,

$$\left|\left\langle[\dots[\mathrm{ad}_{X_0}^{k_1}(X_{i_1}), \mathrm{ad}_{X_0}^{k_2}(X_{i_2})], \dots, \mathrm{ad}_{X_0}^{k_r}(X_{i_r})], b\right\rangle_{\mathcal{B}}\right| \le C(r)^{|b|}. \quad (93)$$
Indeed, for each $j \in [1, r]$, there exists $b_j \in \mathcal{B}$ with $n(b_j) = 1$ and $|b_j| = k_j + 1$ such that $\mathrm{ad}_{X_0}^{k_j}(X_{i_j}) = \pm b_j$ in $\mathcal{L}(X)$. Indeed, the homogeneous part of $\mathcal{L}(X)$ containing $k_j$ times $X_0$ and $X_{i_j}$ once is of dimension one.

Let $M \in \mathbb{N}^*$ and $b \in \mathcal{B}$ be such that $n(b) \le M$. We deduce from (75) that

$$ \eta_b(t, u) = \sum \frac{(-1)^{m-1}}{mr} \int_{\Delta^r(t)} \frac{(\tau_1 - t)^{k_1}}{k_1!} \cdots \frac{(\tau_r - t)^{k_r}}{k_r!} u_{i_1}(\tau_1) \cdots u_{i_r}(\tau_r) \mathrm{d}\tau \quad (94) \\ \left\langle [\cdots[\mathrm{ad}_{X_0}^{k_1}(X_{i_1}), \mathrm{ad}_{X_0}^{k_2}(X_{i_2})], \cdots \mathrm{ad}_{X_0}^{k_r}(X_{i_r})], b \right\rangle $$

where the sum is taken over $r \in [1, \infty]$, $m \in [1, r]$, $\mathbf{r} \in \mathbb{N}_r^m$, $k_1, \dots, k_r \in \mathbb{N}$ and $i_1, \dots, i_r \in [1, q]$. If the summand bra-ket in (94) does not vanish, then $r = n(b)$ and $k_1 + \dots + k_r = n_0(b)$. Thus the sum in (94) is taken over the finite set $r = n(b)$, $m \in [1, n(b)]$, $k_1, \dots, k_r \in \mathbb{N}$ such that $k_1 + \dots + k_r = n_0(b)$ and $i_1, \dots, i_r \in [1, q]$, whose cardinal is bounded by $M 2^{|b|} q^M$. Moreover, for every $r, m, k_1, \dots, k_r, i_1, \dots, i_r$ in this set, the associated term in (94) is bounded, thanks to (93), by

$$ \frac{t^{k_1}}{k_1!} \cdots \frac{t^{k_r}}{k_r!} \|u\|_{L^1(0,t)}^r C(r)^{|b|} \le t^{n_0(b)} \|u\|_{L^1}^r (2^r C(r))^{|b|} \frac{n(b)!}{|b|!} \quad (95) $$

thanks to (117). Thus

$$ |\eta_b(t, u)| \le \frac{1}{|b|!} M! M q^M (2^{M+1} C(M))^{|b|} t^{n_0(b)} \|u\|_{L^1}^{n(b)} \quad (96) $$

which gives the conclusion with, for instance, $C_M := M! M q^M 2^{M+1} C(M)$. $\square$

<figure><img src="image_3.png" /></figure>

## 2.5 Infinite product, coordinates of the second kind

In this section, we present an expansion for the formal power series $x(t)$ solution to (15) as a product of exponentials of the members of a Hall basis of $\mathcal{L}(X)$, multiplied by coefficients that have simple expressions as iterated integrals, called coordinates of the second kind. This infinite product is an extension, suggested in [69], of Sussmann's infinite product on length-compatible Hall bases [100] to all Hall bases (understood in the generalized sense of Viennot [102, Theorem 1.2] or Shirshov [94, Definition 1]).

### 2.5.1 Lazard sets, Hall sets and Hall bases

We start by defining Lazard sets and Hall sets, which are two equivalent notions, as proved by Viennot in [102, Corollary 1.1]. They lead to the essential notion of Hall bases (see Definition 56).

**Definition 53 (Lazard set).** A Lazard set is a subset $B$ of $\mathrm{Br}(X)$, totally ordered by a relation $<$ and such that, for every $M \in \mathbb{N}^*$, the set $B_{[1,M]}$ of elements of $B$ with length at most $M$, labeled as $B_{[1,M]} = \{b_1, \dots, b_{k+1}\}$ with $k \in \mathbb{N}$ and $b_1 < \dots < b_{k+1}$ satisfies

$$ \left\{ \begin{array}{l} b_1 \in Y_0 := X, \\ b_2 \in Y_1 := \{\mathrm{ad}_{b_1}^j(v); j \in \mathbb{N}, v \in Y_0 \setminus \{b_1\}\}, \\ \dots \\ b_{k+1} \in Y_k := \{\mathrm{ad}_{b_k}^j(v); j \in \mathbb{N}, v \in Y_{k-1} \setminus \{b_k\}\} \end{array} \right. \quad (97) $$

and

$$ B_{[1,M]} \cap Y_k = \{b_{k+1}\}, \quad (98) $$

where condition (98) can equivalently be written

$$ B_{[1,M]} \cap Y_{k+1} = \emptyset, \quad (99) $$

where $Y_{k+1} := \{\mathrm{ad}_{b_{k+1}}^j(v); j \in \mathbb{N}, v \in Y_k \setminus \{b_{k+1}\}\}$.
The elements $\mathrm{ad}_{b_\ell}^j(v)$ for $\ell \in \{0, \dots, k+1\}$, $j \in \mathbb{N}$ and $v \in Y_{\ell-1} \setminus \{b_\ell\}$ are all different in $\mathrm{Br}(X)$ (identify their left and right factors iteratively) and all belong to $B$.

**Definition 54 (Hall set).** A Hall set is a subset $B$ of $\mathrm{Br}(X)$, totally ordered by a relation $<$ and such that

* $X \subset B$,

* for $b = (b_1, b_2) \in \mathrm{Br}(X)$, $b \in B$ iff $b_1, b_2 \in B$, $b_1 < b_2$ and either $b_2 \in X$ or $\lambda(b_2) \le b_1$,

* for every $b_1, b_2 \in B$ such that $(b_1, b_2) \in B$ then $b_1 < (b_1, b_2)$.

When $b = (b_1, (b_3, b_4)) \in B$ then $b_1$ is “sandwiched” in between $b_3$ and $b$, since $b_3 \le b_1 < b$.

**Remark 55.** A Hall set can be built by induction on the length. One starts with the set $X$ as well as an order on it. To find all Hall monomials with length $n$ given those of smaller length, one adds first all $(b_1, b_2)$ with $b_1 \in B$, $|b_1| = n-1$, $b_2 \in X$ and $b_1 < b_2$. Then for each bracket $b_2 = (b'_2, b''_2) \in B$ with length $|b_2| < n$ one adds all the $(b_1, b_2)$ with $b_1 \in B$ with $|b_1| = n - |b_2|$ and $b'_2 \le b_1 < b_2$. Finally, one inserts the newly generated monomials of degree $n$ into an ordering, maintaining the condition that $b_1 < (b_1, b_2)$.

Viennot proves in [102, Corollary 1.1] that a subset $B$ of $\mathrm{Br}(X)$ is a Lazard set iff it is a Hall set. He also proves in [102, Proposition 1.1 and Theorem 1.1] that properties (97) and (98) ensure that $\mathrm{E}(B)$ is a linearly independent and generating family of $\mathcal{L}(X)$. This leads to the following definition.

**Definition 56 (Hall basis).** Given $B$ a Hall set (or equivalently a Lazard set), $\mathcal{B} := \mathrm{E}(B)$ is a basis of $\mathcal{L}(X)$. Such bases are called Hall bases.

**Remark 57.** Historically, such bases where introduced by Marshall Hall in [59], based on ideas of Philip Hall in [60]. In his historical narrower definition, the third condition in Definition 54 was replaced by the stronger condition: for every $b_1, b_2 \in B$, $b_1 < b_2 \Rightarrow |b_1| \le |b_2|$. To avoid confusion with the generalized definition, we name them length-compatible Hall bases in the sequel.

Given a Hall set $B$, the evaluation mapping $\mathrm{E}$ is one to one from $B$ to the associated Hall basis $\mathcal{B}$, so that the elements of the basis (belonging to $\mathcal{L}(X)$) can be identified with the bracket of $b \in B \subset \mathrm{Br}(X)$ of which they are the evaluation. We will use this identification in the sequel when no confusion is possible.

Two famous families of Hall bases of $\mathcal{L}(X)$ are the Chen-Fox-Lyndon basis (see [102, Chapter 1]) and the historical length-compatible Hall bases, for which $b_1 < b_2 \Rightarrow |b_1| \le |b_2|$.

**Example 58.** For instance, with $X = \{X_1, X_2\}$, the elements with length at most 4 of each Hall set $B$ of $\mathcal{L}(X)$ with a length-compatible order < such that $X_1 < X_2$ are: $X_1, X_2, (X_1, X_2)$, $\mathrm{ad}_{X_1}^2(X_2)$, $(X_2, (X_1, X_2))$, $\mathrm{ad}_{X_1}^3(X_2)$ and $(X_2, \mathrm{ad}_{X_1}^2(X_2))$, $\mathrm{ad}_{X_2}^2((X_1, X_2))$. Note that, however, $(X_1, (X_2, (X_1, X_2)))$ does not belong to $B$ because $\lambda((X_2, (X_1, X_2))) = X_2$ is not smaller than $X_1$, and the following equality holds in $\mathcal{L}(X)$

$$ [X_1, [X_2, [X_1, X_2]]] = [[X_1, X_2], [X_1, X_2]] + [X_2, [X_1, [X_1, X_2]]] = [X_2, \mathrm{ad}_{X_1}^2(X_2)] \quad (100) $$

This illustrates how Definition 54 prevents elements from $\mathrm{Br}(X)$, whose evaluations in $\mathcal{L}(X)$ are linked by Jacobi relations, to appear simultaneously in $B$.

**Remark 59.** Let $X := \{X_0, X_1\}$ and $B \subset \mathrm{Br}(X)$ be a Hall set with an order such that $X_0 < X_1$. The definition of a Hall set implies that, for every $k \in \mathbb{N}$, $\mathrm{ad}_{X_0}^k(X_1) \in B$. Moreover, these are all the elements of $B$ containing $X_1$ exactly once. Since $\mathrm{E}(B)$ is a basis of $\mathcal{L}(X)$, $\mathrm{E}(B) \cap \dot{S}_1$ is a basis of $\dot{S}_1$ and this provides an alternative proof of Lemma 36.
### 2.5.2 Infinite product on a Hall basis

**Definition 60 (Infinite product).** Let $J$ be a totally ordered set and $(S^j)_{j \in J}$ be a family of $\hat{\mathcal{A}}(X)$ such that

* for every $j \in J$, $\langle S^j, 1 \rangle = 1$

* for every $\sigma \in I^*$ with $\sigma \neq \emptyset$, the set $\{j \in J; \langle S^j, X_\sigma \rangle \neq 0\}$ is finite.

The infinite product $\prod_{j \in J} S^j$ is the element of $\hat{\mathcal{A}}(X)$ defined by

$$ \prod_{j \in J} S^j = \sum_{\sigma \in I^*} P_{\sigma} X_{\sigma}, \qquad (101) $$

where $P_\emptyset = 1$ and $P_\sigma$ is the finite sum

$$ P_{\sigma} := \sum_{n=0}^{|\sigma|} \sum_{\substack{\sigma_1, \dots, \sigma_n \in I^*, \\ X_{\sigma_1} \cdots X_{\sigma_n} = X_{\sigma}}} \sum_{\substack{j_1, \dots, j_n \in J, \\ j_1 > \cdots > j_n}} \langle S^{j_1}, X_{\sigma_1} \rangle \cdots \langle S^{j_n}, X_{\sigma_n} \rangle. \qquad (102) $$

The following lemma is the key point to generalize rigorously Sussmann's infinite product on length-compatible Hall bases, to all Hall bases.

**Lemma 61.** Let $\mathcal{B}$ be a Hall basis and $(\alpha_b)_{b \in \mathcal{B}}$ be a family of $\mathbb{K}$. The infinite product $\prod_{b \in \mathcal{B}} e^{\alpha_b b}$ is well defined in $\hat{\mathcal{A}}(X)$. Moreover, for every $\sigma \in I^*$,

$$ \left\langle \prod_{b \in \mathcal{B}} e^{\alpha_b b}, X_{\sigma} \right\rangle = \left\langle \prod_{b \in \mathcal{B}_{[1,|\sigma|]}} e^{\alpha_b b}, X_{\sigma} \right\rangle \qquad (103) $$

where $\mathcal{B}_{[1,|\sigma|]}$ is ordered by the induced order of $\mathcal{B}$.

*Proof.* $\mathcal{B}$ is a totally ordered set and, for every $b \in \mathcal{B}$, $\langle e^{\alpha_b b}, 1 \rangle = 1$. Let $\sigma \in I^*$ with $|\sigma| \ge 1$. For $\alpha \in \mathbb{K}$ and $b \in \mathcal{B}$, the property $\langle e^{\alpha_b b}, X_\sigma \rangle \neq 0$ requires $|b| \le |\sigma|$. Indeed

$$ e^{\alpha b} - 1 = \sum_{k=1}^{+\infty} \frac{\alpha^k}{k!} b^k \qquad (104) $$

has non vanishing coefficients only on monomials $X_{\sigma'}$ with length $|\sigma'| \ge |b|$. Thus the set $\{b \in \mathcal{B}, \langle e^{\alpha_b b}, X_\sigma \rangle \neq 0\}$ is finite. This proves that the infinite product is well defined in $\hat{\mathcal{A}}(X)$ and, by (102), the formula (103) holds. $\square$

### 2.5.3 Coordinates of the second kind

**Definition 62.** Let $\mathcal{B}$ be a Hall basis of $\mathcal{L}(X)$. The coordinates of the second kind associated to $\mathcal{B}$ is the unique family $(\xi_b)_{b \in \mathcal{B}}$ of functionals $\mathbb{R}_+ \times L_{\text{loc}}^1(\mathbb{R}_+; \mathbb{K}^I) \to \mathbb{K}$ defined by induction in the following way: for every $t > 0$ and $a \in L_{\text{loc}}^1(\mathbb{R}_+; \mathbb{K}^I)$

* $\xi_{X_i}(t, a) := \int_0^t a_i$, for $i \in I$,

* for $b \in \mathcal{B} \setminus X$, there exists a unique pair $(b_1, b_2)$ of elements of $\mathcal{B}$ such that $b_1 < b_2$ and a unique maximal integer $m \in \mathbb{N}^*$ such that $b = \text{ad}_{b_1}^m(b_2)$ and then

$$ \xi_b(t, a) := \frac{1}{m!} \int_0^t \xi_{b_1}^m(\tau, a) \dot{\xi}_{b_2}(\tau, a) \, d\tau. \qquad (105) $$
Formula (105) indeed defines continuous functionals on $L^1$ and the following estimates hold.

**Lemma 63.** Let $a_i \in L_{\text{loc}}^1(\mathbb{R}_+; \mathbb{K})$ for $i \in I$. For every $b \in \mathcal{B}$ and $t \ge 0$,

$$|\dot{\xi}_b(t, a)| \le |b| |a(t)| \|a\|_{L^1(0,t)}^{|b|-1}, \quad (106)$$

$$|\xi_b(t, a)| \le \|a\|_{L^1(0,t)}^{|b|}. \quad (107)$$

*Proof.* Estimate (106) is valid for $b \in X$ because $\dot{\xi}_{X_i}(t) = a_i(t)$ for $i \in I$ and propagated by induction on $b$ using the recursive definition (105). Estimate (107) is obtained by time-integration of (106) for each $b$. $\square$

### 2.5.4 Infinite product expansion of the solution to the formal differential equation

The following result is due to Sussmann in [100]. The proof below follows Sussmann's argument. It is recalled for sake of giving a self consistent presentation and also to treat the extension from length-compatible Hall bases to all Hall bases (which were not included in Sussmann's original statement).

**Theorem 64.** Let $\mathcal{B}$ be a Hall basis of $\mathcal{L}(X)$. Let $T > 0$ and $a_i \in L^1((0, T); \mathbb{K})$ for $i \in I$. For every $x^* \in \hat{\mathcal{A}}(X)$, the solution to the formal diff (15) satisfies, for every $t \in [0, T]$,

$$x(t) = x^* \prod_{b \in \mathcal{B}} e^{\xi_b(t, a)b}. \quad (108)$$

*Proof.* It is sufficient to prove the formula with $x^* = 1$. To simplify the notations in this proof, we write $\xi_b(t)$ instead of $\xi_b(t, a)$. By Lemma 61 it is sufficient to prove that, for every $t \in [0, T]$ and $\sigma \in I^*$

$$\langle x(t), X_{\sigma} \rangle = \left\langle \prod_{b \in \mathcal{B}} e^{\xi_b(t)b}, X_{\sigma} \right\rangle. \quad (109)$$

Let $\sigma \in I^*$, $M := |\sigma|$, $k \in \mathbb{N}$ and $b_1, \dots, b_{k+1}$ and $Y_0, \dots, Y_{k+1}$ be as in (97). The equality (109) can equivalently be written

$$\langle x(t), X_{\sigma} \rangle = \langle e^{\xi_{b_{k+1}}(t)b_{k+1}} \dots e^{\xi_{b_1}(t)b_1}, X_{\sigma} \rangle. \quad (110)$$

We define $x_0(t) := x(t)$ and, for $j \in [1, k+1]$,

$$x_j(t) := x(t)e^{-\xi_{b_1}(t)b_1} \dots e^{-\xi_{b_j}(t)b_j}. \quad (111)$$

We prove by induction on $j \in [0, k+1]$ that

$$\dot{x}_j(t) = x_j(t) \left( \sum_{b \in Y_j} \dot{\xi}_b(t)b \right) \quad \text{and} \quad x_j(0) = 1. \quad (112)$$

It is clear for $j = 0$ because $x_0(t) = x(t)$, $Y_0 = X$ and $\dot{\xi}_{X_i}(t) = a_i(t)$ for $i \in I$. Let $j \in [1, k+1]$. We assume (112) holds at step $j-1$. We deduce from the definition of $x_j(t)$ that

$$x_j(t) = x_{j-1}(t) e^{-\xi_{b_j}(t)b_j}. \quad (113)$$
Since $\xi_{b_j}(0) = 0$, $x_j(0) = 1$. Moreover,

$$
\begin{align}
\dot{x}_j(t) &= x_{j-1}(t) \left( \sum_{b \in Y_{j-1}} \dot{\xi}_b(t)b \right) e^{-\xi_{b_j}(t)b_j} - x_{j-1}(t) \dot{\xi}_{b_j}(t)b_j e^{-\xi_{b_j}(t)b_j} \nonumber \\
&= x_j(t) e^{\xi_{b_j}(t)b_j} \left( \sum_{b \in Y_{j-1} \setminus \{b_j\}} \dot{\xi}_b(t)b \right) e^{-\xi_{b_j}(t)b_j} \tag{114} \\
&= x_j(t) \sum_{m \in \mathbb{N}} \sum_{b \in Y_{j-1} \setminus \{b_j\}} \frac{\xi_{b_j}^m(t)}{m!} \dot{\xi}_b(t) \mathrm{ad}_{b_j}^m(b) \nonumber
\end{align}
$$

which ends the proof by induction.

