# Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schrödinger equation

Théo Gherdaoui

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Théo Gherdaoui. Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schrödinger equation. 2025. hal-04957550v2

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# Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schrödinger equation

Théo Gherdaoui*

March 22, 2025

## Abstract

We investigate the small-time local controllability (STLC) near the ground state of a bilinear Schrödinger equation when the linearized system is not controllable. It is well known that, for single-input systems, quadratic terms in the state expansion can then lead to obstructions to the STLC of the nonlinear system. In this work, we extend this phenomenon to the multi-input setting, presenting the first example of multi-input quadratic obstructions for PDEs. Our results build upon our previous study of such obstructions for ODEs and provide a functional framework for analyzing them in the bilinear Schrödinger equation.

**Keywords:** Bilinear Schrödinger equation, quadratic obstructions, small-time local exact controllability, power series expansion.

# 1 Introduction

## 1.1 Model and problem

Let $r \in \mathbb{N}^*$. In this article, we study the following Schrödinger equation

$$ egin{cases} i\partial_t\psi(t,x) = -\partial_x^2\psi(t,x) - u(t) \cdot \mu(x)\psi(t,x), & t \in (0,T), x \in (0,1), \\ \psi(t,0) = \psi(t,1) = 0, & t \in (0,T), \\ \psi(0,x) = \psi_0(x), & x \in (0,1), \end{cases} \quad (1.1) $$

where $\mu: (0,1) \to \mathbb{R}^r, T > 0, u: (0,T) \to \mathbb{R}^r, \psi: (0,T) \times (0,1) \to \mathbb{C}, \|\psi_0\|_{L^2(0,1)} = 1$, and $(x,y) \in \mathbb{R}^r \mapsto x \cdot y \in \mathbb{R}$ is the canonical scalar product on $\mathbb{R}^r$. We will use the notations $u := (u^1, \dots, u^r)$ and $\mu := (\mu_1, \dots, \mu_r)$ in all the document. When well defined, the solution is denoted $\psi$. When required, we will write $\psi(\cdot; u, \psi_0)$ to refer to this solution to emphasize its dependence on the different parameters.

This equation describes the evolution of the wave function $\psi$ of a quantum particle in a 1D infinite square potential well $(0,1)$, subjected to $r$ electric fields with amplitudes $u^\ell(t)$. The functions $\mu_\ell$, called « dipolar moments », model the interaction between the particle's wave function $\psi$ and the electric fields $u^\ell(t)$.

This is a multi-input nonlinear control system:

- the state is the wave function $\psi: (0,T) \to S$, where $S$ denotes the $L^2(0,1)$ sphere,

- the controls are $u^\ell: (0,T) \to \mathbb{R}$, they act bilinearly on the state.

The fundamental state or ground state is the particular free – with $u \equiv 0$ – trajectory $(t, x) \mapsto \sqrt{2}\sin(\pi x)e^{-i\pi^2 t}$. We are interested in the small-time local controllability around the ground state.

<footnote>*Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France</footnote>
## 1.2 Notations and definitions

Let $T > 0$ and $k \in \mathbb{N}$. We consider the Sobolev space $H^k((0, T), \mathbb{R})$, equipped with its usual norm and $H_0^k((0, T), \mathbb{R})$ the adherence of $C_c^\infty((0, T), \mathbb{R})$ for this norm.

Except explicit precision, we will work with complex valued space-dependent functions. We consider the space $L^2(0, 1)$, endowed with the hermitian scalar product $\langle \cdot, \cdot \rangle$. We define the operator

$$A := -\partial_x^2 \text{ with domain } D(A) := H^2(0, 1) \cap H_0^1(0, 1). \qquad (1.2)$$

The eigenvalues and eigenvectors of A are

$$\lambda_j := (j\pi)^2, \quad \varphi_j := \sqrt{2}\sin(j\pi\cdot), \quad j \ge 1. \qquad (1.3)$$

The family $(\varphi_j)_{j \ge 1}$ is an orthonormal basis of $L^2(0, 1)$. We denote by

$$\psi_j(t, x) := \varphi_j(x)e^{-i\lambda_j t}, \quad t \in (0, T), x \in (0, 1), \quad j \ge 1,$$

the solutions to the free – i.e. with controls $u \equiv 0$ – Schrödinger equation (1.1) with initial data $\varphi_j$ at time $t = 0$ i.e. $\psi(\cdot; u, \varphi_j)$. The solution $\psi_1$ is called the fundamental state, or ground state. We finally define the spaces $H_{(0)}^s(0, 1) := D(A^{\frac{s}{2}})$ for $s \ge 0$, equipped with the norm

$$\|\varphi\|_{H_{(0)}^s} := \left( \sum_{j=1}^{+\infty} |j^s \langle \varphi, \varphi_j \rangle|^2 \right)^{1/2}.$$

With these notations, the equation (1.1) is well-posed, in the sense of the following proposition, proved for the single-input case in [6, Proposition 2] and which can be adapted to our case.

**Proposition 1.1** (Well-posedness). Let $T > 0$, $\mu \in H^3((0, 1), \mathbb{R})^r$, $u \in L^2((0, T), \mathbb{R})^r$ and $\psi_0 \in H_{(0)}^3(0, 1)$. There exists a unique mild solution $\psi \in C^0([0, T], H_{(0)}^3(0, 1))$ to the equation (1.1).

We can now define the notion of small-time local controllability we are working on.