We deduce from (99) and (112) for $j = (k+1)$ that $x_{k+1}(t) - 1$ has non vanishing coefficients only on monomials $X_{\sigma'}$ with $|\sigma'| > |\sigma|$. Therefore, by (102),

$$
\langle x(t), X_{\sigma'} \rangle = \langle x_{k+1}(t) e^{\xi_{b_{k+1}}(t)b_{k+1}} \dots e^{\xi_{b_1}(t)b_1}, X_{\sigma'} \rangle = \langle e^{\xi_{b_{k+1}}(t)b_{k+1}} \dots e^{\xi_{b_1}(t)b_1}, X_{\sigma'} \rangle, \quad (115)
$$

which concludes the proof.

<figure><img src="image_4.png" /></figure>

3 Technical tools about functions and vector fields

In this section, we state classical definitions and technical results about functions and vector fields.
For the sake of completeness, the proofs, although classical, are provided.

Throughout the whole article, $d \in \mathbb{N}^*$ denotes the dimension of the state space for the considered ordinary differential equations. We work locally, in neighborhoods of the origin $0 \in \mathbb{K}^d$. For $\delta > 0$, $B_\delta$ denotes the closed ball of center 0 and radius $\delta$ in the state space $\mathbb{K}^d$.

3.1 Functional spaces for finite or analytic regularity

3.1.1 Conventions for multi-indexes

For $a \in \mathbb{N}^*$ and a multi-index $\alpha = (\alpha^1, \dots, \alpha^a) \in \mathbb{N}^a$, we use the notations $|\alpha| := \alpha^1 + \dots + \alpha^a$, $\partial^\alpha := \partial_1^{\alpha^1} \dots \partial_a^{\alpha^a}$ and $\alpha! := \alpha^1! \dots \alpha^a!$.

**Lemma 65.** The following estimates hold:

$$
\forall n \in \mathbb{N}, \quad n^n e^{-n} e \le n! \le (n+1)^{n+1} e^{-(n+1)} e, \quad (116)
$$

$$
\forall a \in \mathbb{N}^*, \forall \alpha = (\alpha^1, \dots, \alpha^a) \in \mathbb{N}^a, \quad 2^{-(a-1)|\alpha|} |\alpha!| \le \alpha! \le |\alpha!| \quad (117)
$$

Proof. The first inequality can be proved using classical series-integral comparison and the second by iterating $p!q! \ge 2^{-(p+q)}(p+q)!$ for every $p, q \in \mathbb{N}$. $\square$

<figure><img src="image_5.png" /></figure>

3.1.2 Finite-regularity norms

**Definition 66 (Regular functions).** Let $a, b \in \mathbb{N}^*$ and $K$ a compact subset of $\mathbb{K}^a$. Let $k \in \mathbb{N}$. We endow $C^k(K; \mathbb{K}^b)$, the space of functions whose real-derivatives are well-defined and continuous up to order $k$ on an open neighborhood of $K$ to $\mathbb{K}^b$ with the norm

$$
\|f\|_{C^k} := \sum_{j=1}^{b} \sum_{|\alpha| \le k} \frac{1}{\alpha!} \|\partial^{\alpha} f_j\|_{L^{\infty}(K)}, \quad (118)
$$

where the sum ranges over multi-indexes $\alpha \in \mathbb{N}^a$ whose sum is at most $k$ and $f_1, \dots, f_b$ are the components of the vector-valued function $f$. We denote by $C^\infty(K; \mathbb{K}^b)$ the intersection of these spaces over $k \in \mathbb{N}$.
**Lemma 67 (Submultiplicativity).** Let $a \in \mathbb{N}^*$, $K$ a compact subset of $\mathbb{K}^a$ and $k \in \mathbb{N}$. For every $f, g \in C^k(K; \mathbb{K})$, one has

$$ \|fg\|_{C^k} \le \|f\|_{C^k} \|g\|_{C^k}. \qquad (119) $$

*Proof.* Using the multivariate Leibniz formula, one has

$$ \begin{aligned} \|fg\|_{C^k} &= \sum_{|\alpha| \le k} \frac{1}{\alpha!} \|\partial^\alpha (fg)\|_{L^\infty(K)} \\ &\le \sum_{|\alpha| \le k} \frac{1}{\alpha!} \sum_{\beta \le \alpha} \binom{\alpha}{\beta} \|\partial^\beta f\|_{L^\infty(K)} \|\partial^{\alpha-\beta} g\|_{L^\infty(K)} \le \|f\|_{C^k} \|g\|_{C^k}, \end{aligned} \qquad (120) $$

where the sum ranges over all multi-indexes $\beta \in \mathbb{N}^a$ such that $\beta_i \le \alpha_i$ for each $i \in [1, a]$. $\square$

**Lemma 68 (Control of gradients).** Let $a \in \mathbb{N}^*$, $K$ a compact subset of $\mathbb{K}^a$ and $k \in \mathbb{N}$. For every $f \in C^{k+1}(K; \mathbb{K})$ and $j \in [1, a]$, $\square$

$$ \|\partial_j f\|_{C^k} \le (k+1) \|f\|_{C^{k+1}}. \qquad (121) $$

*Proof.* By (118),

$$ \|\partial_j f\|_{C^k} = \sum_{|\alpha| \le k} \frac{1}{\alpha!} \|\partial^{\alpha+e_j} f\|_{L^\infty(K)} = \sum_{|\alpha| \le k} \frac{\alpha_j + 1}{(\alpha + e_j)!} \|\partial^{\alpha+e_j} f\|_{L^\infty(K)} \le (k+1) \|f\|_{C^{k+1}} \qquad (122) $$

since $\alpha_j \le |\alpha| \le k$. $\square$

### 3.1.3 Analytic norms

**Definition 69 (Analytic norms).** Let $a, b \in \mathbb{N}^*$ and $K$ a compact subset of $\mathbb{K}^a$. We define $C^\omega(K; \mathbb{K}^b)$ the space of real-analytic functions defined on an open neighborhood of $K$ to $\mathbb{K}^b$, as the union for $r > 0$ of the spaces $C^{\omega,r}(K; \mathbb{K}^b)$, which are the subsets of $C^\infty(K; \mathbb{K}^b)$ for which the following norm is finite

$$ \|f\|_r := \sum_{i=1}^b \sum_{\alpha \in \mathbb{N}^a} \frac{r^{|\alpha|}}{\alpha!} \|\partial^\alpha f_i\|_{L^\infty(K)}. \qquad (123) $$

**Lemma 70 (Submultiplicativity).** Let $a \in \mathbb{N}^*$, $K$ a compact subset of $\mathbb{K}^a$, $r > 0$. Then, for every $f, g \in C^{\omega,r}(K; \mathbb{K})$, one has

$$ \|fg\|_r \le \|f\|_r \|g\|_r. \qquad (124) $$

*Proof.* Using the multivariate Leibniz formula, one has

$$ \begin{aligned} \|fg\|_r &= \sum_{\alpha \in \mathbb{N}^a} \frac{r^{|\alpha|}}{\alpha!} \|\partial^\alpha (fg)\|_{L^\infty(K)} \\ &\le \sum_{\alpha \in \mathbb{N}^a} \frac{r^{|\alpha|}}{\alpha!} \sum_{\beta \le \alpha} \binom{\alpha}{\beta} \|\partial^\beta f\|_{L^\infty(K)} \|\partial^{\alpha-\beta} g\|_{L^\infty(K)} = \|f\|_r \|g\|_r, \end{aligned} \qquad (125) $$

where the sum ranges over all multi-indexes $\beta \in \mathbb{N}^a$ such that $\beta_i \le \alpha_i$ for each $i \in [1, a]$. $\square$

**Lemma 71 (Control of gradients).** Let $a \in \mathbb{N}^*$, $K$ a compact subset of $\mathbb{K}^a$. For every $r_2 > r_1 > 0$, $f \in C^{\omega,r_2}(K; \mathbb{K})$ and $j \in [1, a]$, $\square$

$$ \|\partial_j f\|_{r_1} \le \frac{1}{r_1} \left( e \ln \frac{r_2}{r_1} \right)^{-1} \|f\|_{r_2}. \qquad (126) $$

In particular, if $r_2 \le er_1$,

$$ \|\partial_j f\|_{r_1} \le \frac{1}{r_2 - r_1} \|f\|_{r_2}. \qquad (127) $$
Proof. We start with the first estimate (126). One has

$$
\begin{align}
\|\partial_j f\|_{r_1} &= \sum_{\alpha \in \mathbb{N}^a} \frac{r_1^{\alpha|\alpha|}}{\alpha!} \|\partial^{\alpha+e_j} f\|_{L^\infty(K)} \nonumber \\
&= \frac{1}{r_1} \sum_{\alpha \in \mathbb{N}^a} \frac{r_1^{\alpha+e_j}}{(\alpha+e_j)!} \frac{(\alpha+e_j)!}{\alpha!} \|\partial^{\alpha+e_j} f\|_{L^\infty(K)} \nonumber \\
&\le \frac{1}{r_1} \|f\|_{r_2} \sup_{\alpha \in \mathbb{N}^a} \left(\frac{r_1}{r_2}\right)^{|\alpha+e_j|} \frac{(\alpha+e_j)!}{\alpha!} \tag{128} \\
&\le \frac{1}{r_1} \|f\|_{r_2} \sup_{n \ge 1} n \left(\frac{r_1}{r_2}\right)^n. \nonumber
\end{align}
$$

For $x \in (0,1)$, let $C(x) := \sup_{n \ge 1} nx^n = \sup_{n \ge 1} \exp(\ln n + n \ln x)$. Considering that $x$ is a fixed parameter and differentiating the argument with respect to $n \in [1, +\infty)$ yields

$$
\frac{d}{dn} (\ln n + n \ln x) = \frac{1}{n} + \ln x. \tag{129}
$$

Since $x < 1$, this derivative is negative for $n$ large enough. For $x \ge 1/e$, the global maximum is for $n = -1/\ln x$. So its value yields the bound

$$
C(x) \le (-e \ln x)^{-1}. \tag{130}
$$

For $x \le 1/e$, the supremum over $n$ is achieved for $n = 1$ and its value is $x$. Since $x \le (-e \ln x)^{-1}$ for $x \in (0, 1)$, the bound (130) is looser and valid for every $x \in (0, 1)$.

The second inequality is a consequence of the estimate $\ln(1+\sigma) \ge \sigma/(e-1)$ for $\sigma \le e-1$. $\square$

**Remark 72.** The first estimate (126) is classical (see e.g. [84]). The second estimate (126) is a simplified version, restricted to the case when the relative radius loss is small enough. This is the form under which we will use Lemma 71 in the sequel since we consider small radius losses.

3.2 Estimates for differential operators and Lie brackets

3.2.1 Vector-valued functions, vector fields and differential operators

As is usual, we will identify each smooth vector-valued function with the associated first-order
linear differential operator, and we will refer to both objects as a *vector field*. Let $K$ be a compact
subset of $K^d$.

**Definition 73** (Vector field). Given coordinates $a_1, \dots, a_d \in C^\infty(K; \mathbb{K})$, we define the associated vector field $f$ indifferently as the vector-valued function $f = (a_1, \dots, a_d)$ of $C^\infty(K; \mathbb{K}^d)$ (mapping each point of the state space to a vector of $\mathbb{K}^d$), or as the first-order linear differential operator $f = a_1 \partial_1 + \dots + a_d \partial_d$ (acting on smooth functions $\phi \in C^\infty(K; \mathbb{K})$).

**Remark 74** (Composition of vector fields). *Seen as linear operators, vector fields can be composed, yielding higher-order differential operators. Let $f, g \in C^\infty(K; \mathbb{K}^d)$ (denoting their coordinates by $a_1, \dots, a_d$ and $b_1, \dots, b_d$) and $\phi \in C^\infty(K; \mathbb{K})$. We will hence write*

$$
f\phi = \sum_{i=1}^{d} a_i \partial_i \phi \tag{131}
$$

and

$$
fg\phi = \sum_{i=1}^{d} \sum_{j=1}^{d} a_i (b_j \partial_i \partial_j \phi + (\partial_i b_j)(\partial_j \phi)). \quad (132)
$$

Similarly, for $k \in \mathbb{N}$, $f^k \phi$ will denote the composition of the linear differential operator $f$ with itself $k$ times, applied to $\phi$. Such formulas still make sense in a finite regularity setting, as long as all derivatives are well-defined.
### 3.2.2 Estimates for products

**Lemma 75.** Let $k \in \mathbb{N}$, $n, b \in \mathbb{N}^*$, $\delta > 0$, $f_1, \dots, f_n \in C^{k+n-1}(B_\delta; \mathbb{K}^d)$ and $\phi \in C^{k+n}(B_\delta; \mathbb{K}^b)$. With the notations of Remark 74,

$$ \|f_n \cdots f_1 \phi\|_{C^k} \le \frac{(k+n)!}{k!} \|f_1\|_{C^{k+n-1}} \cdots \|f_n\|_{C^{k+n-1}} \|\phi\|_{C^{k+n}}. \quad (133) $$

*Proof.* For $n=1$, it is a consequence of (118), Lemma 67 and Lemma 68. For $n > 1$, the estimate follows by induction. $\square$

**Lemma 76.** Let $r_2 > 0$, $r_1 \in [r_2/e, r_2)$, $n, b \in \mathbb{N}^*$ and $\delta > 0$. Let $f_1, \dots, f_n \in C^{\omega, r_2}(B_\delta; \mathbb{K}^d)$ and $\phi \in C^{\omega, r_2}(B_\delta; \mathbb{K}^b)$. With the notations of Remark 74,

$$ \|f_n \cdots f_1 \phi\|_{r_1} \le \frac{n!}{e} \left(\frac{e}{r_2 - r_1}\right)^n \|f_n\|_{r_2} \cdots \|f_1\|_{r_2} \|\phi\|_{r_2}. \quad (134) $$

In particular, under the same assumptions,

$$ \|f_n \cdots f_1 \phi\|_{C^0} \le n! \left(\frac{5}{r_2}\right)^n \|f_n\|_{r_2} \cdots \|f_1\|_{r_2} \|\phi\|_{r_2}. \quad (135) $$

*Proof.* For $n=1$, estimate (134) is a consequence of (123), (124) and (127). For $n > 1$, one applies the $n=1$ estimate $n$ times with a radius increment $(r_2 - r_1)/n$ at each step. This yields more precisely

$$ \begin{aligned} \|f_n \cdots f_1 \phi\|_{r_1} &\le \left(\frac{n}{r_2 - r_1}\right) \|f_n\|_{r_1} \|f_{n-1} \cdots f_1 \phi\|_{r_1 + \frac{r_2 - r_1}{n}} \\ &\le \left(\frac{n}{r_2 - r_1}\right)^n \|\phi\|_{r_2} \prod_{j=1}^{n} \|f_j\|_{r_1 + (n-j) \frac{r_2 - r_1}{n}}, \end{aligned} \quad (136) $$

which concludes the proof because the norm (123) is non-decreasing with respect to $r$, and we can bound $n^n$ using (116). Estimate (135) is a direct consequence for the particular choice $r_1 = r_2/e$, because $e^2/(e-1) \le 5$. $\square$

### 3.2.3 Lie brackets

**Definition 77 (Lie bracket of vector fields).** We define the Lie bracket of smooth vector fields $f$ and $g$ as the usual commutator of the associated linear differential operators: $[f,g] := fg - gf$ (with the notations of Remark 74). By (132) and Schwarz's theorem, one checks that $[f,g]$ is also a first-order differential operator, which, as a vector-valued function, can be computed as $[f,g] = (Dg)f - (Df)g$.

**Definition 78 (Evaluated Lie bracket).** Let $I$ be a finite set of indices, $X = \{X_i; i \in I\}$ be indeterminates and $\{f_i; i \in I\}$ be $C^\infty$ vector fields on a subset $\Omega$ of $\mathbb{K}^d$. For an iterated bracket $b \in \text{Br}(X)$, we define $f_b := \Lambda(\text{E}(b))$, where $\Lambda: \mathcal{L}(X) \to C^\infty(\Omega; \mathbb{K}^d)$ is the unique homomorphism of Lie algebras such that $\Lambda(X_i) = f_i$ for every $i \in I$ (see Remark 9 and Lemma 7).

The vector field $f_b$ is obtained by replacing the indeterminates $X_i$ with the corresponding vector fields $f_i$ in the iterated bracket $b$, for instance $f_{(X_1,(X_2,X_3))} = [f_1, [f_2, f_3]]$.

The notation $f_b$ will sometimes denote the same vector field, seen as a vector-valued function, under weaker regularity assumptions, for instance $f_i \in C^{|b|-1}$ and then $f_b \in C^0$.