**Definition 1.2.** Let $E_T$ be a family of normed vector spaces of functions defined on $[0, T]$ for $T > 0$. The bilinear Schrödinger equation (1.1) is E-STLC in $H_{(0)}^3(0, 1)$ around the ground state if for every $T, \varepsilon > 0$, there exists $\delta > 0$ such that for every $\psi_f \in H_{(0)}^3(0, 1) \cap S$ with $\|\psi_f - \psi_1(T)\|_{H^3} < \delta$, there exists $u \in (E_T)^r \cap L^2((0, T), \mathbb{R})^r$ such that $\|u\|_{E_T} \le \varepsilon$ and $\psi(T; u, \varphi_1) = \psi_f$.

Under appropriate assumptions on the dipolar moments $\mu_\ell$, the STLC around the ground state of (1.1) can be proved by applying the linear test. This is the purpose of the following statement, that can be proved by adapting with several controls the strategy developed in [6, 16] with one control.

**Proposition 1.3.** Let $\mu \in H^3((0, 1), \mathbb{R})^r$ be such that

$$\exists C > 0, \forall j \in \mathbb{N}^*, \sum_{\ell=1}^{r} \left| \langle \mu_{\ell} \varphi_1, \varphi_j \rangle \right| \ge \frac{C}{j^3}. \qquad (1.4)$$

Then, the bilinear Schrödinger equation (1.1) is $L^2$-STLC in $H_{(0)}^3(0, 1)$ around the ground state.

*Idea of proof for Proposition 1.3.* Assumption (1.4) guarantees that the linearized system around the ground state – see (2.2) – is small-time globally controllable with states in $H_{(0)}^3(0, 1) \cap S$ and controls in $L^2$. Moreover, one can prove that the following end-point map is of class $C^1$

$$\Theta_T : u \in L^2((0, T), \mathbb{R})^r \mapsto \psi(T; u, \varphi_1) \in H_{(0)}^3(0, 1) \cap S.$$

By applying the inverse mapping theorem, we obtain the $L^2$-STLC of the nonlinear system (1.1). □
## 1.3 Main result

For all the document, $k, K, r \in \mathbb{N}^*$ are fixed integers. To ensure that equation (1.1) is well-posed in $H_{(0)}^3(0, 1)$ – see Proposition 1.1 – we assume that

$$ (\mathbf{H})_{\text{reg}} : \mu \in H^3((0, 1), \mathbb{R})^r. $$

In this article, we study the case where the linearized system around the ground state – see (2.2) – is not controllable, so we consider an integer $K \ge 1$ and we assume

$$ (\mathbf{H})_{\text{lin}} : \langle \mu_{\ell} \varphi_1, \varphi_K \rangle = 0, \quad 1 \le \ell \le r. $$

In this case, the linearized system misses one direction $\langle \psi(t; u, \varphi_1), \varphi_K \rangle \in \mathbb{C}$, i.e. the first order term in the Taylor expansion at 0 of the map $u \mapsto \langle \psi(t; u, \varphi_1), \varphi_K \rangle$ vanishes. To prove an obstruction to STLC, we will use a « power series expansion » of order 2, i.e. the second order term in the Taylor expansion at 0 of $u \mapsto \langle \psi(t; u, \varphi_1), \varphi_K \rangle$. We make assumptions about the quadratic expansion of the solution. To state them, we need to introduce the following definitions.

**Definition 1.4.** For all $1 \le \ell, L \le r$, we define the following sequences

$$ c^{\ell,L} := \left(c_j^{\ell,L}\right)_{j \ge 1} := \left(\langle \mu_{\ell} \varphi_K, \varphi_j \rangle \langle \mu_L \varphi_1, \varphi_j \rangle\right)_{j \ge 1}. $$

We will denote $c^{\ell} := c^{\ell, \ell}$ and $c_j^{\ell} := c_j^{\ell, \ell}$ for $j \ge 1$ and $1 \le \ell \le r$.

Quadratic hypotheses are formulated as properties on series. To ensure convergence, we assume the following

$$ (\mathbf{H})_{\text{conv}} : \sum_{j=1}^{+\infty} |c_j^{\ell,L}| j^{4k} < +\infty, \quad 1 \le \ell, L \le r. $$

**Remark 1.5.** By Cauchy-Schwarz's inequality, $(\mathbf{H})_{\text{conv}}$ holds when $\mu_{\ell}\varphi_K$ and $\mu_L\varphi_1$ belong to $H_{(0)}^{2k}(0, 1)$, for every $1 \le \ell, L \le r$. In particular, this is the case when $\mu \in H^{2k}(0, 1)^r$ with vanishing odd derivatives, or $\mu \in C_c^{\infty}(0, 1)^r$.

**Definition 1.6 (Quadratic brackets).** Assuming $(\mathbf{H})_{\text{conv}}$, we define the following series which converge,

$$ \gamma_p^{\ell,L} := \sum_{j=1}^{+\infty} \left( (\lambda_K - \lambda_j)^{\lfloor \frac{p+1}{2} \rfloor} (\lambda_j - \lambda_1)^{\lfloor \frac{p}{2} \rfloor} c_j^{\ell,L} - (\lambda_K - \lambda_j)^{\lfloor \frac{p}{2} \rfloor} (\lambda_j - \lambda_1)^{\lfloor \frac{p+1}{2} \rfloor} c_j^{L,\ell} \right), $$

for $0 \le p \le 2k-1$ and $1 \le \ell \le L \le r$. When $\ell = L$, we will note $\gamma_p^{\ell}$ instead of $\gamma_p^{\ell,\ell}$.

**Remark 1.7.** For every $1 \le \ell \le r$, $0 \le p \le k-1$, one has $\gamma_{2p}^{\ell} = 0$

We are now able to state the hypotheses concerning the quadratic terms

$$ (\mathbf{H})_{\text{null}} : \quad \forall 1 \le \ell \le L \le r, \forall 0 \le p \le 2k-2, \quad \gamma_p^{\ell,L} = 0, $$

$$ (\mathbf{H})_{\text{pos}} : \quad q : (a_1, \dots, a_r) \in \mathbb{R}^r \mapsto \sum_{\ell=1}^{r} \gamma_{2k-1}^{\ell} \frac{a_{\ell}^2}{2} + \sum_{1 \le \ell < L \le r} \gamma_{2k-1}^{\ell,L} a_{\ell} a_L \in \mathbb{R} $$

is a definite quadratic form on $\mathbb{R}^r$.

**Remark 1.8.** If $(\mathbf{H})_{\text{pos}}$ holds, $\text{sgn}(\gamma_{2k-1}^1)q$ is a positive-definite quadratic form on $\mathbb{R}^r$.

**Remark 1.9.** When $r = 2$, $(\mathbf{H})_{\text{pos}}$ holds iff $(\gamma_{2k-1}^{1,2})^2 < \gamma_{2k-1}^1 \gamma_{2k-1}^2$.

At first glance, these assumptions may seem mysterious and technical, but they can be interpreted in terms of Lie brackets as detailed in Section 1.5. The main result of this article is the following theorem
**Theorem 1.10.** Let $k, K, r \in \mathbb{N}^*$, $\mu$ satisfying $(\mathbf{H})_{\text{reg}}$, $(\mathbf{H})_{\text{lin}}$, $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$. Then, if $k \ge 2$ (resp. if $k = 1$), the multi-input bilinear Schrödinger equation (1.1) is not $H^{2k-3}$-STLC (resp. $W^{-1,\infty}$-STLC) in $H_{(0)}^3(0,1)$ around the ground state.

Ideas for proving the existence of such functions $\mu_1, \dots, \mu_r$ are given in Appendix A.1.

**Remark 1.11.** When $r = 1$, hypothesis $(\mathbf{H})_{\text{pos}}$ is equivalent to $\gamma_{2k-1}^1 \neq 0$. We then obtain [17, Theorem 1.3].

## 1.4 Heuristic

To simplify the notations, we will use the following definition.

**Definition 1.12.** For $T > 0$ and $f \in L^1((0,T), \mathbb{R})$, one defines the iterated primitives $f_n$ vanishing at 0 by induction as

$$f_0 := f \quad \text{and} \quad \forall n \in \mathbb{N}, t \in [0, T], \quad f_{n+1}(t) := \int_0^t f_n(\tau) d\tau. \qquad (1.5)$$

When $f = (f^1, \dots, f^r) \in L^1((0,T), \mathbb{R})^r$, $f_n$ will denote $(f_n^1, \dots, f_n^r)$.

The essence of Theorem 1.10 can be found in the following example of a finite-dimensional control-affine system

$$\left\{
\begin{aligned}
x_1' &= u^1 \\
x_2' &= x_1 \\
x_3' &= u^2 \\
x_4' &= (x_1^2 + \frac{1}{2}x_1x_3 + 2x_3^2) - 2u^2x_3 - x_2^2 - u^2x_1^2
\end{aligned}
\right.
. \qquad (1.6)$$

Let $T > 0$. Explicit integration from 0 yields

$$\begin{aligned}
x_4(T) &= \int_0^T \left( (u_1^1)^2 + \frac{1}{2}u_1^1 u_1^2 + 2(u_1^2)^2 \right) - u_1^2(T)^2 - \int_0^T \left( \int_0^t u_1^1 \right)^2 dt - \int_0^T u^2 (u_1^1)^2 \\
&\geq \left( \frac{3}{4} - \frac{T^2}{2} - \|u\|_{L^\infty(0,T)} \right) \|u_1\|_{L^2(0,T)}^2 - u_1^2(T)^2 \\
&\geq C \|u_1\|_{L^2(0,T)}^2 - u_1^2(T)^2,
\end{aligned}$$

for any $C \in (0, \frac{3}{4})$, for small enough times and controls in $L^\infty$. As $x_3(T) = u_1^2(T)$, all final state in $\{(x_1, \dots, x_4) \in \mathbb{R}^4; x_4 + x_3^2 < 0\}$ is not reachable from 0. The system (1.6) is not $L^\infty$-STLC around 0 - see Definition 1.13.

To prove Theorem 1.10, we use a « power series expansion » of order 2. Heuristically, it is as if the following terms are the dominant part of the quadratic expansion of the solution in the direction $\varphi_K e^{-i\lambda_1 T}$

$$\sum_{\ell=1}^{r} \sum_{p=1}^{k} i^{2p-1} \gamma_{2p-1}^{\ell} \int_{0}^{T} \frac{u_{p}^{\ell}(t)^{2}}{2} dt + \sum_{1 \le \ell < L \le r} \sum_{p=0}^{2k-1} i^{p} \gamma_{p}^{\ell, L} \int_{0}^{T} u_{\lfloor \frac{p}{2} \rfloor + 1}^{\ell}(t) u_{\lfloor \frac{p+1}{2} \rfloor}^{L}(t) dt. \quad (1.7)$$