**Lemma 79 (Finite regularity estimate).** Let $k \in \mathbb{N}$ and $\delta > 0$. Let $b \in \text{Br}(X)$. For $i \in I$, let $f_i \in C^{k+|b|-1}(B_\delta; \mathbb{K}^d)$. Then,

$$ \|f_b\|_{C^k} \le 2^{|b|-1} \frac{(k + |b| - 1)!}{k!} \prod_{i \in I} \|f_i\|_{C^{k+|b|-1}}^{n_i(b)}. \quad (137) $$
Proof. This follows from (133) because, as can be checked by induction on $|b|$, $f_b$ is a sum of at most $2^{|b|-1}$ terms of the form studied in Lemma 75, where $\phi$ is one of the vector fields $f_i$. $\square$

**Lemma 80** (Analytic estimate). Let $r_2 > 0$, $r_1 \in [r_2/e, r_2)$ and $\delta > 0$. Let $b \in \text{Br}(X)$. For $i \in I$, let $f_i \in C^{\omega,r_2}(B_\delta; \mathbb{K}^d)$. Then,

$$ \|f_b\|_{r_1} \le \frac{(|b|-1)!}{e} \left(\frac{2e}{r_2-r_1}\right)^{|b|-1} \prod_{i \in I} \|f_i\|_{r_2}^{n_i(b)}. \quad (138) $$

In particular, under the same assumptions,

$$ \|f_b\|_{C^0} \le (|b|-1)! \left(\frac{9}{r_2}\right)^{|b|-1} \prod_{i \in I} \|f_i\|_{r_2}^{n_i(b)}. \quad (139) $$

$$ \|f_b\|_{C^1} \le \max \left\{ 1, \frac{1}{r_2} \right\} (|b|-1)! \left(\frac{9}{r_2}\right)^{|b|-1} \prod_{i \in I} \|f_i\|_{r_2}^{n_i(b)}. \quad (140) $$

Proof. Estimate (138) stems from (134) because, as can be checked by induction on $|b|$, $f_b$ is a sum of at most $2^{|b|-1}$ terms of the form studied in Lemma 76, where $\phi$ is one of the vector fields $f_i$. Estimates (139) and (140) are direct consequences of (138) for the particular choice $r_1 = r_2/e$ because $2e^2/(e-1) \le 9$ and, for every $r_1 > 0$, $\|f_b\|_{C^1} \le \max\{1, \frac{1}{r_1}\} \|f_b\|_{r_1}$. $\square$

**Remark 81.** The fact that estimate (138) scales like the factorial of the length of the Lie bracket is optimal, as illustrated by the following vector fields. For $x \in \mathbb{R}^2$ with $|x| < 1$, define

$$ f_0(x) := e_1 \quad \text{and} \quad f_1(x) := \frac{1}{1-x_1} e_2. \quad (141) $$

Using (123), one checks that these vector fields belong in particular to $C^{\omega,r}(B_\delta; \mathbb{R}^2)$ for $r = \frac{1}{4}$ and $\delta = \frac{1}{2}$, with $\|f_0\|_r = 1$ and $\|f_1\|_r = 2$. For $k \in \mathbb{N}$, one has

$$ \mathrm{ad}_{f_0}^k(f_1)(x) = \partial_1^k \left( \frac{1}{1-x_1} \right) e_2 = \frac{k!}{(1-x_1)^{k+1}} e_2. \quad (142) $$

Moreover, since $f_0$ is constant and $f_1$ depends only on $x_1$ but is a multiple of $e_2$, every Lie bracket involving $f_1$ at least twice vanishes identically. Since these analytic vector fields “saturate” the bounds and exhibit such a nice structure, we will use them repeatedly in our counter-examples.

## 3.3 Well-posedness of ordinary differential equations

The nonlinear differential equations

$$ \dot{x}(t) = f(t, x(t)) \quad \text{and} \quad x(0) = p \quad (143) $$

will be studied in the following classical frameworks.

**Lemma 82.** Let $\delta, T > 0$ and $f \in L^1((0, T); C^1(B_{2\delta}; \mathbb{K}^d))$ such that $\|f\|_{L^1((0,T);C^0)} < \delta$.

1. For each $p \in B_\delta$, there exists a unique function $x(\cdot; f, p) \in C^0([0, T]; B_{2\delta})$ such that

$$ \forall t \in [0, T], \quad x(t; f, p) = p + \int_0^t f(\tau, x(\tau; f, p)) \, d\tau. \quad (144) $$

2. If $f \in C^0([0, T] \times B_{2\delta}; \mathbb{K}^d)$ then $x(\cdot; f, p) \in C^1([0, T]; B_{2\delta})$ and satisfies (143) pointwise.
3. If $f \in C^\infty([0, T] \times B_{2\delta}; \mathbb{K}^d)$, the map $p \in B_\delta \mapsto x(\cdot; f, p) \in C^0([0, T]; B_{2\delta})$ is smooth.

4. If $g$ satisfies the same assumptions as $f$, then, for each $p \in B_\delta$ and $t \in [0, T]$,

$$|x(t; f, p) - x(t; g, p)| \le \|f - g\|_{L^1((0,t);C^0)} \exp (\|f\|_{L^1((0,t);C^1)}). \quad (145)$$

*Proof.* We proceed step by step. Let $E := C^0([0, T]; B_{2\delta})$.

1. Define $\Theta: E \to E$ by $\Theta(x)(t) := p + \int_0^t f(\tau, x(\tau)) d\tau$ for $x \in E$. Thanks to the smallness assumption on $f$, $\Theta(x)(t) \in B_{2\delta}$. Let $n \in \mathbb{N}^*$ be such that $\|f\|_{L^1((0,T);C^1)}^n/n! < 1$. By the Banach fixed-point theorem, $\Theta^n$ has a unique fixed point, which is also a fixed point of $\Theta$.

2. If $f$ is continuous, then $t \mapsto \Theta(x(t; f, p))$ belongs to $C^1([0, T]; B_{2\delta})$ and its derivative at time $t$ is $f(t, x(t; f, p))$.

3. If $f$ is smooth, let $\bar{p} \in B_\delta$, $\bar{x} := x(\cdot; f, \bar{p})$ and define $F: B_\delta \times E \to E$ by

$$\forall t \in [0, T], \quad F(p, x)(t) := x(t) - p - \int_0^t f(\tau, x(\tau)) \, d\tau \quad (146)$$

Then $F$ is of class $C^\infty$, vanishes at $(\bar{p}, \bar{x})$ and $\partial_x F(\bar{p}, \bar{x})$ is a bijection on $E$. By the implicit function theorem, the map $p \mapsto x(\cdot; f, p)$ is $C^\infty$ on a neighborhood of $\bar{p}$.

4. This follows from a standard Grönwall's lemma argument. $\square$

**Lemma 83.** Let $\delta, \delta_u > 0$, $q \in \mathbb{N}^*$ and $f \in C^\omega(B_{2\delta} \times B_{\mathbb{K}^q}(0, \delta_u); \mathbb{K}^d)$. Let $T := \delta/\|f\|_{C^0}$. For each $p \in B_\delta$ and $u \in L^\infty((0, T); \mathbb{K}^q)$ with $\|u\|_{L^\infty} \le \delta_u$, there exists a unique solution $x \in C^0([0, T]; B_{2\delta})$ to

$$\begin{cases} \dot{x}(t) = f(x(t), u(t)), \\ x(0) = p, \end{cases} \quad (147)$$

denoted $x(t; f, u, p)$. Moreover, the map $(u, p) \mapsto x(\cdot; f, u, p) \in C^0([0, T]; B_{2\delta})$ is real-analytic on $B_\delta \times B_{L^\infty((0,T))}(0, \delta_u)$.

*Proof.* Existence stems from Lemma 82. Analyticity is a consequence of the implicit function theorem, which yields the analyticity of the implicit function when the direct function is analytic (see e.g. [27, Theorem 4.5.4]). $\square$

## 3.4 Flows, compositions and pushforwards

Here and in the sequel, when we manipulate flows of vector fields, we always make sure that an appropriate smallness assumption ensures that the local flow is well-defined up to the time at which we evaluate it.

### 3.4.1 Definitions and approximations

By applying Lemma 82 to a time-independent vector field we obtain the following object.

**Definition 84** (Flow of time-independent vector fields). Let $\delta > 0$. Let $f \in C^1(B_{2\delta}; \mathbb{K}^d)$ such that $\|f\|_{C^0} < \delta$. We denote by $e^f$ the flow at time one of the vector field $f$,

$$e^f: \begin{cases} B_\delta \to B_{2\delta}, \\ p \mapsto x(1; f, p), \end{cases} \quad (148)$$

with the notations of Section 3.3. We write $e^f p$ instead of $e^f(p)$ to allow easier composition of flows. When moreover $f \in C^\infty(B_{2\delta}; \mathbb{K}^d)$, $e^f$ can also be seen as the zero-order linear operator on $C^\infty(B_{2\delta}; \mathbb{K})$ defined by $e^f \phi : p \mapsto \phi(e^f p)$.
**Lemma 85.** Let $\delta > 0$ and $f \in C^1(B_\delta; \mathbb{K}^d)$. Assume that $\delta' := \delta - \|f\|_{C^0(B_\delta)} > 0$. For each $p \in B_{\delta'}$, $e^f p$ is well-defined and $e^f p \in B_\delta$. Moreover,

$$|e^f p - p| \le \|f\|_{C^0(B_\delta)}, \qquad (149)$$

and

$$\|D(e^f)\|_{C^0(B_{\delta'})} \le e^{\|Df\|_{C^0(B_\delta)}} \le e^{\|f\|_{C^1(B_\delta)}}. \qquad (150)$$

*Proof.* The second estimate comes from the fact that $D(e^f)|_p = R(1)$ where

$$\dot{R}(t) = Df(e^{tf}p)R(t) \quad \text{and} \quad R(0) = \text{Id}. \qquad (151)$$

Thus, by Grönwall's lemma,

$$\|R(1)\| \le \|\text{Id}\| e^{\int_0^1 \|Df(e^{tf}p)\| dt} \le e^{\|Df\|}, \qquad (152)$$

which concludes the proof. $\square$

The exponential notation is motivated by the possibility to approximate $e^f$ by partial sums of the exponential series of the linear differential operator $f$. It is completely legitimate in the analytic setting, as underlined by the following result.

**Lemma 86** (Approximation of autonomous flows). Let $\delta > 0$, $f \in C^1(B_{2\delta}; \mathbb{K}^d)$ with $\|f\|_{C^0} < \delta$. Using the notations of Remark 74:

1. For each $M \in \mathbb{N}$, if $f \in C^M(B_{2\delta}; \mathbb{K}^d)$ and $\phi \in C^{M+1}(B_{2\delta}; \mathbb{K})$, for each $p \in B_\delta$,

$$\left| \left( e^f - \sum_{k=0}^{M} \frac{f^k}{k!} \right) (\phi)(p) \right| \le \|f\|_{C^M}^{M+1} \|\phi\|_{C^{M+1}}. \qquad (153)$$

2. If $f \in C^\omega(B_{2\delta}; \mathbb{K}^d)$ and $\phi \in C^\omega(B_{2\delta}; \mathbb{K})$, for $t$ small enough, for each $p \in B_\delta$,

$$e^{tf}(\phi)(p) = \sum_{k=0}^{+\infty} \frac{t^k}{k!} f^k \phi(p) \qquad (154)$$

and the sum converges absolutely in the sense of analytic functions.

*Proof.* First statement. By the first point of Lemma 82, $e^{tf}(p)$ is well defined for every $t \in [0, 1]$ and takes values in $B_{2\delta}$. For $t \in [0, 1]$ and $k \in [0, M+1]$, we have

$$\frac{\mathrm{d}^k}{\mathrm{d}t^k} [\phi(e^{tf}(p))] = (f^k \phi)(e^{tf}(p)). \qquad (155)$$

Thus, the considered sum is the Taylor expansion of order $M$ of the map $t \mapsto \phi(e^{tf}(p))$ at $t = 0$ and

$$\left( e^f - \sum_{k=0}^{M} \frac{f^k}{k!} \right) (\phi)(p) = \int_{0}^{1} \frac{(1-s)^M}{M!} (f^{M+1}\phi)(e^{sf}(p)) \mathrm{d}s. \qquad (156)$$

This concludes the proof of (153) thanks to the integration in (156) and Lemma 75.

Second statement. Let $r > 0$ be such that $f \in C^{\omega,r}(B_{2\delta}; \mathbb{K}^d)$ and $\phi \in C^{\omega,r}(B_{2\delta}; \mathbb{K})$. Let $r' \in [r/e, r)$. By (134), for each $k \in \mathbb{N}$,

$$\left\| \frac{t^k}{k!} f^k \phi \right\|_{r'} \le \frac{|t|^k}{k!} \frac{k!}{e} \left( \frac{e}{r-r'} \right)^k \|f\|_r^k \|\phi\|_r, \qquad (157)$$
so that the sum converges absolutely in $C^{\omega,r'}$ when $|t|e \|f\|_r < r - r'$. Moreover, by (156) with $f \leftarrow tf$ and (134),

$$ \left\| \left( e^{tf} - \sum_{k=0}^{M} \frac{t^k}{k!} f^k \right) (\phi) \right\|_{C^0} \le \frac{|t|^{M+1}}{(M+1)!} \|f^{M+1} \phi\|_{C^0}, \quad (158) $$

where, using (157), the right-hand side tends to zero as $M \to +\infty$ under the same smallness condition; so that the sum converges towards $e^{tf}\phi$ in $C^{\omega,r'}$ when $|t|e \|f\|_r < r - r'$. $\square$

### 3.4.2 Pushforwards of vector fields by diffeomorphisms

**Definition 87** (Pushforward of a vector field by a diffeomorphism). Let $\Omega, \Omega'$ be open subsets of $K^d$. Let $\theta \in C^1(\Omega; \Omega')$ be a local diffeomorphism from $\Omega$ to $\Omega'$. Let $f \in C^0(\Omega; K^d)$ be a vector field. We define $\theta_*f \in C^0(\Omega'; K^d)$ the pushforward of $f$ by $\theta$ as

$$ (\theta_*f)(q) := (D\theta)|_{\theta^{-1}(q)} f(\theta^{-1}(q)) = (D\theta^{-1})_{|q}^{-1} f(\theta^{-1}(q)). \quad (159) $$

**Lemma 88** (Chain rule for pushforwards). Let $\Omega, \Omega', \Omega''$ be open subsets of $K^d$. Let $\theta \in C^1(\Omega; \Omega')$ be a local diffeomorphism from $\Omega$ to $\Omega'$ and $\theta' \in C^1(\Omega'; \Omega'')$ be a local diffeomorphism from $\Omega'$ to $\Omega''$. Let $f \in C^0(\Omega; K^d)$ be a vector field. Then, on $\Omega''$,

$$ \theta'_*(\theta_*f) = (\theta' \circ \theta)_*f. \quad (160) $$

*Proof.* This is a consequence of the chain rule for differentiation, see e.g. [77, Problem 12-10]. $\square$

**Lemma 89** (Lie brackets of pushforwards). Let $\Omega, \Omega'$ be open subsets of $K^d$. Let $\theta \in C^2(\Omega; \Omega')$ be a local diffeomorphism from $\Omega$ to $\Omega'$. Let $f, g \in C^1(\Omega; K^d)$ be two vector fields. Then, on $\Omega'$,

$$ [\theta_*f, \theta_*g] = \theta_*[f, g]. \quad (161) $$

*Proof.* This is a consequence of the chain rule for differentiation, see e.g. [77, Corollary 8.31]. $\square$

### 3.4.3 Composition of vector fields with flows

**Lemma 90.** Let $\delta > 0$, $f_0 \in C^1(B_{2\delta}; K^d)$ and $t \in \mathbb{R}$ such that $|t|\|f_0\|_{C^0} < \delta$. Denote by $\Phi_0(t, p) := e^{tf_0}(p)$ the associated flow for $p \in B_\delta$.

1. For each $M \in \mathbb{N}$, if $f_0, f_1 \in C^{M+1}(B_{2\delta}; K^d)$, then, for each $p \in B_\delta$,

$$ \left| (\partial_p \Phi_0(t, p))^{-1} f_1 (\Phi_0(t, p)) - \sum_{k=0}^{M-1} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k (f_1)(p) \right| \le \frac{t^M}{M!} \left\| \mathrm{ad}_{f_0}^M (f_1) \right\|_{C^0}. \quad (162) $$

2. For each $M \in \mathbb{N}$, if $f_0, f_1 \in C^{M+1}(B_{2\delta}; K^d)$ and $\mathrm{ad}_{f_0}^M(f_1) \equiv 0$, then, for each $p \in B_\delta$,

$$ (\Phi_0(-t)_*f_1)(p) = (\partial_p\Phi_0(t,p))^{-1}f_1(\Phi_0(t,p)) = \sum_{k=0}^{M-1} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k(f_1)(p). \quad (163) $$

This holds in particular when $\mathcal{L}(\{f_0, f_1\})$ is nilpotent with index at most $(M+1)$.

3. If $r > 0$, $f_0, f_1 \in C^{\omega,r}(B_{2\delta}; K^d)$, then, for $|t| < \frac{r}{6\|f_0\|_r}$, for each $p \in B_\delta$,

$$ (\Phi_0(-t)_*f_1)(p) = (\partial_p\Phi_0(t,p))^{-1}f_1(\Phi_0(t,p)) = \sum_{k=0}^{+\infty} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k(f_1)(p), \quad (164) $$

where, for every $r' \in [r/e, r)$ the series converges in $C^{\omega,r'}(B_{2\delta}; K^d)$ when $|t| < \frac{r-r'}{6\|f_0\|_r}$.
4. Let $H_0, H_1 \in \mathcal{M}_d(\mathbb{K}^d)$ and $M \in \mathbb{N}^*$. Then

$$ \left\| e^{H_0} H_1 e^{-H_0} - \sum_{k=0}^{M-1} \frac{1}{k!} \mathrm{ad}_{H_0}^k (H_1) \right\| \le \frac{(2\|H_0\|)^M}{M!} \|H_1\| e^{2\|H_0\|} \quad (165) $$

and

$$ e^{H_0} H_1 e^{-H_0} = \sum_{k=0}^{+\infty} \frac{1}{k!} \mathrm{ad}_{H_0}^k (H_1), \quad (166) $$

where ad is the commutator of matrices $\mathrm{ad}_A(B) := [A, B] = AB - BA$ and $\| \cdot \|$ a sub-multiplicative norm on $\mathcal{M}_d(\mathbb{K})$ such that $\|\mathrm{Id}_d\| = 1$.

*Proof.* We proceed step by step.

1. First, for each $\tau \in [0, t]$, $\Phi_0(\tau, p)$ is well-defined. Taking into account that

$$ \begin{aligned} \frac{d}{d\tau} \left[ (\partial_p \Phi_0(t, p))^{-1} \right] &= -(\partial_p \Phi_0(t, p))^{-1} \frac{d}{d\tau} [\partial_p \Phi_0(\tau, p)] (\partial_p \Phi_0(t, p))^{-1} \\ &= -(\partial_p \Phi_0(t, p))^{-1} D f_{0|\Phi_0(\tau,p)} \end{aligned} \quad (167) $$

one obtains by induction on $k \in [0, M+1]$ that

$$ \frac{d^k}{d\tau^k} \left[ (\partial_p \Phi_0(t, p))^{-1} f_1 (\Phi_0(\tau, p)) \right] = (\partial_p \Phi_0(t, p))^{-1} \mathrm{ad}_{f_0}^k (f_1) (\Phi_0(\tau, p)). \quad (168) $$

The Taylor formula

$$ \begin{aligned} & (\partial_p \Phi_0(t, p))^{-1} f_1 (\Phi_0(t, p)) - \sum_{k=0}^{M-1} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k (f_1)(p) \\ & \qquad = \int_0^t \frac{(t-s)^{M-1}}{(M-1)!} (\partial_p \Phi_0(s, p))^{-1} \mathrm{ad}_{f_0}^M (f_1) (\Phi_0(s, p)) \, ds \end{aligned} \quad (169) $$

proves the first statement.

2. Equation (169) yields the conclusion.

3. Let $r' \in [r/e, r)$. Thanks to (138),

$$ \left\| \frac{t^k}{k!} \mathrm{ad}_{f_0}^k (f_1) \right\|_{r'} \le \frac{|t|^k k!}{k! e} \left( \frac{2e}{r-r'} \right)^k \|f_0\|_r^k \|f_1\|_r, \quad (170) $$

so the series converges absolutely in $C^{\omega,r'}$ when $2e|t| \|f_0\|_r < r-r'$, which is the case when $6|t| \|f_0\|_r < r-r'$ because $2e < 6$. The weakest bound, for $r' = r/e$ is $2e|t| \|f_0\|_r < (1-1/e)r$ and it holds when $9|t| \|f_0\|_r < r$ because $2e/(1-1/e) < 9$.