The cancellation assumption $(\mathbf{H})_{\text{null}}$ reduces this sum to the simpler expression

$$i(-1)^{k+1} \sum_{\ell=1}^{r} \gamma_{2k-1}^{\ell} \int_{0}^{T} \frac{u_{k}^{\ell}(t)^{2}}{2} dt + i(-1)^{k+1} \sum_{1 \le \ell < L \le r} \gamma_{2k-1}^{\ell, L} \int_{0}^{T} u_{k}^{\ell}(t) u_{k}^{L}(t) dt, \quad (1.8)$$

and $(\mathbf{H})_{\text{pos}}$ allows use to take advantage of a signed term because (1.8) and $(\mathbf{H})_{\text{lin}}$ lead to

$$(-1)^{k+1} \operatorname{sgn}(\gamma_{2k-1}^1) \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle \approx \operatorname{sgn}(\gamma_{2k-1}^1) \int_0^T q(u_k(t)) dt \ge 0.$$
## 1.5 Comparison with the finite-dimensional case

The main result of this article is an adaptation to (1.1) of a theorem that proves quadratic obstructions to STLC for multi-input control-affine systems in [34] for $r = 2$. Here is the framework and key definitions. One considers the system

$$x'(t) = f_0(x(t)) + \sum_{\ell=1}^{r} u^{\ell}(t) f_{\ell}(x(t)), \quad (1.9)$$

where the state is $x(t) \in \mathbb{R}^d$, the controls are scalar functions $u^\ell(t)$ and $f_\ell$ are real-analytic vector fields on a neighborhood of 0 in $\mathbb{R}^d$ such that $f_0(0) = 0$. The last hypothesis ensures that 0 is an equilibrium of the free system – i.e. with $u \equiv 0$.

For each $t > 0$, $u \in L^1((0, t), \mathbb{R})^r$, there exists a unique maximal mild solution to (1.9) with initial data 0 at time $t = 0$, which we will denote by $x(\cdot; u)$. As we are interested in small-time and small controls, the solution is well-defined up to time $t$.

**Definition 1.13 (W$^{m,p}$-STLC).** Let $m \in [-1, +\infty[$ and $p \in [1, +\infty[$. The system (1.9) is W$^{m,p}$-STLC (around 0) if for every $T, \varepsilon > 0$, there exists $\delta > 0$, such that, for every $x_f \in B(0, \delta)$, there exist $u \in W^{m,p}((0, T), \mathbb{R})^r \cap L^1((0, T), \mathbb{R})^r$ with $\|u\|_{W^{m,p}} \le \varepsilon$ s.t. $x(T; u) = x_f$.

The historical definition of STLC is the case where $m = 0$ and $p = \infty$ – see [25]. Let $X := \{X_0, \dots, X_r\}$ be a set of $r + 1$ non-commutative indeterminates.

**Definition 1.14 (Free Lie algebra).** We consider $\mathcal{A}(X)$ the unital associative algebra of polynomials of the indeterminates in $X$. For $a, b \in \mathcal{A}(X)$, one defines the Lie bracket of a and $b$ as $[a, b] := ab - ba$. We denote by $\mathcal{L}(X)$ the free Lie algebra generated by $X$ over the field $\mathbb{R}$.

**Definition 1.15 (Lie bracket of vector fields).** Let $f, g : \Omega \to \mathbb{R}^d$ be $\mathcal{C}^\infty$ vector fields on an open subset $\Omega$ of $\mathbb{R}^d$. One defines

$$[f, g] : x \in \Omega \mapsto Dg(x) \cdot f(x) - Df(x) \cdot g(x).$$

**Definition 1.16 (Evaluated Lie bracket).** Let $f_0, \dots, f_r$ be $\mathcal{C}^\infty(\Omega, \mathbb{R}^d)$ vector fields on an open subset $\Omega$ of $\mathbb{R}^d$. For $b \in \mathcal{L}(X)$, we define $f_b := \Lambda(b)$, where $\Lambda : \mathcal{L}(X) \to \mathcal{C}^\infty(\Omega, \mathbb{R}^d)$ is the unique homomorphism of Lie algebras such that $\Lambda(X_\ell) = f_\ell$ for $0 \le \ell \le r$.

For example, if $b = [[[X_1, X_r], X_0], [X_{r-1}, X_0]]$, one has $f_b = [[[f_1, f_r], f_0], [f_{r-1}, f_0]]$.

**Definition 1.17 (Bracket integration $b0^\nu$).** For $b \in \mathcal{L}(X)$ and $\nu \in \mathbb{N}$, we use the short-hand $b0^\nu$ to denote the right-iterated bracket $[\dots[b, X_0], \dots, X_0]$, where $X_0$ appears $\nu$ times.

For example, if $b = [[X_r, X_0], [X_0, X_r]]$, then $b0^2 = [[[[X_r, X_0], [X_0, X_r]], X_0], X_0]$.

**Definition 1.18.** We define the following brackets

$$a. M_l^\ell := X_\ell 0^l; \quad (\ell, l) \in [1, r] \times \mathbb{N},$$

$$b. W_{p,l}^\ell := [M_{p-1}^\ell, M_p^\ell] 0^l; \quad (\ell, p, l) \in [1, r] \times \mathbb{N}^* \times \mathbb{N},$$

$$c. C_{p,l}^{\ell,L} := (-1)^p [M_{\lfloor \frac{p+1}{2} \rfloor}^\ell, M_{\lfloor \frac{p}{2} \rfloor}^L] 0^l; \quad (\ell, L) \in [1, r]^2 \text{ such that } \ell < L, (p, l) \in \mathbb{N}^2.$$

The following theorem generalizes [34, Theorem 1.15], originally stated for $r = 2$. By applying the same method as the one introduced in [8], we obtain the extended result presented here, which relies on a Magnus-type representation formula of the state.<fnref n="1" /> The proof is designed to prepare an easier transfer to PDEs.