Moreover, thanks to (169) and (170),

$$ \begin{aligned} \left| (\Phi_0(-t)_* f_1)(p) - \sum_{k=0}^{M-1} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k (f_1)(p) \right| &\le \frac{|t|^M}{M!} \| \mathrm{ad}_{f_0}^M (f_1) \|_{C^0} \sup_{s \in [0,t]} \|(\partial_p \Phi_0(s, \cdot))^{-1}\|_{C^0} \\ &\le A_0 \|f_1\|_r \left( \frac{2e|t| \|f_0\|_r}{r-r'} \right)^M, \end{aligned} \quad (171) $$

where $A_0$ denotes the supremum in the right-hand side of (171) which is finite. So the sum converges towards the pushforward under the same smallness assumption on time.

4. The last statement is proved similarly, by considering the function $t \mapsto e^{tH_0} H_1 e^{-tH_0}$. $\square$
### 3.4.4 Partial derivative of a flow with respect to a parameter

In this paragraph, we compute the partial derivative of a flow with respect to a parameter on which the vector field depends, under a particular nilpotency assumption.

**Lemma 91.** Let $J$ an interval of $\mathbb{R}$. Let $\delta > 0$ and $f \in C^\infty(J \times B_{4\delta}; \mathbb{K}^d)$ such that $\|f\|_{C^0} < \delta$. Let $\lambda_0 \in J$, $M \in \mathbb{N}$ and assume that, for each $\lambda \in J$, $\text{ad}_{f(\lambda_0)}^M(f(\lambda)) \equiv 0$. Then, for each $p \in B_{\delta}$,

$$ \frac{d}{d\lambda} \left( e^{f(\lambda)} p \right) \Big|_{\lambda=\lambda_0} = \sum_{k=0}^{M-1} \frac{(-1)^k}{(k+1)!} \text{ad}_{f(\lambda_0)}^k (\partial_\lambda f(\lambda_0)) \left( e^{f(\lambda_0)} p \right). \quad (172) $$

This holds in particular when $\mathcal{L}(f(J))$ is nilpotent with index at most $M+1$.

*Proof.* Let $\Theta \in C^\infty([0, 1] \times J \times B_\delta)$ defined by $\Theta(t, \lambda, p) := e^{tf(\lambda)}(p)$. Let $p_0 \in B_\delta$ and $\lambda_0 \in J$. Let $x_0(t) := e^{tf(\lambda_0)}(p_0)$ for $t \in [0, 1]$. Then, the desired derivative is $\partial_\lambda \Theta(1, \lambda_0, p_0) = z(1)$ where $z$ is the solution to $z(0) = 0$ and

$$ \dot{z}(t) = \partial_x f(\lambda_0, x_0(t)) z(t) + \partial_\lambda f(\lambda_0, x_0(t)). \quad (173) $$

Let $R: (t, s) \in [0, 1]^2 \to \mathcal{M}_d(\mathbb{K})$ be the resolvent associated with the linearized system at $p_0$, which is the solution to $R(s, s) = \text{Id}$ and

$$ \partial_t R(t, s) = \partial_x f(\lambda_0, x_0(t)) R(t, s), \quad (174) $$

i.e. $R(t, s) = \partial_p \Theta(t - s, \lambda_0, x_0(s))$. Then by Duhamel's principle

$$ \begin{aligned} z(1) &= \int_0^1 R(\tau, 1)^{-1} \partial_\lambda f(\lambda_0, x_0(\tau)) \, d\tau \\ &= \int_0^1 (\partial_p \Theta(\tau - 1, \lambda_0, x_0(1)))^{-1} \partial_\lambda f(\lambda_0, \Theta(\tau - 1, \lambda_0, x_0(1))) \, d\tau. \end{aligned} \quad (175) $$

By (163) of Lemma 90 with $t \leftarrow \tau - 1$, $f_0 \leftarrow f(\lambda_0, \cdot)$, $f_1 \leftarrow \partial_\lambda f(\lambda_0, \cdot)$ and $p \leftarrow x_0(1)$,

$$ z(1) = \int_0^1 \sum_{k=0}^{M-1} \frac{(\tau - 1)^k}{k!} \text{ad}_{f(\lambda_0)}^k (\partial_\lambda f(\lambda_0)) (x_0(1)) \, d\tau, \quad (176) $$

which gives the conclusion. $\square$

# 4 Error estimates in time for nonlinear vector fields

Using a classical linearization principle (see Section 4.1), we show that the formal expansions for linear equations of Section 2 can yield approximate formulas in the context of nonlinear ordinary differential equations. We derive rigorous error bounds at every fixed order with respect to time, involving finite sums or products.

## 4.1 Linearization principle for nonlinear vector fields

We explain how, by seeing vector fields as first-order differential operators and points on the manifold as the operator of evaluation at this point, one classically recasts a nonlinear ODE driven by smooth vector fields to a linear equation set on a larger space of operators on smooth functions. This approach is notably used in [3, 4] (replacing nonlinear objects by infinite-dimensional linear ones is the foundation of the “chronological calculus”) and in [97]. More generally, the idea of replacing the study of a space by the study of the ring of functions on that space is reminiscent of the representation results of [50]. For readers with a background in PDE analysis, this linearization principle can be seen as a “reversed method of characteristics”: it transforms a nonlinear ODE into a linear transport PDE, considered at the level of evolution operators.
### 4.1.1 Definition of an operator acting on smooth functions

When $T > 0$ and $f \in C_c^\infty([0, T] \times \mathbb{K}^d; \mathbb{K}^d)$ satisfies $\|f\|_{L^1([0,T];C^0)} < 1$, we take the nonlinear ODE (143) back to a linear framework by considering, for every $t \in [0, T]$ the linear operator $L(t)$ on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$ defined, for $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$, by

$$L(t)\varphi : p \mapsto \varphi(x(t; f, p)). \qquad (177)$$

$L(t)\varphi$ is of class $C^\infty$ as a composition of $C^\infty$ functions, by the third statement of Lemma 82. $L(t)\varphi$ is compactly supported in $\mathbb{K}^d$ because $\varphi$ is and $|x(t; f, p) - p| \le 1$ for every $p \in \mathbb{K}^d$, by the first statement of Lemma 82 (which is of course invariant by translation of the origin). We don't specify the dependence of $L(t)$ with respect to $f$ to simplify the notations.

For every $p \in \mathbb{K}^d$, the map $t \in [0, T] \mapsto (L(t)\varphi)(p)$ belongs to $C^1([0, T]; \mathbb{K})$ and satisfies, for every $t \in [0, T]$, using the notations of Remark 74,

$$\frac{d}{dt}(L(t)\varphi)(p) = D\varphi(x(t; f, p))f(t, x(t; f, p)) = (L(t)f(t)\varphi)(p). \qquad (178)$$

Thus, $L$ satisfies the following linear equation

$$\frac{d}{dt}L(t) = L(t)f(t) \qquad (179)$$

in the weak sense explicated above. For every fixed $t \in [0, T]$,

$$\forall \varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K}), \forall p \in \mathbb{K}^d, \quad (L(t)\varphi)(p) = \varphi(p) + \int_0^t (L(\tau)f(\tau)\varphi)(p) \, d\tau, \qquad (180)$$

where the symbol $\int_0^t$ is the Lebesgue integral on $L^1((0, t); \mathbb{K})$. We will use the following notation to refer to this property:

$$L(t) = \mathrm{Id} + \int_0^t L(\tau)f(\tau) \, d\tau. \qquad (181)$$

In the sequel, all integral equalities between operators on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$ should be understood in this weak sense (after evaluation on a test function and at a point). The right-hand side refers to the composition of two operators on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$: $L(\tau)$ and $f(\tau)$, seen as a first-order differential operator on smooth functions.

Equation (179) is now a linear differential equation satisfied by the object $L(t)$ (in a much larger space), so one can hope to apply the linear results of the previous sections.

### 4.1.2 Approximating sequence

In order to approximate the operator $L(t)$, it is natural to introduce the sequence $(L_j)_{j \in \mathbb{N}}$ of time-dependent operators on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$ defined, for every $t \in [0, T]$, by $L_0(t) := \mathrm{Id}$ and, for $j \in \mathbb{N}$,

$$L_{j+1}(t) := \int_{0}^{t} L_{j}(\tau)f(\tau) \, d\tau, \qquad (182)$$

where this definition should be understood in the weak sense. Hence, recalling Remark 74,

$$L_j(t) = \int_{\Delta^j(t)} f(\tau_1) \cdots f(\tau_j) \, d\tau, \qquad (183)$$

where the integration domain is the ordered simplex of Definition 15. Then, for every $j \in \mathbb{N}$, $L_j$ is “of order $j$ with respect to $f$”, and a differential operator of order at most $j$ (with respect to $x$) on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$. And this sequence indeed allows to approximate $L(t)$ in the sense exposed in Proposition 92 below, in a finite regularity setting.
## 4.2 Chen-Fliess expansion

The approximating sequence for the operator $L(t)$ yields the following straight-forward estimate for the Chen-Fliess expansion of the state, which can also be found in [3, Proposition 2.3].

**Proposition 92.** For every $M \in \mathbb{N}$, $\delta > 0$, $T > 0$, $f \in L^1((0,T); C^{\max(M,1)}(B_{2\delta}; \mathbb{K}^d))$, with $\|f\|_{L^1((0,T);C^0)} < \delta$ and $\varphi \in C^{M+1}(B_{2\delta}; \mathbb{K})$, for each $t \in [0,T]$, with the notations of Remark 74,

$$ \left| \varphi(x(t; f, p)) - \sum_{j=0}^{M} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j)\varphi)(p) \, d\tau \right| \le (M+1)! \|f\|_{L^1((0,t);C^M)}^{M+1} \|\varphi\|_{C^{M+1}}. \quad (184) $$

In particular, for each $p \in B_\delta$,

$$ \left| x(t; f, p) - \sum_{j=0}^{M} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j) \mathrm{Id}_d)(p) \, d\tau \right| \le (M+1)! \|f\|_{L^1((0,t);C^M)}^{M+1}. \quad (185) $$

Hence, if $f \in L^\infty((0,T); C^M)$, both estimates correspond to a bound scaling like $t^{M+1}$.

*Proof.* Let $p \in B_\delta$. Thanks to Lemma 82, $x(\tau; f, p)$ is well-defined for $\tau \in [0,T]$ and $x(\cdot; f, p) \in C^1([0,T]; \mathbb{K}^d)$. Thus, for each $\tau \in [0,T]$,

$$ \varphi(x(\tau; f, p)) = \varphi(p) + \int_0^\tau (f(\tau_1)\varphi)(x(\tau_1; f, p)) \, d\tau_1. \quad (186) $$

By iterating this formula, we obtain for $t \in [0,T]$,

$$ \begin{aligned} \varphi(x(t; f, p)) - \varphi(p) & - \sum_{j=1}^{M} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j)\varphi)(p) \, d\tau \\ & = \int_{\Delta^{M+1}(t)} (f(\tau_1) \cdots f(\tau_{M+1})\varphi)(x(\tau_{M+1}; f, p)) \, d\tau, \end{aligned} \quad (187) $$

which concludes the proof of (184) using Lemma 75. Then (185) follows by applying (184) to coordinate functions. $\square$

## 4.3 Magnus expansion in the usual setting

In Section 4.3.1, we state a precise estimate of the difference between the exact flow and the exponential of its truncated logarithm. In Section 4.3.2, we show that this estimate implies a similar estimate for the CBHD formula. Section 4.3.3 is devoted to a technical result used in the proof, which transposes to vector fields a formal integral identity.

### 4.3.1 Standard error estimate in time

The following estimate can be viewed as a refined version of classical time-focused estimates (see e.g. [87, Proposition 4.3]). It bears a lot of similarity with [38, Theorem 1.32], but is both easier to state and to prove in our flat setting since [38] is concerned with the truncated logarithm of flows in general Riemannian manifolds. We propose a proof for sake of completeness, and because this precise estimate is the founding principle of the new estimate, proved in the next section. It relies on the usual arguments, used for instance in [97] and [3, Proposition 4.1] (which states a slightly tighter estimate).
**Proposition 93.** For every $M \in \mathbb{N}$, there exists $\delta_M$, $C_M > 0$ such that, for every $\delta > 0$, $T > 0$, $f \in L^1((0,T); C^{\max(M^2,1)}(B_{2\delta}; \mathbb{K}^d))$ with $\|f\|_{L^1((0,T);C^{M^2})} \le \delta_M \min\{1; \delta\}$, $p \in B_\delta$ and $t \in [0,T]$,

$$|x(t; f, p) - e^{Z_M(t, f)} p| \le C_M \|f\|_{L^1((0,t); C^{M^2})}^{M+1}, \quad (188)$$

where $Z_M(t, f) := \text{Log}_M\{f\}(t)$ is the vector field introduced in Definition 23.

Hence, if $f \in L^\infty((0,T); C^{M^2}(B_{2\delta}; \mathbb{K}^d))$ this estimate corresponds to a bound scaling like $t^{M+1}$.

Moreover, if $f(t, x) = \sum_{i \in I} u_i(t) f_i(x)$ with $u_i \in L^1((0,T); \mathbb{K})$ and $f_i \in C^{M^2}(B_{2\delta}; \mathbb{K}^d)$, then, for each monomial basis $\mathcal{B}$ of $\mathcal{L}(\bar{X})$,

$$Z_M(t, f) = \sum_{b \in \mathcal{B}_{[1,M]}} \zeta_b(t, u) f_b \quad (189)$$

where the functionals $\zeta_b$ are the associated coordinates of the first kind and $f_b$ are the evaluated Lie brackets (see Definitions 28, 30 and 78).

*Proof.* For $M = 0$, $Z_0(t, f) = 0$ thus (188) holds with $C_0 = 1$ because $|x(t; f, p) - p| \le \|f\|_{L^1(C^0)}$. From now on $M \in \mathbb{N}^*$ is fixed. By Definition 23, there exists $C'_M > 0$ such that, for every $\delta > 0$, $T > 0$, $f \in L^1((0,T); C^{M-1})$ with $\|f\|_{L^1(C^{M-1})} \le 1$ and $t \in [0,T]$,

$$\|\text{Log}_M\{f\}(t)\|_{C^0} \le C'_M \|f\|_{L^1(C^{M-1})}. \quad (190)$$

In particular, for every $\delta > 0$, $T > 0$, $f \in L^1((0,T); C^{M-1})$ with $\|f\|_{L^1(C^{M-1})} \le \min\{1; \delta; \delta/C'_M\}$, for every $p \in B_\delta$ and $t \in [0,T]$,

* $x(t; f, p)$ is well defined and belongs to $B_{2\delta}$,

* for every $s \in [0, 1]$, $e^{s\text{Log}_M\{f\}(t)}p$ is well defined belongs to $B_{2\delta}$.

This happens, in particular, when $\|f\|_{L^1(C^{M-1})} \le \delta_M \min\{1; \delta\}$ with $\delta_M := \min\{1; 1/C'_M\}$.

From now on, we fix $\delta, T > 0$ and $f \in L^1((0,T); C^{M^2})$ with $\|f\|_{L^1(C^{M^2})} \le \delta_M \min\{1; \delta\}$.

In order to use the operators $L(t)$ defined in Section 4.1, we assume that $f \in C_c^\infty([0, T] \times \mathbb{K}^d; \mathbb{K}^d)$. This is not restrictive because this space is dense in $L^1((0,T); C^{M^2}(B_{2\delta}; \mathbb{K}^d))$ and both sides of (188) are continuous for the $L^1((0,T); C^{M^2})$ topology on $f$ (see the fourth item of Lemma 82). Moreover, this regularization procedure is merely an heuristical convenience, since all the computations performed below make perfect sense even in our finite regularity setting.

**Step 1: Construction of the formal logarithm.** We introduce $Z_M(t, f)$ the finite sum of terms “of order at most $M$ with respect to $f$” in the following formal power series (recall the formal power series for $\log(1+x)$):

$$\log L(t) = \sum_{m \in \mathbb{N}^*} \frac{(-1)^{m-1}}{m} \left( \sum_{j \in \mathbb{N}^*} L_j(t) \right)^m, \quad (191)$$

with the notation of (182), i.e. we define

$$Z_M(t, f) := \sum_{r=1}^{M} \sum_{m=1}^{M} \frac{(-1)^{m-1}}{m} \sum_{\mathbf{r} \in \mathbb{N}_r^m} L_{\mathbf{r}_m}(t) \cdots L_{\mathbf{r}_1}(t), \quad (192)$$

where $\mathbb{N}_r^m$ is defined in (31). For instance,

$$Z_3 = L_1 + \left(L_2 - \frac{1}{2} L_1^2\right) + \left(L_3 - \frac{1}{2}(L_1 L_2 + L_2 L_1) + \frac{1}{3} L_1^3\right). \quad (193)$$
Then, by (183),

$$Z_M(t, f) = \sum_{r=1}^{M} \sum_{m=1}^{M} \frac{(-1)^{m-1}}{m} \sum_{\mathbf{r} \in \mathbb{N}_m^d} \int_{\Delta^r(t)} f(\tau_1) \cdots f(\tau_r) \, d\tau, \quad (194)$$

A priori, $Z_M(t, f)$ is thus an inhomogeneous differential operator on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$, of order at most $M$. Using Lemma 96 (see below in the next paragraph) and Definition 23, $Z_M(t, f) = \text{Log}_M\{f\}(t)$ and satisfies (189). Thus $Z_M(t, f)$ is a smooth vector field, i.e. both a vector-valued function and a first-order differential operator.

**Step 2: Strategy for the proof of the estimate.** The key observation is that it is sufficient to prove that there exists $C_M > 0$ (independent of $\delta, T, f$) such that, for every $p \in B_\delta$, $t \in [0, T]$ and $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$,

$$\left| (L(t) - e^{Z_M(t,f)}) (\varphi)(p) \right| \le C_M \|f\|_{L^1(C^{M2})}^{M+1} \|\varphi\|_{C^{M2+1}}. \quad (195)$$

Then, the conclusion follows by considering an appropriate $C_c^\infty$ truncation of the coordinate functions $\varphi_j : x \in \mathbb{K}^d \mapsto x_j \in \mathbb{K}$. To prove (195), we will decompose the difference in three terms

$$L - e^{Z_M} = \left( L - \sum_{j=0}^{M} L_j \right) + \left( \sum_{j=0}^{M} L_j - \sum_{k=0}^{M} \frac{Z_M^k}{k!} \right) + \left( \sum_{k=0}^{M} \frac{Z_M^k}{k!} - e^{Z_M} \right), \quad (196)$$

with the notation of (182). The first term is estimated in Proposition 92.