<footnote n="1">¹A proof of this theorem is available in my thesis manuscript.</footnote>
**Theorem 1.19.** Let $k, r \in \mathbb{N}^*$ and $f_0, \dots, f_r$ be real-analytic vector fields on a neighborhood of 0 in $\mathbb{R}^d$ such that $f_0(0) = 0$. We define the vector subspace of $\mathbb{R}^d$

$$
\mathcal{N}_k := \operatorname{Span} \left\{ f_{M_l^\ell}(0); (\ell, l) \in [1, r] \times \mathbb{N} \right\} + \operatorname{Span} \left\{ f_{W_{p,l}^\ell}(0); (\ell, p, l) \in [1, r] \times [1, k-1] \times \mathbb{N} \right\} \\
+ \operatorname{Span} \left\{ f_{C_{p,l}^{\ell,L}}(0); (\ell, L) \in [1, r]^2 \text{ such that } \ell < L, (p, l) \in [0, 2k-2] \times \mathbb{N} \right\}.
$$

Assume that there exists a linear form $\mathbb{P}$ on $\mathbb{R}^d$ such that $\mathbb{P}|_{\mathcal{N}_k} \equiv 0$ and

$$
(a_1, \dots, a_r) \in \mathbb{R}^r \mapsto \sum_{\ell=1}^{r} \mathbb{P} \left( f_{W_{k,0}^{\ell}}(0) \right) \frac{a_{\ell}^2}{2} + \sum_{1 \le \ell < L \le r} \mathbb{P} \left( f_{C_{2k-1,0}^{\ell,L}}(0) \right) a_{\ell} a_L \in \mathbb{R} \text{ is a definite} \\
\text{quadratic form on } \mathbb{R}^r.
$$

Then, the affine system (1.9) is not $H^{2k}$-STLC around 0.

**Remark 1.20.** Using Theorem 1.19, (1.6) can be studied again with $r = 2$ and $k = 1$. One has $[f_1, f_2](0) = 0$. Then, for every $l \in \mathbb{N}$, $f_{C_{0,l}^{1,2}}(0) = 0$. Thus, $\mathcal{N}_k = \operatorname{Span}(e_1, e_2, e_3)$. Finally, one has $f_{W_{1,0}^1}(0) = 2e_4$, $f_{W_{1,0}^2}(0) = 4e_4$ and $f_{C_{1,0}^{1,2}}(0) = \frac{1}{2}e_4$. Then, the linear form $\mathbb{P} := \langle \cdot, e_4 \rangle$ is suitable because, for every $(x, y) \in \mathbb{R}^2$,

$$
x^2 + \frac{1}{2}xy + 2y^2 \geq \frac{3}{4}x^2 + \frac{7}{4}y^2,
$$

so the associated quadratic form is positive-definitive on $\mathbb{R}^2$.

Let us make the link between Theorems 1.10 and 1.19. When $\ell < L$, the Lie brackets $\gamma_p^{\ell,L}$ are specific to the multi-input system (1.1) with $r > 1$, they can be used to recover a complex lost direction at linear order – see [35] with $r = 2$. They can be considered as good in a STLC point of view. A *contrario*, the Lie brackets $\gamma_{2k-1}^{\ell,\ell}$ are known to be potential obstructions to STLC for the bilinear Schrödinger equation in the single-input case, in an appropriate functional setting – see [17] with $r = 1$.

We now give an interpretation of the series $\gamma_p^{\ell,L}$ in terms of Lie brackets. For $A$ and $B$ two operators, we define $\mathrm{ad}_A^l(B)$ as $\mathrm{ad}_A^0(B) := B$ and $\mathrm{ad}_A^{l+1}(B) := \mathrm{ad}_A^l(B)A - A\mathrm{ad}_A^l(B)$. Under appropriate assumptions on the functions $\mu_\ell$ for domain compatibility (a finite number of derivatives of odd order have to vanish on the boundary) the assumption $(\mathbf{H})_{\text{null}}$ can be interpreted as Lie brackets because

$$
\forall 0 \le p \le 2k-1, \ 1 \le \ell \le L \le r, \quad \gamma_p^{\ell,L} = (-1)^p \langle [\mathrm{ad}_A^{\lfloor \frac{p+1}{2} \rfloor}(B_\ell), \mathrm{ad}_A^{\lfloor \frac{p}{2} \rfloor}(B_L)]\varphi_1, \varphi_K \rangle, \quad (1.10)
$$

where $A$ is defined in (1.2) and $B_\ell$ is the multiplication operator by $\mu_\ell$ in $L^2(0,1)$. We refer to [35, Propositions A.7, A.8, A.10 and A.11] for a precise proof.

Using the interpretation $\gamma_p^{\ell,L}$ in terms of Lie brackets given by (1.10), we can recognize in the expansion (1.7) the leading terms of the Magnus-type representation formula, that is used to prove Theorem 1.19. See [7] for more details.

Assume that $\mu \in \mathcal{C}_c^\infty(0,1)^r$ are smooth functions. Heuristically, we can think of the bilinear Schrödinger equation (1.1) as a control-affine system on the form (1.9) with $f_0 = A$, where the operator $A$ is defined in (1.2) and $f_\ell = B_\ell$ with $1 \le \ell \le r$, where $B_\ell$ is the multiplication operator by $\mu_\ell$ in $L^2(0,1)$. The equilibrium is no longer 0 but the free trajectory $\psi(T; 0, \varphi_1) = \varphi_1 e^{-i\lambda_1 T}$. In this situation, the linear form is $\mathbb{P} := \langle \cdot, \varphi_K \rangle$. The hypotheses are the same in the finite and infinite-dimensional cases.