**Step 3: Bound for** $\sum L_j - \sum \frac{Z_M^k}{k!}$. By (192), this operator is a (finite) linear combination of terms of the form $L_{j_1}(t) \cdots L_{j_p}(t)$ where $p \in \mathbb{N}^*$, $j_1, \dots, j_p \in [1, M]$ and $M+1 \le j_1 + \dots + j_p \le M^2$. Indeed, $Z_M(t, f)$ is also the finite sum of terms “of order at most $M$ with respect to $f$” in the formal power series (191). Thus, there exists $C_M'' > 0$ (independent of $\delta, T, f$) such that, for every $p \in B_\delta$, $t \in [0, T]$ and $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$,

$$\left| \left( \sum_{j=0}^{M} L_j(t) - \sum_{k=0}^{M} \frac{Z_M(t, f)^k}{k!} \right) (\varphi)(p) \right| \le C_M'' \|f\|_{L^1(C^{M2-1})}^{M+1} \|\varphi\|_{C^{M2}}. \quad (197)$$

**Step 4: Bound for** $\sum \frac{Z_M^k}{k!} - e^{Z_M}$. Using Lemma 86 for the time-independent vector field $Z_M(t, f)$ (where $t \in [0, T]$ has been fixed), estimate (153) yields for every $p \in B_\delta$, $t \in [0, T]$ and $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$,

$$\left| \left( e^{Z_M(t,f)} - \sum_{k=0}^{M} \frac{Z_M(t,f)^k}{k!} \right) (\varphi)(p) \right| \le \|Z_M(t,f)\|_{C^M}^{M+1} \|\varphi\|_{C^{M+1}}. \quad (198)$$

We deduce from (36) the existence of $C_M''' > 0$ (independent of $\delta, T, f$) such that for every $t \in [0, T]$

$$\|Z_M(t, f)\|_{C^M} \le C_M''' \|f\|_{L^1((0,t);C^{2M-1})}. \quad (199)$$

Hence, for every $p \in B_\delta$, $t \in [0, T]$ and $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$

$$\left| \left( e^{Z_M(t,f)} - \sum_{k=0}^{M} \frac{Z_M(t,f)^k}{k!} \right) (\varphi)(p) \right| \le (C_M''')^{M+1} \|f\|_{L^1(C^{2M-1})}^{M+1} \|\varphi\|_{C^{M+1}}. \quad (200)$$

Gathering (184), (197) and (200) concludes the proof of (188). □
### 4.3.2 Campbell Baker Hausdorff Dynkin formula

We deduce from Proposition 93 the following estimate for the classical CBHD formula with $q$ time-independent vector fields.

**Corollary 94.** For every $M \in \mathbb{N}^*$, there exists $\delta_M, C_M > 0$ such that, for every $\delta > 0$, $q \in \mathbb{N}^*$, $f_1, \dots, f_q \in C^{M^2}(B_{2\delta}; \mathbb{K}^d)$ with $\sum_{1 \le j \le q} \|f_j\|_{C^{M^2}} \le \delta_M \min\{1; \delta\}$,

$$ \left\| e^{f_q} \dots e^{f_1} - e^{\mathrm{CBHD}_M(f_1, \dots, f_q)} \right\|_{C^0} \le C_M \|f\|^{M+1} \quad (201) $$

where $\mathrm{CBHD}_M(f_1, \dots, f_q) = \mathrm{Log}_M\{f\}(q)$, where the time-dependent vector field $f$ is defined by $f : (t, x) \in [0, q] \times B_{2\delta} \mapsto \sum_{j=1}^q 1_{[j-1, j]}(t)f_j(x)$ and $\|f\| := \|f\|_{L^1(C^{M^2})} = \sum_{1 \le j \le q} \|f_j\|_{C^{M^2}}$.

Moreover, for each monomial basis $\mathcal{B}$ of $\mathcal{L}(\{X_1, \dots, X_q\})$

$$ \mathrm{CBHD}_M(f_1, \dots, f_q) = \sum_{b \in \mathcal{B}_{[1, M]}} \alpha_b f_b \quad (202) $$

where $(\alpha_b)_{b \in \mathcal{B}} \subset \mathbb{K}^{\mathcal{B}}$ is given by Corollary 33.

*Proof.* Because of the particular form of $f$, we have $x(t; f, p) = e^{f_q} \dots e^{f_1} p$. Thus the estimate (201) is an application of Proposition 93. Let $\Lambda : \mathcal{L}(\{X_1, \dots, X_q\}) \to \mathcal{L}(\{f_1, \dots, f_q\})$ be the homomorphism of Lie algebras such that $\Lambda(X_j) = f_j$. The map $\mathrm{CBHD}_M$ is defined by a finite sum of Lie brackets, thus it commutes with $\Lambda$

$$ \mathrm{CBHD}_M(f_1, \dots, f_q) = \Lambda(\mathrm{CBHD}_M(X_1, \dots, X_q)) = \Lambda \left( \sum_{b \in \mathcal{B}_{[1, M]}} \alpha_b b \right) = \sum_{b \in \mathcal{B}_{[1, M]}} \alpha_b \Lambda(b), \quad (203) $$

which proves (202). $\square$

### 4.3.3 Replacing products with brackets in logarithm integrals

The goal of this section is to prove Lemma 96, which is a key point in the proof of Proposition 93, as it allows to replace products of differential operators with Lie brackets in the integrals involved in the computation of the logarithm of the flow.

We first state and prove a corollary of Theorem 27 in algebras. Indeed, Theorem 27 is a statement about formal differential equations, but it has consequences for concrete realizations, e.g. for systems governed by vector fields or matrices (this will be used in Section 5.2.2).

**Corollary 95.** Let $A$ be a unital associative algebra over $\mathbb{K}$ and $A_1$ be a finite dimensional linear subspace of $A$. Then, for every $r \in \mathbb{N}^*$, $t > 0$ and $a \in L^1((0, t); \mathbb{K})$ and $y_i \in A$ then, for each monomial basis $\mathcal{B}_r$ of $\mathcal{L}_r(X)$,

$$ \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} a(\tau_1)a(\tau_2) \cdots a(\tau_r) \, d\tau = \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} [\cdots[a(\tau_1), a(\tau_2)], \dots, a(\tau_r)] \, d\tau, \quad (204) $$

where the equality should be seen as an equality between elements of a finite dimensional linear subspace of $A$ (generated by monomials of terms in $A_1$ of degree $r$), so that one can give a meaning to the integrals without introducing any topology on $A$.

Moreover, if $a(\tau) = \sum_{i \in I} \alpha_i(\tau)y_i$ with $\alpha_i \in L^1((0, t); \mathbb{K})$ and $y_i \in A$ then, for each monomial basis $\mathcal{B}_r$ of $\mathcal{L}_r(X)$,

$$ \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} [\cdots[a(\tau_1), a(\tau_2)], \dots, a(\tau_r)] \, d\tau = \sum_{b \in \mathcal{B}_r} \zeta_b(t, \alpha)y_b, \quad (205) $$
where the functionals $\zeta_b$ are the associated coordinates of the first kind and $y_b = \Upsilon(b)$ where $\Upsilon: A(X) \to A$ is the homomorphism of algebras such that $\Upsilon(X_i) = y_i$ (see Definition 30 and Lemma 7).

**Proof.** Let $q \in \mathbb{N}^*$ be the dimension of $A_1$ (as a linear subspace) and $y_1, \dots, y_q$ be a linear basis of $A_1$. Let $\alpha_i \in L^1((0, t); \mathbb{K})$ denote the components of $a(\cdot)$ in the basis $y_1, \dots, y_q$, i.e. $a(\tau) = \alpha_1(\tau)y_1 + \dots + \alpha_q(\tau)y_q$ for almost every $\tau \in [0, t]$. Then $a(t) = \Upsilon(a(t))$ where $a(\tau) := \alpha_1(\tau)X_1 + \dots + \alpha_q(\tau)X_q \in A_1(X)$. From (40) and (41), one obtains that (204) holds for $a(\cdot)$. Applying the homomorphism $\Upsilon$ of algebras to both sides proves (204) for $a(\cdot)$. The same strategy proves (205). $\square$

**Lemma 96.** For every $r \in \mathbb{N}^*$, $t > 0$ and $f \in C_c^\infty([0, t] \times \mathbb{K}^d; \mathbb{K}^d)$,

$$ \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} f(\tau_1) f(\tau_2) \cdots f(\tau_r) \, d\tau = \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} [\cdots[f(\tau_1), f(\tau_2)], \dots, f(\tau_r)] \, d\tau, \quad (206) $$

which should be seen as an equality between linear operators on $C_c^\infty(\mathbb{K}^d; \mathbb{K})$, hence only valid after evaluation at a function $\varphi$ at a point $p$, so that the integrals are integrals of scalar-valued functions. Moreover, if $f(\tau, x) = \sum_{i \in I} u_i(\tau) f_i(x)$ with $u_i \in L^1((0, t); \mathbb{K})$ and $f_i \in C_c^\infty(\mathbb{K}^d; \mathbb{K}^d)$ then

$$ \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} [\cdots[f(\tau_1), f(\tau_2)], \dots, f(\tau_r)] \, d\tau = \sum_{b \in \mathcal{B}_r} \zeta_b(t, u) f_b, \quad (207) $$

where $\mathcal{B}_r$ is a monomial basis of $\mathcal{L}_r(X)$, the functionals $\zeta_b$ are the associated coordinates of the first kind and $f_b$ are the evaluated Lie brackets (see Definitions 4, 30 and 78).

**Proof.** Let $(f_n)_{n \in \mathbb{N}^*}$ be a sequence of functions in $C_c^\infty([0, t] \times \mathbb{K}^d; \mathbb{K}^d)$ such that $f_n$ takes values in an at-most $n$-dimensional vector subspace $E_n$ of $C_c^\infty(\mathbb{K}^d; \mathbb{K}^d)$ and $\|f_n - f\|_{L^1((0,t);\mathcal{C}^r)} \to 0$ when $n \to \infty$. For example, one can choose an $n$-points trapezoidal approximation of $f$. For each fixed $n$, applying Corollary 95 with $A = \text{Op}(C_c^\infty(\mathbb{K}^d; \mathbb{K}))$ and $A_1 = E_n$ proves (206) for $f_n$. Let $\varphi \in C_c^\infty(\mathbb{K}^d; \mathbb{K})$ and $p \in \mathbb{K}^d$. For each $n \in \mathbb{N}^*$, we deduce that

$$ \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} (f_n(\tau_1) \cdots f_n(\tau_r)\varphi)(p) \, d\tau = \frac{1}{r} \sum_{m=1}^{r} \sum_{\mathbf{r} \in \mathbb{N}_{r}^{m}} \frac{(-1)^{m-1}}{m} \int_{\Delta^{\mathbf{r}}(t)} ([\cdots[f_n(\tau_1), f_n(\tau_2)], \dots, f_n(\tau_r)]\varphi)(p) \, d\tau. \quad (208) $$

For each fixed $\varphi$ and $p$, both sides converge as $n \to +\infty$ towards the same quantities for $f$. This proves that (206) holds as an equality between linear operators. Applying (205) gives (207). $\square$

**Remark 97.** Although most algebraic results of Section 2 remain valid for infinite alphabets (sets of indeterminates), there is a difficulty when one wishes to “evaluate” equalities in the free algebra over an infinite alphabet towards some target algebra (one must somehow introduce compatible topologies on both sides). Our approach to prove Lemma 96, where $f$ is allowed to take values in the infinite-dimensional space $C_c^\infty$, therefore relies on a discretization scheme to return to a finite alphabet, and the convergence of the involved integrals in a weak sense. Another approach, followed in [91, 92], consists in introducing definitions allowing an infinite (continuous) number of generators and proving analogous algebraic results in such a setting.
## 4.4 Magnus expansion in the interaction picture

In this section, we consider the nonlinear ordinary differential equation

$$ \dot{x}(t) = f_0(x) + f_{\sharp}(t, x) \qquad (209) $$

We show how the formal expansion introduced in Section 2.4 allows to obtain error bounds at every order in the size of the time-varying perturbation $f_{\sharp}$, provided that the flow of $f_0$ is known. Such estimates can be useful for example to design splitting methods in the case of a small perturbation (see e.g. [20, Section 3.6] or [21, Section 2]).

The results of this section can be seen as quite natural, but we are not aware of references containing the same statements. We adopt the notations specified in the following definition.

**Definition 98.** Let $M \in \mathbb{N}$, $\delta > 0$, $T > 0$ and $f_0 \in C^{M^2+1}(B_{5\delta}; \mathbb{K}^d)$ such that $T\|f_0\|_{C^0} < \delta$. Let $f_{\sharp} \in L^1((0, T); C^{M^2}(B_{5\delta}; \mathbb{K}^d))$ and $t \in [0, T]$. We consider

* $\Phi_0 \in C^{M^2+1}([0, T] \times B_{4\delta}; B_{5\delta})$ the flow associated with $f_0$ i.e. $\Phi_0(\tau; p) = e^{\tau f_0}(p)$,

* $g_t \in L^1((0, T); C^{M^2}(B_{4\delta}; \mathbb{K}^d))$ defined by

$$ g_t(\tau, y) := (\Phi_0(t - \tau)_{*} f_{\sharp}(\tau))(y) = (\partial_p \Phi_0(\tau - t, y))^{-1} f_{\sharp}(\tau, \Phi_0(\tau - t, y)), \qquad (210) $$

* $\mathcal{Z}_M(t, f_0, f_{\sharp}) := \text{Log}_M\{g_t\}(t) \in C^{M^2-M+1}(B_{4\delta}; \mathbb{K}^d)$ in the sense of Definition 23.

### 4.4.1 Error bound

**Proposition 99.** Let $M, \delta, T, f_0, f_{\sharp}$ as in Definition 98. There exists $\gamma = \gamma(M, \delta, \|f_0\|_{C^{M^2+1}}) > 0$ such that, if

$$ \|f_{\sharp}\|_{L^1((0,T);C^{M^2})} \le \gamma \qquad (211) $$

then, for every $p \in B_{\delta}$ and $t \in [0, \gamma]$,

$$ \left| x(t; f_0 + f_{\sharp}, p) - e^{\mathcal{Z}_M(t, f_0, f_{\sharp})} e^{tf_0} p \right| \le C_M \|g_t\|_{L^1((0,t);C^{M^2})}^{M+1} \qquad (212) $$

where $C_M > 0$ is the constant of Proposition 93.

Hence, if $f_{\sharp} \in L^{\infty}((0, T); C^{M^2}(B_{5\delta}; \mathbb{K}^d))$, estimate (212) scales like $t^{M+1}$.

**Proof.** Let $\eta_M, C_M > 0$ be as in Proposition 93. There exists $T^* = T^*(\delta, \|f_0\|_{C^{M^2+1}}) \in (0, T]$ such that, for every $f_{\sharp} \in L^1((0, T); C^{M^2}(B_{5\delta}; \mathbb{K}^d))$ and $t \in [0, T^*]$

$$ \|g_t\|_{L^1((0,t),C^{M^2})} \le 2 \|f_{\sharp}\|_{L^1((0,t),C^{M^2})}. \qquad (213) $$

Let $\gamma := \min\{T^*, \delta, \frac{\delta_M}{2} \min\{1, \delta\}\}$. Let $f_{\sharp} \in L^1((0, T); C^{M^2}(B_{5\delta}; \mathbb{K}^d))$ with $\|f_{\sharp}\|_{L^1(C^{M^2})} < \gamma$. Then, for every $p \in B_{\delta}$ and $\tau \in [0, T]$, $x(\tau; f_0 + f_{\sharp}, p)$ is well defined and belongs to $B_{3\delta}$. To simplify the notations in this proof, we write $x(\tau)$ instead of $x(\tau; f_0 + f_{\sharp}, p)$. Let $t \in [0, \gamma]$. The function $y: [0, t] \to \mathbb{K}^d$ defined by

$$ y(\tau) := \Phi_0(t - \tau; x(\tau)) \qquad (214) $$

takes values in $B_{4\delta}$ and satisfies, for every $\tau \in [0, t]$,

$$ \dot{y}(\tau) = g_t(\tau, y(\tau)). \qquad (215) $$

By (213), $\|g_t\|_{L^1(C^{M^2})} < 2\eta \le \delta_M \min\{1, \delta\}$ thus, by Proposition 93

$$ |y(t) - e^{\mathcal{Z}_M(t, f_0, f_{\sharp})} y(0)| \le C_M \|g_t\|_{L^1((0,t);C^{M^2})}^{M+1} \qquad (216) $$

which is exactly (212) because $y(t) = x(t)$ and $y(0) = e^{tf_0}p$. $\square$
### 4.4.2 Expansions of $Z_M$

**Proposition 100.** Let $M, \delta, T, f_0, f_\sharp$ as in Definition 98. Let $r > 0$. If $f_0 \in C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$ and $f_\sharp \in C^0([0, T]; C^{\omega,r}(B_{5\delta}; \mathbb{K}^d))$ then, for $0 \le \tau \le t \le \min\{T; \frac{r}{9\|\|f_0\|\|_r}\}$

$$g_t(\tau, \cdot) = e^{(\tau-t)\operatorname{ad}_{f_0}(f_\sharp(\tau))} = \sum_{k=0}^{+\infty} \frac{(\tau-t)^k}{k!} \operatorname{ad}_{f_0}^k(f_\sharp(\tau)) \quad (217)$$

and

$$Z_M(t, f_0, f_\sharp) = \sum \frac{(-1)^{m-1}}{rm} \int_{\Delta r(t)} \frac{(\tau_1 - t)^{k_1}}{k_1!} \cdots \frac{(\tau_r - t)^{k_r}}{k_r!} \left[ \cdots \left[ \operatorname{ad}_{f_0}^{k_1}(f_\sharp(\tau_1)), \operatorname{ad}_{f_0}^{k_2}(f_\sharp(\tau_2)) \right], \dots, \operatorname{ad}_{f_0}^{k_r}(f_\sharp(\tau_r)) \right] \mathrm{d}\tau, \quad (218)$$

where the sum is taken over $r \in [1, M]$, $m \in [1, r]$, $\mathbf{r} \in \mathbb{N}_r^m$, and $k_1, \dots, k_r \in \mathbb{N}$. Moreover, for every $r' \in [r/e, r)$ and $0 \le \tau \le t \le \min\{T; \frac{r-r'}{6\|\|f_0\|\|_r}\}$, the series (217) and (218) converge absolutely in $C^{\omega,r'}(B_{5\delta}; \mathbb{K}^d)$.

*Proof.* We apply the third statement of Lemma 90 to $f_0$ and $f_\sharp(\tau)$ to get (217). The absolute convergence in this series allows to interchange the sums and the integrals. $\square$

When the perturbation $f_\sharp(t, x)$ is affine, i.e. of the form $\sum_{i=1}^q u_i(t)f_i(x)$, by analogy with Theorem 41, we use the notation $Z_M(t, f, u)$ instead of $Z_M(t, f_0, \sum_{i=1}^q u_i f_i)$, with $f = (f_0, f_1, \dots, f_q)$ and $u = (u_1, \dots, u_q)$. In this context, we have the following result, that emphasizes that $Z_M$ is a truncated version of $Z_\infty$.

**Proposition 101.** Let $M, \delta, T, f_0, f_\sharp$ as in Definition 98. Let $r > 0$. If $f_0 \in C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$ and $f_\sharp(t, x) = \sum_{i=1}^q u_i(t)f_i(x)$ where $u_i \in L^1(0, T)$ and $f_i \in C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$. Then

$$Z_M(t, f, u) = \lim_{N \to \infty} \sum_{\substack{n(b) \le M \\ n_0(b) \le N}} \eta_b(t, u) f_b \quad (219)$$

where, for every $r' \in [r/e, r]$ the limit holds in $C^{\omega,r'}(B_{5\delta}; \mathbb{K}^d)$ when $0 \le t \le \min\{T; \frac{r-r'}{6\|\|f_0\|\|_r}\}$.

*Proof.* Let $X = \{X_0, X_1, \dots, X_q\}$ and $\Lambda : \mathcal{L}(X) \to C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$ be the homomorphism of Lie algebras such that $\Lambda(X_i) = f_i$ for $i \in [0, q]$ (see Lemma 7). By applying $\Lambda$ to each term in the equality (84) (where $Z_\infty^{r,\nu}(t, X, a)$ is the finite sum defined in (82)), we obtain for every $r \in \mathbb{N}^*$ and $\nu \in \mathbb{N}$

$$Z_\infty^{r,\nu}(t, f, u) = \sum_{b \in \mathcal{B}_{r,\nu}} \eta_b(t, u) f_b. \quad (220)$$

By Proposition 100

$$Z_M(t, f, u) = \lim_{N \to \infty} \sum_{\nu=0}^{N} \sum_{r=1}^{M} Z_\infty^{r,\nu}(t, f, u) \quad (221)$$

where for every $r' \in [r/e, r]$ the limit holds in $C^{\omega,r'}(B_{5\delta}; \mathbb{K}^d)$ when $0 \le t \le \min\{T; \frac{r-r'}{6\|\|f_0\|\|_r}\}$. This proves (219). $\square$

**Remark 102.** Although the family $\eta_b(t, u)f_b$ for $b \in \mathcal{B} \cap S_M = \{b \in \mathcal{B}; n(b) \le M\}$ (using Definition 10) is not proved to be absolutely summable, equality (219) gives a sense to the expression

$$Z_M(t, f, u) = \sum_{b \in \mathcal{B} \cap S_M} \eta_b(t, u) f_b. \quad (222)$$

Indeed, the proof above justifies the absolute summability of appropriate packages $Z_\infty^{r,\nu}(t, f, u)$ for $r \in [1, M]$ and $\nu \in \mathbb{N}$ of this family. The full absolute summability over $\mathcal{B} \cap S_M$ is investigated in the next subsection.
### 4.4.3 Absolute convergence for coordinates of the pseudo-first kind

Continuing the discussion started in Section 2.4.3 we state a criterion on the basis $\mathcal{B}$ which entails the absolute summability for analytic vector fields of the family $\eta_b(t, u)f_b$ for $b \in \mathcal{B} \cap S_M = \{b \in \mathcal{B}; n(b) \le M\}$ (using Definition 10).

**Proposition 103.** Let $q \in \mathbb{N}^*$, $X = \{X_0, X_1, \dots, X_q\}$ and $\mathcal{B}$ a Hall basis of $\mathcal{L}(X)$; or more generally a monomial basis of $\mathcal{L}(X)$ with geometric growth with respect to $X_0$ (see Definition 48 and Remark 50).

Let $M, \delta, T, f_0, f_\sharp$ as in Definition 98. Let $r > 0$. We assume $f_0 \in C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$ and $f_\sharp(t, x) = \sum_{i=1}^q u_i(t)f_i(x)$ where $u_i \in L^1(0, T)$ and $f_i \in C^{\omega,r}(B_{5\delta}; \mathbb{K}^d)$.

Let $r' \in [r/e, r)$. There exists $T^* = T^*(M, q, r, r', \|f_0\|_r) > 0$ such that, for every $t \in (0, T^*)$ and $u \in L^1((0, t), \mathbb{K}^q)$

$$Z_M(t, f, u) = \sum_{b \in \mathcal{B} \cap S_M} \eta_b(t, u) f_b \qquad (223)$$

where the series converges absolutely in $C^{\omega,r'}(B_\delta; \mathbb{K}^d)$.

*Proof.* By (92) of Proposition 52 and (138), for every $b \in \mathcal{B} \cap S_M$ and $t \in [0, T]$

$$|\eta_b(t, u)| \|f_b\|_{r'} \le \frac{r-r'}{2e^2} \left( \frac{2eC_M t \|f_0\|_r}{r-r'} \right)^{n_0(b)} \left( \frac{2eC_M}{r-r'} \|u\|_{L^1(0,t)} \|f\|_r \right)^{n(b)} \quad (224)$$

where $\|f\|_r := \max\{\|f_j\|_r; j \in [0, q]\}$. In particular, if $|t| < T^*(M, r, r') := \frac{r-r'}{4(q+1)eC_M \|f_0\|_r}$ then the series $\sum \eta_b(t, a)f_b$ converges absolutely in $C^{\omega,r'}$ because

$$\sum_{b \in \mathcal{B} \cap S_M} (2(q+1))^{-n_0(b)} \le \sum_{n=1}^{M} \sum_{n_0=0}^{+\infty} (q+1)^{n_0+n} (2(q+1))^{-n_0} \le M(q+1)^M. \quad (225)$$

□

## 4.5 Sussmann's infinite product expansion

Let $T > 0$. In this section, we consider affine systems of the form

$$\dot{x}(t) = \sum_{i \in I} u_i(t) f_i(x(t)) \quad \text{and} \quad x(0) = p, \qquad (226)$$

where, for $i \in I$, $f_i$ is a vector field and $u_i \in L^1((0, T); \mathbb{K})$. When well-defined, its solution is denoted $x(t; f, u, p)$. For every norm $\| \cdot \|$ on vector fields, $\|f\|$ denotes $\sum_{i \in I} \|f_i\|$.

**Proposition 104.** Let $\mathcal{B}$ be a Hall basis of $\mathcal{L}(X)$ and $(\xi_b)_{b \in \mathcal{B}}$ be the associated coordinates of the second kind. For every $M \in \mathbb{N}^*$, there exist $C_M, \eta_M > 0$ such that the following property holds. Let $T, \delta > 0$, $f_i \in C^{2M}(B_{3\delta}; \mathbb{K}^d)$ and $u_i \in L^1((0, T); \mathbb{K})$ for $i \in I$. Assume that

$$\|u\|_{L^1(0,T)} \|f\|_{C^M} \le \eta_M \min\{1, \delta\}. \qquad (227)$$

Then, for each $t \in [0, T]$ and $p \in B_\delta$,

$$\left| x(t; f, u, p) - \prod_{b \in \mathcal{B}_{[1,M]}} e^{\xi_b(t, u) f_b} p \right| \le C_M \|u\|_{L^1(0,t)}^{M+1} \|f\|_{C^{2M}}^{M+1} (1 + \|f\|_{C^{2M}}^{M-1}), \quad (228)$$

where the arrow above the product symbol designates the order for the product, i.e. with the notations of Definition 53

$$\prod_{b \in \mathcal{B}_{[1,M]}} e^{\xi_b(t, u) f_b} = e^{\xi_{b_1}(t, u) f_{b_1}} \dots e^{\xi_{b_{k+1}}(t, u) f_{b_{k+1}}}. \qquad (229)$$
Proof. Let $M \in \mathbb{N}^*$. We adopt the notations $b_1, \dots, b_{k+1}$ and $Y_0, \dots, Y_{k+1}$ of Definition 53. For $j \in [1, k+1]$, we denote by $\Phi_j$ the flow associated with $f_{b_j}$, i.e. $\Phi_j(t, p) := e^{t f_{b_j}(p)}$. To simplify the notations in this proof, we write $x(t)$ and $\xi_b(t)$ instead of $x(t; f, u, p)$ and $\xi_b(t, u)$. Let $\eta_M := 1/(4|I|M!)$.

Step 1: Well-definition of the flows. Using (227),

$$ \left\| \sum_{i \in I} u_i f_i \right\|_{L^1((0,T); \mathcal{B}^0)} \le \eta_M \min\{1, \delta\} \le \delta. \quad (230) $$

Thus, for $t \in [0, T]$, $x(t)$ is well-defined and $x(t) \in B_{2\delta}$. For $b \in \mathcal{B}$, using (107) and Lemma 79, we obtain, for each $t \in [0, T]$,

$$ \|\xi_b(t)f_b\|_{C^1} \le \ell! 2^{\ell-1} \|u\|_{L^1(0,t)}^{\ell} \|f\|_{C^{\ell}}^{\ell}, \quad (231) $$

where $\ell := |b|$. Hence, using the crude estimate $|\mathcal{B}_\ell| \le |I|^{\ell}$, we obtain, for each $t \in [0, T]$,

$$ \begin{aligned} \sum_{b \in \mathcal{B}_{[1,M]}} \|\xi_b(t)f_b\|_{C^1} & \le \sum_{\ell=1}^{M} |\mathcal{B}_\ell| \ell! 2^{\ell-1} \|u\|_{L^1}^{\ell} \|f\|_{C^\ell}^{\ell} \\ & \le M! \sum_{\ell=1}^{+\infty} (2|I| \|u\|_{L^1} \|f\|_{C^\ell})^\ell \\ & \le \frac{2M!|I| \|u\|_{L^1} \|f\|_{C^M}}{1-2|I| \|u\|_{L^1} \|f\|_{C^M}} \le \min\{1, \delta\}. \end{aligned} \quad (232) $$

Thus, for every $j \in [1, k+1]$,

$$ x_j(t) := e^{-\xi_{b_j}(t)f_{b_j}} \dots e^{-\xi_{b_1}(t)f_{b_1}}(x(t)) \quad (233) $$

is well-defined and belongs to $B_{3\delta}$.

Step 2: Estimates along a Lazard elimination. We prove by induction on $j \in [0, k+1]$ the existence of a numerical constant $C_j > 0$ such that

$$ (\mathcal{H}_j) : \begin{cases} \dot{x}_j(t) = \sum_{b \in \mathcal{B}_{[1,M]} \cap Y_j} \dot{\xi}_b(t) f_b(x_j(t)) + \varepsilon_j(t), \\ x_j(0) = p, \end{cases} \quad (234) $$

where

$$ |\varepsilon_j(t)| \le C_j |u(t)| \|u\|_{L^1}^M F^{M+1} (1 + F^{M-1}). \quad (235) $$

First, letting $x_0(t) := x(t)$ by convention, $(\mathcal{H}_0)$ holds with $\varepsilon_0 = 0$, $C_0 = 0$ because $\dot{\xi}_{X_i}(t) = u_i(t)$ for $i \in I$. Let $j \in [1, k+1]$ and assume that $(\mathcal{H}_{j-1})$ holds. We deduce from the definition of $x_j(t)$ that

$$ x_j(t) = e^{-\xi_{b_j}(t)f_{b_j}}(x_{j-1}(t)) = \Phi_j(-\xi_{b_j}(t), x_{j-1}(t)) \quad (236) $$

and, using $(\mathcal{H}_{j-1})$, that

$$ \dot{x}_j(t) = -\dot{\xi}_{b_j}(t) f_{b_j}(x_j(t)) + \sum_{b \in \mathcal{B}_{[1,M]} \cap Y_{j-1}} \dot{\xi}_b(t) \partial_p \Phi_j(-\xi_{b_j}(t), x_{j-1}(t)) f_b(x_{j-1}(t)) + \tilde{\varepsilon}_j(t) \quad (237) $$

where $\tilde{\varepsilon}_{j-1}(t) := \partial_p \Phi_j(-\xi_{b_j}(t), x_{j-1}(t)) \varepsilon_{j-1}(t)$. By (232), $\|\xi_{b_j}(t)f_{b_j}\|_{C^1} \le 1$, so, using (150),

$$ |\tilde{\varepsilon}_{j-1}(t)| \le e |\varepsilon_{j-1}(t)|. \quad (238) $$

Moreover, for each $b \in \mathcal{B}$

$$ \partial_p \Phi_j(-\xi_{b_j}(t), x_{j-1}(t)) f_b(x_{j-1}(t)) = (\Phi_j(-\xi_{b_j}(t))_* f_b)(x_j(t)), \quad (239) $$
thus,

$$ \dot{x}_j(t) = \sum_{b \in \mathcal{B}_{\llbracket 1,M \rrbracket} \cap Y_{j-1} \setminus \{b_j\}} \dot{\xi}_b(t) (\Phi_j(-\xi_{b_j}(t))_* f_b)(x_j(t)) + \tilde{\varepsilon}_j(t). \quad (240) $$

For $b \in \mathcal{B}_{\llbracket 1,M \rrbracket} \cap Y_j \setminus \{b_j\}$, we introduce the maximal integer $h(b) \in \mathbb{N}^*$ such that

$$ |b| + (h(b) - 1)|b_j| \leq M. \quad (241) $$

Then, by the first statement of Lemma 90 and Definition 62

$$ \begin{aligned} \dot{\xi}_b(t) (\Phi_j(-\xi_{b_j}(t))_* f_b)(x_j(t)) &= \sum_{m=1}^{h(b)-1} \frac{\xi_{b_j}^m(t)}{m!} \dot{\xi}_b(t) f_{\text{ad}_{b_j}^m(b)}(x_j(t)) + \bar{\varepsilon}_b^j(t) \\ &= \sum_{m=1}^{h(b)-1} \dot{\xi}_{\text{ad}_{b_j}^m(b)}(t) f_{\text{ad}_{b_j}^m(b)}(x_j(t)) + \bar{\varepsilon}_b^j(t) \end{aligned} \quad (242) $$

where

$$ |\bar{\varepsilon}_b^j(t)| \leq |\dot{\xi}_b(t)| \frac{|\xi_{b_j}(t)|^{h(b)}}{h(b)!} \|f_{\text{ad}_{b_j}^{h(b)}(b)}\|_{C^0}. \quad (243) $$

By definition of $h(b)$ we have $M+1 \leq |b| + h(b)|b_j| \leq M + |b_j| \leq 2M$. Thus, using Lemma 79, (106) and (107), we get

$$ \begin{aligned} |\bar{\varepsilon}_b^j(t)| &\leq |u(t)|||u||_{L^1}^{|b|+h(b)|b_j|-1} \frac{|b|}{h(b)!} 2^{2M}(2M-1)!F^{M+1}(1+F^{M-1}) \\ &\leq |u(t)|||u||_{L^1}^M M 2^{2M}(2M-1)!F^{M+1}(1+F^{M-1}). \end{aligned} \quad (244) $$

By definition of $Y_j$ in Definition 53, we obtain $(\mathcal{H}_j)$ with

$$ \varepsilon_j(t) := \tilde{\varepsilon}_{j-1}(t) + \sum_{b \in \mathcal{B}_{\llbracket 1,M \rrbracket} \cap Y_{j-1} \setminus \{b_j\}} \bar{\varepsilon}_b^j(t). \quad (245) $$

that satisfies (235) with, for instance $C_{j+1} := eC_j + |I|^{M+1}M2^{2M}(2M-1)!$.

**Step 3: Conclusion.** Taking into account that $\mathcal{B}_{\llbracket 1,M \rrbracket} \cap Y_{k+1} = \{0\}$, we get $\dot{x}_{k+1}(t) = \varepsilon_{k+1}(t)$ thus $|x_{k+1}(t) - p| \leq C_{k+1} \|u\|_{L^1}^{M+1} F^{M+1}(1+F^{M-1})$, i.e.

$$ \left| \left. \stackrel{\leftarrow}{\Pi}_{b \in \mathcal{B}_{\llbracket 1,M \rrbracket}} e^{-\xi_b(t,u)} f_b x(t) - p \right| \leq C_{k+1} \|u\|_{L^1}^{M+1} F^{M+1}(1+F^{M-1}). \quad (246) $$

Applying the locally Lipschitz map $e^{\xi_{b_1}(t,u)f_{b_1}} \dots e^{\xi_{b_{k+1}}(t,u)f_{b_{k+1}}}$ to the two terms in the left-hand side, we get another constant $C_M > 0$ such that (228) holds. Note that (232) and (150) ensure that $C_M \leq eC_{k+1}$, so that $C_M$ depends indeed only on $M$. $\square$

# 5 Convergence results and issues

The formal expansions of Section 2 generally exhibit poor convergence properties for smooth vector fields without any additional assumption. Nevertheless, one can hope to obtain convergence results in the following particular contexts:

* **Nilpotent Lie algebras.** Here, one assumes that the Lie algebra generated by the set of smooth vector fields $\{f(t, \cdot); t \in [0, T]\}$ is nilpotent (see Definition 5). This structural assumption turns most of the involved infinite expansions into finite ones, and it is thus reasonable to expect convergence properties.
* **Banach algebras.** Here, one assumes that the vector fields are actually linear in the space variable, e.g. that $f(t, x) = A(t)x$ for some $A(t) \in \mathcal{M}_d(\mathbb{K})$. This assumption yields better estimates for Lie brackets (since products of matrices behave more nicely than differentiation of nonlinear vector fields) and it is thus reasonable to expect convergence properties. In this section, we give statements for matrices for consistency, but similar results can be obtained for linear operators in a Banach algebra.

* **Analytic vector fields.** Here, one assumes that the vector fields are locally real-analytic, i.e. than their $k$-th derivative grows roughly no more than $k!$. This bound is compatible with the $1/k!$ factors which come out of the corresponding time integrals, and it is thus reasonable to expect convergence properties.

In the following paragraphs, we investigate the convergence properties of each expansion in each of these three reasonable contexts and encounter some surprises. We summarize the results in the following table.

<table><thead><tr><th>Expansion</th><th>Lie-Nilpotent</th><th>Banach</th><th>Analytic</th></tr></thead><tbody><tr><td>Chen-Fliess</td><td>No<br>(Section 5.1.1)</td><td>Global<br>(Section 5.1.2)</td><td>Yes<br>(Section 5.1.3)</td></tr><tr><td>Magnus in the<br>usual setting</td><td>Yes for <i>C</i><sup>∞</sup><br>(Section 5.2.1)</td><td>Small time<br>(Section 5.2.2)</td><td>No<br>(Section 5.2.3)</td></tr><tr><td>Magnus in the<br>interaction picture</td><td>Yes for <i>C</i><sup>ω</sup><br>(Section 5.3.1)</td><td>Small perturbation<br>(Section 5.3.2)</td><td>No<br>(Section 5.3.3)</td></tr><tr><td>Sussmann's<br>infinite product</td><td>Yes for <i>C</i><sup>∞</sup><br>(Section 5.4.1)</td><td>Small time<br>(Section 5.4.2)</td><td>Open problem<br>(Section 5.4.3)</td></tr></tbody></table>

# 5.1 Chen-Fliess expansion

## 5.1.1 Counter-example for nilpotent vector fields

As already discussed in Remark 17, the Chen-Fliess expansion is not an intrinsic representation of the flow and involves quantities which are not Lie brackets of the dynamics. Therefore, this expansion is not expected to converge under a Lie-nilpotency assumption. The following counter-example (where the dynamic does not depend on time, thereby obviously generating a nilpotent Lie algebra of order 2) proves that this expansion indeed relies on quantities which are not Lie brackets.