1. The equality (1.10), [35, Lemma A.9] and the assumptions $(\mathbf{H})_{\text{lin}}$ and $(\mathbf{H})_{\text{null}}$ give

$$
\forall (\ell, l) \in [1, r] \times \mathbb{N},
$$

$$
\mathbb{P} \left( \underline{\mathrm{ad}}_A^l (B_\ell) \varphi_1 \right) = \left\langle \underline{\mathrm{ad}}_A^l (B_\ell) \varphi_1, \varphi_K \right\rangle = (\lambda_1 - \lambda_K)^l \langle \mu_\ell \varphi_1, \varphi_K \rangle = 0,
$$
$\forall (\ell, L) \in [1, r]^2$ such that $\ell \le L, \forall (p, l) \in [0, 2k - 2] \times \mathbb{N},$

$$ \mathbb{P} \left( \mathrm{ad}_A^l \left( [\mathrm{ad}_A^{\lfloor \frac{p+1}{2} \rfloor} (B_\ell), \mathrm{ad}_A^{\lfloor \frac{p}{2} \rfloor} (B_L)] \right) \varphi_1 \right) = (\lambda_1 - \lambda_K)^l (-1)^p \gamma_p^{\ell,L} = 0. $$

This is the equivalent of the condition $\mathbb{P}|_{\mathcal{N}_k} = 0$.

2. The equality (1.10) and the hypothesis $(\mathbf{H})_{\text{pos}}$ ensure that the hypotheses about the quadratic forms are the same.

**Remark 1.21.** The functional setting is different. In Theorem 1.19, we deny $H^{2k}$-STLC, whereas in Theorem 1.10, we disprove the $H^{2k-3}$-STLC. These choices of functional settings are determined through interpolation theory by the competition between the negative Sobolev norm quantifying the quadratic drift and the size of the cubic remainder. For (1.1), we obtain a very good error estimate involving a Lebesgue norm of the cube of the control's primitive – see Proposition 2.2. While classical for scalar-input systems – see [7, Section 7.5] – such an estimate fails in general for multi-input systems, even for ODEs – see [7, Section 7.5]. The key point is that, for (1.1), the operators $B_\ell$ commute. Taking advantage of this property allows us to obtain an error estimate similar to the scalar-input case by using an auxiliary system.

# 1.6 State of the art

## 1.6.1 Topological obstructions to exact controllability

In [1], Ball, Marsden and Slemrod proved obstructions to local exact controllability of linear PDEs with bilinear controls. For instance, if the multiplicative operators $\mu_\ell$ are bounded on $H_{(0)}^s(0, 1)$, then system (1.1) is not exactly controllable in $S \cap H_{(0)}^s(0, 1)$, with controls $u \in L_{\text{loc}}^p(\mathbb{R}, \mathbb{R})^r$ and $p > 1$. The fundamental reason behind is that, under these assumptions, the reachable set has empty interior in $H_{(0)}^s(0, 1)$. The case of $L_{\text{loc}}^1$-controls ($p = 1$) was incorporated in [19] and extensions to nonlinear equations were proved in [23, 24]. Turicini adapted this statement to Schrödinger equations in [44]. After the important work [1], different notions of controllability were studied for the single-input bilinear Schrödinger equation such as

- exact controllability in more regular spaces, on which the $\mu_\ell$ do not define bounded operators – see [6, 18, 35],

- approximate controllability.

## 1.6.2 Exact controllability in more regular spaces, by linear test

For the single-input bilinear Schrödinger equation, local exact controllability was first proved in [2, 3] with Nash-Moser techniques, to deal with an apparent derivative loss problem and then in [6] with a classical inverse mapping theorem, thanks to a regularizing effect. By grafting other ingredients onto this core strategy, global (resp. local) exact controllability in regular spaces was proved for different models in [39, 41] (resp. [16]). This strategy has also been used to obtain local controllability results for nonlinear Schrödinger equations, as in [30], or for coupled Schrödinger equations, as in [38].

## 1.6.3 Power series expansion of order 2 or 3

First, let us look at the case of **single-input systems**. A power series expansion of order 2 allows to recover a lost direction in large time: this strategy is used in [12] for the single-input bilinear Schrödinger equation. This method is also used for other equations, such as KdV, in [20]. If the order 2 vanishes, a power series of order 3 can be used to recover the small-time local controllability, for example in [27], for KdV. If the order 2 doesn't cancel out, but the term of order 3 is strong enough, this expansion can also give the small-time local controllability – see [18], for the single-input bilinear Schrödinger equation.

In the context of **multi-input systems**, we use a power series expansion of order 2 in [35] to recover in small-time a direction that is lost at the linear order. The purpose of this article is to give other algebraic assumptions to prove obstructions for STLC.
### 1.6.4 Quadratic obstructions to STLC

In [26], Coron denied the $L^\infty$-small-time local controllability result for the single-input bilinear Schrödinger equation, with a particular dipolar moment $\mu_\ell$, thanks to a drift. In [12], Beauchard and Morancey gave general assumptions on $\mu_1$ to deny $L^\infty$-STLC for the same system. In [17], Bournissou proved that this drift also occurs with small control in $W^{-1,\infty}$. In [11], Beauchard, Marbach and Perrin proved an obstruction for the single-input bilinear Schrödinger equation with Neumann boundary conditions, using a power series of order 2. Quadratic terms have also been used to create obstructions to the controllability of other single-input systems, for example in [9] for ODEs, in [36] for the Burgers' equation, [10] for the heat equation, [28] for KdV and [42] for KdV-Neumann. With the exception of a non-physical PDE designed for, [10, Section 5 and 6] for single-input systems, the quadratic terms generally do not recover small-time controllability.

In [34], we prove quadratic obstructions for multi-input control-affine systems in the finite-dimensional case (ODEs). Our strategy is to adapt [34, Theorem 1.15] for the multi-input bilinear Schrödinger equation. The strong analogy has already been developed in Section 1.5.

### 1.6.5 Approximate controllability

The first results of global approximate controllability of bilinear Schrödinger equations were obtained in large time – see [14, 21, 32, 40, 37]. For particular systems, a large time is indeed necessary for the approximate controllability – see [5, 4]. Small-time approximate controllability between eigenstates for Schrödinger equations on the torus is proved by Duca and Nersesyan in [29], by means of an infinite-dimensional geometric control approach (satura- tion argument). Related results have been subsequently established in [15, 22, 31]. Recently, Beauchard and Pozzoli provided the first examples of small-time globally approximately con- trollable bilinear Schrödinger equations in [13].

## 2 Proof of the main theorem

One recalls that $\lambda_j$ and $\varphi_j$ are defined in (1.3). In all the document, for $j \in \mathbb{N}^*$, we will note

$$ \omega_j := \lambda_j - \lambda_1 \quad \text{and} \quad \nu_j := \lambda_K - \lambda_j. \qquad (2.1) $$

### 2.1 Expansion of the solution

We are going to make an asymptotic expansion of the solution to (1.1). Let $u \in L^2((0,T), \mathbb{R})^r$ be fixed controls. The first-order term $\Psi \in C^0([0,T], H_{(0)}^3(0,1))$ is the solution to the linearized system of (1.1) around the free trajectory $(\psi_1, u \equiv 0)$, i.e.

$$ \left\{ \begin{array}{l} i \partial_t \Psi(t, x) = - \partial_x^2 \Psi(t, x) - u(t) \cdot \mu(x) \psi_1(t, x), \quad t \in (0, T), x \in (0, 1), \\ \Psi(t, 0) = \Psi(t, 1) = 0, \quad t \in (0, T), \\ \Psi(0, x) = 0, \quad x \in (0, 1). \end{array} \right. \qquad (2.2) $$

The solution is given by: $\forall t \in [0, T]$,

$$ \Psi(t) = i \sum_{\ell=1}^{r} \sum_{j=1}^{+\infty} \left( \langle \mu_{\ell} \varphi_{1}, \varphi_{j} \rangle \int_{0}^{t} u^{\ell}(\tau) e^{i \omega_{j} \tau} d\tau \right) \psi_{j}(t). \qquad (2.3) $$

We expand the development of the solution to the quadratic term. The second-order term $\xi \in C^0([0,T], H_{(0)}^3(0,1))$ is the solution to the following system

$$ \left\{ \begin{array}{l} i \partial_t \xi(t, x) = - \partial_x^2 \xi(t, x) - u(t) \cdot \mu(x) \Psi(t, x), \quad t \in (0, T), x \in (0, 1), \\ \xi(t, 0) = \xi(t, 1) = 0, \quad t \in (0, T), \\ \xi(0, x) = 0, \quad x \in (0, 1). \end{array} \right. \qquad (2.4) $$
The idea is that $\psi(T; u, \varphi_1) \simeq \psi_1(T) + \Psi(T) + \xi(T)$. Thus,

$$ \mathfrak{F}\langle\psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T}\rangle \simeq 0 + 0 + \mathfrak{F}\langle\xi(T), \varphi_K e^{-i\lambda_1 T}\rangle, \quad (2.5) $$

the first term being 0 since it is real and the second one by hypothesis $(\mathbf{H})_{\text{lin}}$. For $1 \le \ell, L \le r$, one defines $H_{\ell,L} : (t,s) \in [0,T]^2 \mapsto -e^{-i\omega_K T} \sum_{j=1}^{+\infty} c_j^{\ell,L} e^{i(\nu_j t + \omega_j s)} \in \mathbb{C}$.

**Remark 2.1.** The kernels introduced in this article differ by a phase $e^{-i\omega_K T}$, compared with those introduced in [35] in the case where $r = 2$.

We finally use the notation, for $f,g \in L^2((0,T), \mathbb{R})$, $1 \le \ell, L \le r$,

$$ \mathcal{F}_T^{\ell,L}(f,g) := \int_0^T f(t) \left( \int_0^t H_{\ell,L}(t,\tau)g(\tau)d\tau \right) dt. $$

With these notations and (2.3),

$$ \langle \xi(T), \varphi_K e^{-i\lambda_1 T} \rangle = \sum_{\ell=1}^{r} \mathcal{F}_T^{\ell,\ell}(u^\ell, u^\ell) + \sum_{1 \le \ell < L \le r} \left( \mathcal{F}_T^{\ell,L}(u^\ell, u^L) + \mathcal{F}_T^{L,\ell}(u^L, u^\ell) \right). \quad (2.6) $$

## 2.2 Error estimates

We recall that the iterated primitives of a function are defined in (1.5). We use sharp error estimates to prove that the remainder term of the expansion (2.5) can be neglected compared to the drift $\|u_k\|_{L^2(0,T)}^2$. A rough error estimate – see for example [35, Lemma 3.7] – involves the $L^2$-norm of the controls. As shown in [17, 18] in the single-input case, one can prove a better estimate involving the $L^2$-norm of the primitive $u_1$ of the controls by introducing the new state

$$ \tilde{\psi}((t,x); u, \varphi_1) := \psi((t,x); u, \varphi_1)e^{-iu_1(t)\cdot\mu(x)}. $$

which is the weak solution to

$$ \left\{ \begin{array}{l} i \partial_t \tilde{\psi} = - \partial_x^2 \tilde{\psi} - i [u_1(t) \cdot \mu''(x) + 2(u_1(t) \cdot \mu'(x)) \partial_x] \tilde{\psi} + (u_1(t) \cdot \mu'(x))^2 \tilde{\psi}, \\ \tilde{\psi}(t, 0) = \tilde{\psi}(t, 1) = 0, \\ \tilde{\psi}(0, x) = \varphi_1(x). \end{array} \right. \quad (2.7) $$