**Proposition 105.** There exists $f_0 \in \mathcal{C}^\infty(\mathbb{R}; \mathbb{R})$ such that, for every $t \in (0, 1]$, the solution $x(t; f, 0)$ to (143) with $f(t, x) := f_0(x)$ satisfies, with the notations of Remark 74,

$$ \lim_{N \to +\infty} \left| x(t; f, 0) - \sum_{n=0}^{N} \frac{t^n}{n!} (f_0^n \mathrm{Id}_1)(0) \right| = +\infty. \quad (247) $$

*Proof.* For every sequence $(\alpha_n)_{n \in \mathbb{N}} \in \mathbb{R}^\mathbb{N}$, there exists $f_\alpha \in \mathcal{C}^\infty(\mathbb{R}; \mathbb{R}) \cap L^\infty(\mathbb{R}; \mathbb{R})$ with $f_\alpha(0) = 1$ such that

$$ \forall n \ge 2, \quad (f_\alpha^n \mathrm{Id}_1)(0) = \alpha_n. \quad (248) $$

This is an easy consequence of Borel's lemma. Indeed, for $n \ge 2$ and $f_\alpha(0) = 1$,

$$ (f_\alpha^n \mathrm{Id}_1)(0) = f_\alpha^{(n-1)}(0) + P_n(f_\alpha(0), \dots, f_\alpha^{(n-2)}(0)), \quad (249) $$

for some polynomial $P_n$. Thus, given a sequence $(\alpha_n)_{n \in \mathbb{N}}$, one can prescribe an appropriate value for $f_\alpha^{(n-1)}$ and recursively ensure (248). Let $f_0$ be a vector field constructed following this process
for $\alpha_n := n!^2$. On the one hand, since $f_0 \in L^\infty(\mathbb{R}; \mathbb{R})$, $x(t; f, 0)$ is bounded for $t \in [0, 1]$. On the other hand, thanks to (248), for each $t > 0$

$$ \sum_{n=0}^{N} \frac{t^n}{n!} (f_0^n \mathrm{Id}_1)(0) = \sum_{n=0}^{N} n! t^n \to +\infty, \qquad (250) $$

which proves (247).

<figure><img src="image_6.png" /></figure>

**Remark 106.** In this counter-example, the local change of coordinates which transforms $f_0(x)e_1$ into the constant vector field $e_1$ allows to transform the ODE on $x$ to a new ODE for which the Chen-Fliess expansion is finite (and thus convergent). It would be even more interesting to construct a counter-example, probably in dimension $d \ge 2$, for which no local change of coordinates can restore the convergence of the Chen-Fliess expansion.

### 5.1.2 Global convergence for matrices

Let $T > 0$. In this paragraph, we study linear systems of the form

$$ \dot{x}(t) = A(t)x(t) \quad \text{and} \quad x(0) = p, \qquad (251) $$

where $A \in L^1((0, T); \mathcal{M}_d(\mathbb{K}))$. The solution is denoted $x(t; A, p)$.

**Proposition 107.** Let $T > 0$ and $A \in L^1((0, T); \mathcal{M}_d(\mathbb{K}))$. For each $t \in [0, T]$ and $p \in \mathbb{K}^d$,

$$ x(t; A, p) = p + \sum_{j=1}^{+\infty} \int_{\Delta^j(t)} A(\tau_j) \cdots A(\tau_1) p \, d\tau, \qquad (252) $$

where the series converges absolutely.

*Proof.* To simplify the notations, we write $x(t)$ instead of $x(t; A, p)$. By Grönwall's lemma, we have $|x(\tau)| \le |p|e^{\|A\|_{L^1(0,\tau)}}$ for every $\tau \in [0, T]$. By iterating the formula

$$ x(\tau) = p + \int_{0}^{\tau} A(\tau')x(\tau') \, d\tau' \qquad (253) $$

we obtain, for every $M \in \mathbb{N}^*$

$$ \left| x(t) - p - \sum_{j=1}^{M-1} \int_{\Delta^j(t)} A(\tau_j) \cdots A(\tau_1) p \, d\tau \right| = \left| \int_{\Delta^M(t)} A(\tau_M) \cdots A(\tau_1) x(\tau) \, d\tau \right| \qquad (254) \\ \le \int_{\Delta^M(t)} \|A(\tau_M)\| \cdots \|A(\tau_1)\| \, d\tau |p| e^{\|A\|_{L^1(0,t)}} = \frac{\|A\|_{L^1(0,t)}^M}{M!} |p| e^{\|A\|_{L^1(0,t)}} $$

which proves the convergence. Similar estimates prove the absolute convergence. □

<figure><img src="image_7.png" /></figure>

### 5.1.3 Local convergence for analytic vector fields

For analytic vector fields, it is well known that the Chen-Fliess series (also called “(right) chronological exponential” in [3, Section 2.1]) converges locally in time (see e.g. [3, Proposition 2.1], or [99, Proposition 4.3] for slightly different assumptions). The analyticity assumption is necessary, as highlighted by the counter-example of Section 5.1.1.
**Proposition 108.** Let $T, \delta, r > 0$. There exists $\eta > 0$ such that, for $f \in L^1([0, T]; C^{\omega,r}(B_{2\delta}; \mathbb{K}^d))$ with $\|f\|_{L^1(C^{\omega,r})} \le \eta$, $\varphi \in C^{\omega,r}(B_{2\delta}; \mathbb{K}^d)$, $t \in [0, T]$ and $p \in B_{\delta}$,

$$ \varphi(x(t; f, p)) = \varphi(p) + \sum_{j=1}^{+\infty} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j))(\varphi)(p) \, d\tau, \quad (255) $$

where the sum converges absolutely. In particular,

$$ x(t; f, p) = p + \sum_{j=1}^{+\infty} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j))(\mathrm{Id}_d)(p) \, d\tau. \quad (256) $$

*Proof.* Let $\eta := \min\{\delta/2, r/10\}$. By Lemma 82, $x(t; f, p)$ is well defined for $t \in [0, T]$, $p \in B_{\delta}$ and belongs to $B_{2\delta}$. Moreover, by Lemma 76, we have, for every $j \in \mathbb{N}^*$

$$ \int_{\Delta^j(t)} |(f(\tau_1) \cdots f(\tau_j))(\varphi)(p)| \, d\tau \le j! \left(\frac{5}{r}\right)^j \cdot \frac{\|f\|^j}{j!} \|\varphi\|_r, \quad (257) $$

where $\|f\| := \|f\|_{L^1((0,t);C^{\omega,r})}$, which proves the absolute convergence because the right-hand side is bounded by $2^{-j} \|\varphi\|_r$. Eventually, we deduce from (187) and Lemma 76 that

$$ \left| \varphi(x(t; f, p)) - \sum_{j=0}^{M-1} \int_{\Delta^j(t)} (f(\tau_1) \cdots f(\tau_j))(\varphi)(p) \, d\tau \right| \le 2^{-M} \|\varphi\|_r, \quad (258) $$

which proves (255). □

## 5.2 Magnus expansion in the usual setting

### 5.2.1 Equality for nilpotent systems

The goal of this section is to prove that the Magnus expansion is an exact expansion for regular vector fields generating a nilpotent Lie algebra (see Proposition 110).

If the vector fields are analytic in space, a simple proof can be given (see e.g. [66, Remark A.1] for the case of the CBHD formula), with the following steps. First, by density, one can assume that the dynamic depends analytically on time. Then, the maps $t \mapsto x(t)$ and $t \mapsto e^{Z_M}(t)$ are analytic. Because of the nilpotency assumption, $Z_M = Z_{M'}$ for every $M' \ge M$ and estimate (188) proves that both functions have the same Taylor expansion at $t = 0$, and are thus equal.

For non-analytic vector fields, the proof is much more intricate. A sketch of proof is briefly suggested in [4, Proposition 2.4]. In this paragraph, we write the proof completely. The difficulty is to formulate the question in the nilpotent Lie algebra generated by the vector fields, in order to conclude with the universal property of free nilpotent Lie algebras (Lemma 7).

To that end, we start with the following statement.

**Lemma 109.** Let a be given by (14), $M \in \mathbb{N}^*$, $Z_M(t) := \mathrm{Log}_M\{a\}(t)$ with the notation of Definition 23. Then for every $t \in \mathbb{R}$, the following equality holds in $\mathcal{N}_{M+1}(X)$

$$ \sum_{n=0}^{M-1} \frac{(-1)^n}{(n+1)!} \mathrm{ad}_{Z_M(t)}^n (\dot{Z}_M(t)) = a(t), \quad (259) $$

where $Z_M(t)$ belongs to the space $\bigoplus_{r \in [1, M]} \mathcal{L}(X)^r$ which is identified with $\mathcal{N}_{M+1}(X)$ as a vector space.

*Proof.* The canonical surjection $\sigma_{M+1} : \mathcal{L}(X) \to \mathcal{N}_{M+1}(X)$ is an homomorphism of Lie algebras. Applying this homomorphism to (22), where $z(t) = \mathrm{Log}_\infty\{a\}(t)$ thanks to Theorem 27, proves (259). □
**Proposition 110.** Let $M \in \mathbb{N}^*$. There exists $\eta_M > 0$ such that, for every $T, \delta > 0$ and every time-varying vector field $f: [0, T] \to \mathcal{C}^\infty(B_{4\delta}; \mathbb{K}^d)$ such that $\mathcal{L}(f([0, T]))$ is nilpotent with index at most $M+1$ and $f \in L^1((0, T); \mathcal{C}^M(B_{4\delta}; \mathbb{K}^d))$ with

$$ \|f\|_{L^1((0,T); \mathcal{C}^M)} + \|f\|_{L^1((0,T); \mathcal{C}^M)}^M \le \eta_M \delta, \qquad (260) $$

then, for each $p \in B_\delta$ and $t \in [0, T]$, one has $x(t; f, p) = e^{Z_M(t,f)}(p)$ where $Z_M(t, f) := \text{Log}_M\{f\}(t)$ is the vector field defined in Definition 23.

*Proof.* Let $M \in \mathbb{N}^*$. By Definition 23 and Lemma 79, there exists $\eta_M > 0$ such that,

$$ \|Z_M(t, f)\|_{C^1} \le \frac{1}{\eta_M} \left( \|f\|_{L^1(\mathcal{C}^M)} + \|f\|_{L^1(\mathcal{C}^M)}^M \right). \qquad (261) $$

In particular, for every $p \in B_\delta$, $e^{Z_M(t,f)}(p)$ is well-defined thanks to Eq. (260).

*Step 1: Proof for $f(t, x) = \sum_{j=1}^q a_j(t)f_j(x)$ with $q \in \mathbb{N}^*$, $a_j \in \mathcal{C}^\infty([0, T]; \mathbb{K})$ and $f_j \in \mathcal{C}_c^\infty(\mathbb{K}^d; \mathbb{K}^d)$. By uniqueness in the Cauchy-Lipschitz theorem, it is sufficient to prove that for every $t \in [0, T]$ and $p \in B_\delta$,

$$ \frac{d}{dt}(e^{Z_M(t,f)}(p)) = f(t, e^{Z_M(t,f)}(p)). \qquad (262) $$

By Definition 23, the map $(t, p) \mapsto Z_M(t, f)(p)$ belongs to $\mathcal{C}^\infty([0, T] \times B_{4\delta}; \mathbb{K}^d)$. Thanks to the nilpotency assumption, $\text{ad}_{Z_M(t,f)}^M(Z_M(\tau, f)) = 0$ on $B_{4\delta}$ for every $t, \tau \in [0, T]$. Thus Lemma 91 yields

$$ \frac{d}{dt} (e^{Z_M(t,f)}(p)) = \sum_{k=0}^{M} \frac{(-1)^k}{(k+1)!} \text{ad}_{Z_M(t,f)}^k (\dot{Z}_M(t,f)) (e^{Z_M(t,f)}(p)). \qquad (263) $$

Let $\Lambda: \mathcal{N}_{M+1}(X) \to \mathcal{L}(f_1, \dots, f_q)$ be the homomorphism of nilpotent Lie algebras such that $\Lambda(X_j) = f_j$ for $j = 1, \dots, q$. By applying $\Lambda$ to the equality (259), we obtain that the right-hand side of the above equality is $f(t, e^{Z_M(t,f)}(p))$.

*Step 2: Proof for a general time-dependent vector field $f$. We apply Step 1 to a sequence $f_n$ of simple functions, taking values in $f([0, T])$, converging towards $f$ in $L^1((0, T); \mathcal{C}^M(B_{4\delta}; \mathbb{K}^d))$. We get the conclusion by passing to the limit in both sides, using the fourth item of Lemma 82. $\square$*

### 5.2.2 Convergence for linear systems

In this paragraph, we consider linear systems of the form (251). Since the Magnus expansion was designed for linear systems, its convergence in this context has received much attention. Depending on the exact convergence notion that one considers and on the way one groups terms, different sufficient conditions for the convergence can be derived. In [97], $T \|A\|_{L^\infty(0,T)} \le 1$ is shown to be a sufficient condition for convergence on $[0, T]$ thanks to a careful estimate of the combinatorial terms. In [83], $\|A\|_{L^1(0,T)} < \pi$ is shown to be a sufficient condition for convergence using complex analysis.

We give below a short elementary proof with a sub-optimal constant, for the sake of completeness and because it will be used in the sequel. Let $\| \cdot \|$ be a sub-multiplicative norm on $\mathcal{M}_d(\mathbb{K})$.

**Proposition 111.** Let $T > 0$ and $A \in L^1((0, T); \mathcal{M}_d(\mathbb{K}))$ such that $\|A\|_{L^1(0,T)} < \frac{1}{4}$. For each $t \in [0, T]$,

$$ Z_\infty(t) := \sum_{r=1}^{+\infty} \frac{1}{r} \sum_{m=1}^{r} \frac{(-1)^{m-1}}{m} \sum_{\mathbf{r} \in \mathbb{N}_m^d} \int_{\Delta^r(t)} [\dots[A(\tau_1), A(\tau_2)], \dots A(\tau_r)] d\tau \qquad (264) $$

is well defined in $\mathcal{M}_d(\mathbb{K})$ and, for every $p \in \mathbb{K}^d$, $x(t; A, p) = e^{-Z_\infty(t)}p$, where the brackets refer to commutators of matrices, i.e. $[A, B] = AB - BA$.
Proof. Step 1: Absolute convergence of $Z_\infty(t)$. Let $r \in \mathbb{N}^*$. For every $m \in [1, r]$ and $\mathbf{r} \in \mathbb{N}_r^m$,

$$ \begin{aligned} & \int_{\Delta^r(t)} \|[\cdots[A(\tau_1), A(\tau_2)], \dots, A(\tau_r)]\| \, d\tau \\ & \le \int_{\Delta^r(t)} 2^r \|A(\tau_1)\| \dots \|A(\tau_r)\| \, d\tau \le 2^r \left( \int_0^t \|A(\tau)\| \, d\tau \right)^r. \end{aligned} \quad (265) $$

Moreover, recalling the definition of (31), $|\mathbb{N}_r^m| = \binom{r-1}{m-1}$ and $\sum_{m=1}^r \binom{r-1}{m-1} = 2^{r-1}$. Thus,

$$ \sum_{r=1}^{+\infty} \frac{1}{r} \sum_{m=1}^{r} \frac{1}{m} \sum_{\mathbf{r} \in \mathbb{N}_r^m} \int_{\Delta^r(t)} \|[\cdots[A(\tau_1), A(\tau_2)], \dots, A(\tau_r)]\| \, d\tau \le \sum_{r=1}^{+\infty} (4\|A\|_{L^1})^r < \infty. \quad (266) $$

Step 2: Formula for the solution $L \in C^1([0, t]; \mathcal{M}_d(\mathbb{K}))$ of

$$ \left\{ \begin{array}{l} L'(\tau) = L(\tau)A(\tau) \\ L(0) = \mathrm{Id}_d. \end{array} \right. \quad (267) $$

By working as in the proof of Proposition 107, we obtain

$$ L(t) = \mathrm{Id}_d + \sum_{r=1}^{+\infty} \int_{\Delta^r(t)} A(\tau_1) \cdots A(\tau_r) \, d\tau \quad (268) $$

where the series converges absolutely. Moreover, we have

$$ \left\| \sum_{r=1}^{+\infty} \int_{\Delta^r(t)} A(\tau_1) \cdots A(\tau_r) \, d\tau \right\| \le \sum_{r=1}^{+\infty} \frac{\|A\|_{L^1}^r}{r!} < e^{\frac{1}{4}} - 1 < 1. \quad (269) $$

Thus

$$ \log(L(t)) = \sum_{m=1}^{+\infty} \frac{(-1)^m}{m} \left( \sum_{r=1}^{+\infty} \int_{\Delta^r(t)} A(\tau_1) \cdots A(\tau_r) \, d\tau \right)^m \quad (270) $$

is well defined in $\mathcal{M}_d(\mathbb{K})$ and $L(t) = e^{\log(L(t))}$. By applying Corollary 95 with $A = A_1 = \mathcal{M}_d(\mathbb{K})$, we get $\log(L(t)) = Z_\infty(t)$.

Step 3: Conclusion. The resolvent $R(\tau)$ associated to the linear system $\dot{x} = A(\tau)x$ with initial condition at $\tau = 0$ is $R(\tau) = L(\tau)^{-1}$. Thus $x(t) = R(t)p = e^{-Z_\infty(t)}p$. $\square$

**Remark 112.** For $X, Y \in \mathcal{M}_d(\mathbb{K})$ such that $\|X\| + \|Y\| < \frac{1}{8}$, the previous statement implies the convergence of the CBHD formula, yielding a matrix $Z_\infty$ such that $e^X e^Y = e^{Z_\infty}$. Some authors have investigated the optimal convergence domain in different contexts for the CBHD formula. Such a domain sometimes depends on the summation process (i.e. the way terms are grouped together) and the exact question one asks (existence of a logarithm, absolute summability of the series, convergence of the remainder, etc.). Better sufficient conditions than ours can be found for instance in [18], for instance, $\|X\| + \|Y\| < \frac{\ln 2}{2}$. We refer to [16] for a nice survey of the convergence questions regarding the CBHD formula.

**Remark 113.** The smallness assumption (on time or on the matrices) is in general necessary, both for the CBHD formula (see [16, Example 2.3] or [103, Section II]) and for the Magnus expansion (see [83], where the authors also prove that, although the condition $\|A\|_{L^1(0,T)} < \pi$ is not necessary for convergence, there exists $A$ with $\|A\|_{L^1(0,T)} = \pi$ for which the Magnus series at time $\pi$ does not converge).
### 5.2.3 Divergence for arbitrarily small analytic vector fields

The convergence of Magnus expansions is deeply linked with the convergence of the CBHD series. For analytic vector fields, it is expected that both series diverge (see e.g. [4, p.1671] or [97, p.335] for statements without examples). Some authors nevertheless suggested that, despite the divergence of the series, the flows could converge for analytic vector fields (see [97, p.335] and [71, p.241]).

In this paragraph, we give explicit counter-examples to the convergence, even in the weak sense of the flows, for arbitrarily small analytic vector fields, of both the CBHD series and the Magnus expansion. Similarly to counter-examples concerning the convergence of the CBHD series for large matrices (see e.g. [16, Theorem 2.5]), our construction relies on the choice of generators for which many brackets vanish thanks to their particular structure, and the remaining non-vanishing brackets are associated with coordinates of the first kind involving Bernoulli numbers.

**Proposition 114.** There exists $\delta > 0$ and $f_0, f_1 \in C^\omega(B_\delta; \mathbb{R}^2)$ such that,

$$ \forall M \in \mathbb{N}, \exists C_M, \varepsilon_M > 0, \forall \varepsilon \in [0, \varepsilon_M], \quad \left| e^{\varepsilon f_0} e^{\varepsilon f_1}(0) - e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(0) \right| \le C_M \varepsilon^{M+1}, \quad (271) $$

where $\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)$ is defined in Corollary 94, but, simultaneously, for every $\varepsilon > 0$,

$$ \lim_{M \to +\infty} |\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)(0)| = +\infty \quad (272) $$

and

$$ \lim_{M \to +\infty} \left| e^{\varepsilon f_0} e^{\varepsilon f_1}(0) - e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(0) \right| = +\infty. \quad (273) $$

*Proof.* Let $f_0, f_1$ as in Remark 81. For these vector fields, estimate (271) comes from Corollary 94. Due to their structure, the only non vanishing brackets are those containing $f_1$ at most once. Therefore, formula (69) of Corollary 39 yields, for $M \ge 1$,

$$ \text{CBHD}_M(\varepsilon f_1, \varepsilon f_0) = \varepsilon f_0 + \sum_{k=0}^{M-1} \frac{B_k}{k!} \varepsilon^{k+1} \text{ad}_{f_0}^k(f_1). \quad (274) $$

Hence, using (142),

$$ \text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)(x) = \varepsilon e_1 + \varepsilon \Theta_M^\varepsilon(x_1)e_2, \quad (275) $$

where we introduce, for $q \in \mathbb{R}$,

$$ \Theta_M^\varepsilon(q) := \sum_{k=0}^{M-1} B_k \varepsilon^k (1-q)^{-k-1}. \quad (276) $$

In particular,

$$ |\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)(0)| \ge |\varepsilon \Theta_M^\varepsilon(0)|. \quad (277) $$

Since the odd Bernoulli numbers except $B_1$ are zero, when $M = 2M' + 2$ with $M' \ge 1$, $\Theta_{2M'+2}^\varepsilon = \Theta_{2M'+1}^\varepsilon$. Then,

$$ \Theta_{2M'+1}^{\varepsilon}(q) = \frac{1}{1-q} - \frac{\varepsilon}{2(1-q)^2} + \sum_{k=1}^{M'} B_{2k} \varepsilon^{2k} (1-q)^{-2k-1}. \quad (278) $$

In particular, using (531),

$$ \Theta_{2M'+1}^{\varepsilon}(0) = 1 - \frac{\varepsilon}{2} + \sum_{k=1}^{M'} B_{2k} \varepsilon^{2k} = 1 - \frac{\varepsilon}{2} + \sum_{k=1}^{M'} (-1)^{k+1} \frac{2(2k)!}{(2\pi)^{2k}} \zeta(2k) \varepsilon^{2k}. \quad (279) $$

Thus, for every fixed $\varepsilon > 0$, $|\Theta_M^\varepsilon(0)| \to +\infty$ when $M \to +\infty$, because it involves a sum of the form $\sum_{k=1}^{M'} a_k$ where $|a_{k+1}|/|a_k| \to +\infty$ when $k \to +\infty$. Using (277), this proves (272).
For $p \in \mathbb{R}^2$ close enough to the origin, one can also compute the flow $e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(p)$, which is $y(1)$ where $y$ is the solution to the ODE $y(0) = p$ and

$$ \dot{y}_1(s) = \varepsilon \quad \text{and} \quad \dot{y}_2(s) = \varepsilon \Theta_M^\varepsilon(y_1(s)). \qquad (280) $$

Solving successively for $y_1$ then $y_2$ yields $y_1(s) = p_1 + s\varepsilon$ and

$$ y_2(s) = p_2 + \int_{y_1(0)}^{y_1(s)} \Theta_M^\varepsilon(h) \, dh. \qquad (281) $$

Thus,

$$ e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(p) = (p_1 + \varepsilon)e_1 + \left( p_2 + \int_{p_1}^{p_1+\varepsilon} \Theta_M^\varepsilon(h) \, dh \right) e_2. \qquad (282) $$

In particular,

$$ e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(0) = \varepsilon e_1 + \left( \int_0^\varepsilon \Theta_M^\varepsilon(h) \, dh \right) e_2. \qquad (283) $$

When $M = 2M' + 2$ with $M' \ge 1$, using (278), we get

$$ e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(0) = \varepsilon e_1 + \left( -\ln(1-\varepsilon) - \frac{\varepsilon}{2} \left( \frac{1}{1-\varepsilon} - 1 \right) + \sum_{k=1}^{M'} \frac{B_{2k}}{2k} \varepsilon^{2k} \left( \frac{1}{(1-\varepsilon)^{2k}} - 1 \right) \right) e_2. \qquad (284) $$

Hence, for the same reason as above, the flow satisfies $|e^{\text{CBHD}_M(\varepsilon f_1, \varepsilon f_0)}(0)| \to +\infty$ when $M \to +\infty$, which proves (273). $\square$

**Remark 115.** If one sees $(x_1, x_2)$ as $(q, p)$ in an Hamiltonian setting, one checks that the vector fields defined in (141) and used in this counter-example are associated with the Hamiltonians $\mathcal{H}_0(q, p) := p$ and $\mathcal{H}_1(q, p) := \ln(1-q)$. Therefore, assuming an Hamiltonian structure on the considered vector fields does not provide enough structure to yield convergence.

One could wonder if assuming even more structure on the dynamics, for example assuming that it is time-reversible, prevents the construction of such counter-examples.

**Open problem 116.** Do there exist Hamiltonians $\mathcal{H}_0$ and $\mathcal{H}_1$ on $\mathbb{R}^{2d}$, which are time-reversible (i.e. satisfy $\mathcal{H}_i(q, p) = \mathcal{H}_i(q, -p)$ for every $q, p \in \mathbb{R}^d$), locally real-analytic near zero and for which the convergence of the CBHD series fails as in Proposition 114?

The counter-example of Proposition 114 for the convergence of the CBHD series allows to build counter-examples to the convergence of the Magnus expansion which blow up instantly, despite analytic regularity in both time and space.

**Proposition 117.** There exist $T, \delta > 0$ and $f \in C^\omega([0, T] \times B_\delta; \mathbb{R}^2)$ such that, for every $\varepsilon > 0$ and $t \in (0, T]$,

$$ \lim_{M \to +\infty} |Z_M(t, \varepsilon f)(0)| = +\infty \qquad (285) $$

and

$$ \lim_{M \to +\infty} \left| x(t; \varepsilon f, 0) - e^{Z_M(t, \varepsilon f)}(0) \right| = +\infty, \qquad (286) $$

where $x$ is the solution to $\dot{x}(t) = \varepsilon f(t, x(t))$ with $x(0) = 0$ and $Z_M(t, \varepsilon f) = \text{Log}_M\{\varepsilon f\}(t)$.

*Proof.* Let $T = 1$. We define $f(t, x) := f_0(x) + tf_1(x)$, where $f_0$ and $f_1$ are defined in Remark 81. Similarly as for the previous construction, only Lie brackets involving $f_1$ at most once are non-vanishing. Moreover, the coordinates of the first kind associated with the controls $a_0(t) = 1$ and $a_1(t) = t$ have been computed in Example 40. Hence, recalling (142), we have

$$ Z_M(t, \varepsilon f) = \varepsilon t e_1 + \sum_{k=0}^{M-1} \varepsilon^{k+1} \frac{(-1)^{k+1} t^{k+2}}{(k+1)!} B_{k+1} \frac{k!}{(1-x_1)^{k+1}} e_2. \qquad (287) $$
Proceeding along the same lines as in the proof of Proposition 114 allows to conclude that both $Z_M(t, \varepsilon f)(0)$ and $e^{Z_M(t, \varepsilon)}(0)$ diverge when $M \to +\infty$. $\square$

# 5.3 Magnus expansion in the interaction picture

## 5.3.1 Nilpotent systems

For ODEs of the form (209), the starting point of the interaction picture is to factorize the flow of $f_0$. Hence, the roles of $f_0$ and $f_1$ are asymmetric. One can expect that, under the assumption that Lie brackets of $f_0$ and $f_1$ containing at least $M + 1$ times $f_1$ identically vanish, the Magnus expansion in the interaction picture should yield an equality of the form

$$x(t; f_0 + f_1, p) = e^{Z_M(t, f_0, f_1)} e^{tf_0} p, \qquad (288)$$

where $Z_M(t, f_0, f_1)$ is defined in Proposition 99. We prove in this paragraph that it is indeed the case, when $f_0$ and $f_1$ are analytic. However, contrary to the case of the usual Magnus expansion (see Section 5.2.1), we give examples highlighting the fact that the analyticity assumption cannot be removed, which is quite surprising but stems from the mixing induced by pushforwards.

We therefore start with the following definition.

**Definition 118** (Semi-nilpotent family of vector fields). Let $\Omega$ be an open subset of $\mathbb{K}^d$. Let $\mathcal{F} \subset C^\infty(\Omega; \mathbb{K}^d)$, $f_0 \in C^\infty(\Omega; \mathbb{K}^d)$ and $M \in \mathbb{N}^*$. We say that the family of vector fields $\mathcal{F}$ is semi-nilpotent of index $M$ with respect to $f_0$ if every bracket of elements of $\mathcal{F} \cup \{f_0\}$ involving $M$ elements of $\mathcal{F}$ vanishes identically on $\Omega$ and $M$ is the smallest positive integer for which this property holds.

**Remark 119.** Some authors (see e.g. [62, Section 3]) refer to this situation by saying that $\mathcal{L}(S_1)$ is nilpotent of index $M$, where $S_1 := \{\mathrm{ad}_{f_0}^k(f); f \in \mathcal{F}, k \in \mathbb{N}\}$. Both definitions are equivalent, thanks to the Jacobi identity for Lie brackets.

**Proposition 120.** Let $T, \delta > 0$. Let $M \in \mathbb{N}$. Let $f_0 \in C^\infty(B_{4\delta}; \mathbb{K}^d)$ with $T\|f_0\|_{C^0} \le \delta$. There exists $\eta > 0$ such that, for every $f_1: [0, T] \to C^\infty(B_{4\delta}; \mathbb{K}^d)$ with $f_1 \in L^1([0, T]; C^{M+1}(B_{4\delta}; \mathbb{K}^d))$ and $\|f_1\|_{L^1(C^M)} \le \eta$, the following family is well-defined

$$\mathcal{G} := \{\Phi_0(-t)_* f_1(t); t \in [0, T]\} \subset C^\infty(B_\delta; \mathbb{K}^d). \qquad (289)$$

and, assuming moreover that $\mathcal{G}$ is nilpotent of index $M + 1$, then, for each $t \in [0, T]$ and $p \in B_\delta$, the solution to (209) satisfies (288).

*Proof.* Let $t > 0$. As in the proof of Proposition 99, we introduce the new variable $y(s) := \Phi_0(t-s, x(s))$. Then $\dot{y}(s) = g_t(s, y(s))$, where $g_t$ is defined in (210). Thanks to Lemma 88, $g_t(s) = \Phi_0(t)_* \Phi_0(-s)_* f_1(s)$. Thanks to the assumption and to Lemma 89, the family $\{g_t(s); s \in [0, t]\}$ is nilpotent of index $M + 1$. Thus, by Proposition 110, $y(t) = e^{Z_M(t, f_0, f_1)} y(0)$. Since $x(t) = y(t)$ and $y(0) = \Phi_0(t, p)$, this concludes the proof of (288). $\square$

**Lemma 121.** Let $T, \delta > 0$, $\mathcal{F} \subset C^\infty(B_{4\delta}; \mathbb{K}^d)$, $f_0 \in C^\infty(B_{4\delta}; \mathbb{K}^d)$ such that $T\|f_0\|_{C^0} \le \delta$. The following family is well-defined

$$\mathcal{G} := \{\Phi_0(-t)_* f; t \in [0, T], f \in \mathcal{F}\} \subset C^\infty(B_\delta; \mathbb{K}^d). \qquad (290)$$

Assume that the family $\mathcal{F}$ is semi-nilpotent of index $M$ with respect to $f_0$ and that there exists $r > 0$ such that $\mathcal{F} \cup \{f_0\} \subset C^{\omega, r}(B_\delta; \mathbb{K}^d)$. Then $\mathcal{G}$ is nilpotent of index $M$.

*Proof.* For $t \in [0, T]$ and $f \in \mathcal{F}$, equation (164) of Lemma 90 implies that

$$\Phi_0(-t)_* f = \sum_{k=0}^{+\infty} \frac{t^k}{k!} \mathrm{ad}_{f_0}^k(f) \qquad (291)$$
and that the series converges absolutely in $C^M(B_\delta; \mathbb{K}^d)$ (in particular). Hence, if $t_1, \dots, t_M \in [0, T]$ and $f_1, \dots, f_M \in \mathcal{F}$, the bracket

$$[\Phi_0(-t_M)_*f_M, [\dots[\Phi_0(-t_2)_*f_2, \Phi_0(-t_1)_*f_1]\dots]] = \sum_{k_1, \dots, k_M \in \mathbb{N}} \frac{t_1^{k_1} \dots t_M^{k_M}}{k_1! \dots k_M!} [\mathrm{ad}_{f_0}^{k_M}(f_M), [\dots[\mathrm{ad}_{f_0}^{k_2}(f_2), \mathrm{ad}_{f_0}^{k_1}(f_1)]\dots]] \quad (292)$$

vanishes thanks to the assumption and the absolute convergence of the sums. The same is true for every other bracket structure, which proves that $\mathcal{G}$ is nilpotent of index $M$. $\square$

**Corollary 122.** Let $T, \delta, r > 0$. Let $f_0 \in C^{\omega,r}(B_{4\delta}; \mathbb{K}^d)$ such that $T\|f_0\|_{C^0} \le \delta$ and $f_1 \in L^1([0, T]; C^{\omega,r}(B_{4\delta}; \mathbb{K}^d))$. Assume moreover that $\mathcal{F} := \{f_1(t, \cdot); t \in [0, T]\}$ is semi-nilpotent of index $M+1$ with respect to $f_0$. Then, for each $t \in [0, T]$ and $p \in B_\delta$, the solution to (209) satisfies (288), where $\mathcal{Z}_M(t, f_0, f_1)$ is defined in Proposition 99.

*Proof.* This corollary is a direct consequence of Proposition 120 and Lemma 121. $\square$

The analyticity assumption in Lemma 121 is necessary, as illustrated by the following counter-example for smooth functions.

**Example 123.** We consider smooth vector fields on $\mathbb{R}^3$. Let $\chi \in C^\infty(\mathbb{R}; \mathbb{R})$ with $\chi \equiv 0$ on $\mathbb{R}_-$ and $\chi(x) > 0$ for $x > 0$. Let $f_0$ and $\mathcal{F} := \{f_1, f_2\}$ where

$$f_0(x) := e_2, \quad (293)$$

$$f_1(x) := \chi(x_2)x_1e_3, \quad (294)$$

$$f_2(x) := \chi(-x_2)e_1. \quad (295)$$

Heuristically, $f_1$ and $f_2$ commute because they have disjoint (touching) supports, but the flow of $f_0$ involved in (290) mixes these supports for every positive time. This is possible only because $\chi$ is not analytic.

First, we check that $\mathcal{F}$ is semi-nilpotent of order 2 with respect to $f_0$. Indeed, for every $j \in \mathbb{N}$,

$$\mathrm{ad}_{f_0}^j(f_1)(x) = \chi^{(j)}(x_2)x_1e_3, \quad (296)$$

$$\mathrm{ad}_{f_0}^j(f_2)(x) = (-1)^j \chi^{(j)}(-x_2)e_1. \quad (297)$$

Thus, for $j, k \in \mathbb{N}$, $[\mathrm{ad}_{f_0}^j(f_1), \mathrm{ad}_{f_0}^k(f_1)]$ (resp. $[\mathrm{ad}_{f_0}^j(f_2), \mathrm{ad}_{f_0}^k(f_2)]$) vanishes because both vector fields are multiples of $e_3$ but independent of $x_3$ (resp. multiples of $e_1$ but independent of $x_1$). Moreover,

$$[\mathrm{ad}_{f_0}^j(f_1), \mathrm{ad}_{f_0}^k(f_2)](x) = -(-1)^k \chi^{(k)}(-x_2)\chi^{(j)}(x_2)e_3 = 0, \quad (298)$$

because the supports of $\chi(\cdot)$ and $\chi(-\cdot)$ only touch at $x_2 = 0$ where all derivatives vanish.

Second, let us check however that the family $\mathcal{G}$ defined in (290) is not nilpotent of index 2. Indeed, for $t \ge 0$ and $x \in \mathbb{R}^3$, $\Phi_0(t)(x) = x + te_2$. Thus, for $f \in C^\infty(\mathbb{R}^3; \mathbb{R}^3)$, $(\Phi_0(-t)_*f)(x) = f(x + te_2)$. Therefore, for every $T > 0$, $\mathcal{G}$ is well-defined on $\mathbb{R}^3$. Moreover,

$$[f_2, (\Phi_0(-t)_*f_1)](x) = \chi(-x_2)\chi(x_2 + t)e_3. \quad (299)$$

In particular, for every $\varepsilon > 0$, $[f_2, (\Phi_0(-2\varepsilon)_*f_1)](-\varepsilon e_2) = \chi(\varepsilon)^2 e_3 \ne 0$, which prevents the family $\mathcal{G}$ from being nilpotent of index 2 (even locally in time and space).

The analyticity assumption in Corollary 122 is also necessary, as illustrated by the following counter-example for smooth functions, inspired by the previous one.