The well-posedness of the auxiliary system (2.7) is stated in [17, Proposition 4.2] in the case where $r = 1$ and can be adapted, without any additional difficulty, to our setting. We want to study the linear and quadratic expansions of (2.7) around the ground state. Linearizing (2.7), the first-order term $\tilde{\Psi}$ is given by

$$ \tilde{\Psi}(t, x) = \Psi(t, x) - i u_1(t) \cdot \mu(x) \psi_1(t, x), \quad (2.8) $$

where $\Psi$ is the solution to (2.2). Thus, $\tilde{\Psi} \in C^0([0, T], H^3 \cap H_0^1(0, 1))$ is a weak solution to

$$ \left\{ \begin{array}{l} i \partial_t \tilde{\Psi} = - \partial_x^2 \tilde{\Psi} - i [u_1(t) \cdot \mu''(x) + 2(u_1(t) \cdot \mu'(x)) \partial_x] \psi_1, \quad (0, T) \times (0, 1), \\ \tilde{\Psi}(t, 0) = \tilde{\Psi}(t, 1) = 0, \quad (0, T), \\ \tilde{\Psi}(0, x) = 0, \quad (0, 1). \end{array} \right. $$

Doing an expansion of order 2 of (2.7), the second-order term $\tilde{\xi}$ is given by

$$ \tilde{\xi}(t, x) = \xi(t, x) - i u_1(t) \cdot \mu(x) \Psi(t, x) - \frac{(u_1(t) \cdot \mu(x))^2}{2} \psi_1(t, x), \quad (2.9) $$

where $\xi$ is the solution to (2.4). Then, $\tilde{\xi} \in C^0([0, T], H^3 \cap H_0^1(0, 1))$ is a weak solution to

$$ \left\{ \begin{array}{l} i \partial_t \tilde{\xi} = - \partial_x^2 \tilde{\xi} - i [u_1(t) \cdot \mu''(x) + 2(u_1(t) \cdot \mu'(x)) \partial_x] \tilde{\Psi} + (u_1(t) \cdot \mu'(x))^2 \psi_1, \\ \tilde{\xi}(t, 0) = \tilde{\xi}(t, 1) = 0, \\ \tilde{\xi}(0, x) = 0. \end{array} \right. $$
In [17, Lemma 4.6, Propositions 4.7 and 4.8], Bournissou proved estimates for the auxiliary system when $r = 1$. These results can be adapted, without any additional difficulty, to our case to our case, giving the following proposition.

**Proposition 2.2.** For every $T > 0$, $u \in L^2((0, T), \mathbb{R})^r$, $\mu$ satisfying $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$ and $p \in \mathbb{N}^*$ one has, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \begin{aligned} \langle e^{iu_1(T)\cdot\mu} (\tilde{\psi}(\cdot; u, \varphi_1) - \psi_1 - \tilde{\Psi}) (T), \varphi_p \rangle &= \mathcal{O} (\|u_1\|_{L^2(0,T)}^2), \\ \langle e^{iu_1(T)\cdot\mu} (\tilde{\psi}(\cdot; u, \varphi_1) - \psi_1 - \tilde{\Psi} - \tilde{\xi}) (T), \varphi_p \rangle &= \mathcal{O} (\|u_1\|_{L^2(0,T)}^3). \end{aligned} $$

From these estimates, proved for the auxiliary system, we derive the following error estimates for the bilinear Schrödinger equation (1.1). This result can be adapted from [17, Proposition 4.9] in the multi-input case.

**Proposition 2.3.** For every $T > 0$, $u \in L^2((0, T), \mathbb{R})^r$, $\mu$ satisfying $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$ and $p \in \mathbb{N}^*$, the following error estimates hold, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \langle (\psi(\cdot; u, \varphi_1) - \psi_1 - \Psi)(T), \varphi_p \rangle = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + |u_1(T)|^2 \right), \quad (2.10) $$

$$ \langle (\psi(\cdot; u, \varphi_1) - \psi_1 - \Psi - \xi)(T), \varphi_p \rangle = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^3 + |u_1(T)|^3 \right). \quad (2.11) $$

The final step is to show that the boundary terms $(u_1^1(T), \dots, u_1^r(T))$ can be neglected, as they are part of the dynamics. Unlike the previous steps, this one does not follow directly from Bournissou's work [17, Proposition 4.19], which was done for $r = 1$; therefore, we will explain the details. We recall that the sequences $c^{\ell,L}$ are introduced in Definition 1.4 and the quantities $\omega_j$ and $\nu_j$ are defined in (2.1).

**Proposition 2.4.** For every $T > 0$, $u \in L^2((0, T), \mathbb{R})^r$, $\mu$ satisfying $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$, $(\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$, the following error estimate holds, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \sum_{\ell=1}^{r} |u_1^{\ell}(T)| = \mathcal{O} \left( \sqrt{T} \|u_1\|_{L^2(0,T)} + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)} \right). \quad (2.12) $$

*Proof.* Let $(\alpha_\ell)_{\ell \in [1, r]} \in \mathbb{R}^r$ be such that, for all $j \ge 1$, $\sum_{\ell=1}^r \alpha_\ell \langle \mu_\ell \varphi_1, \varphi_j \rangle = 0$. Then,

$$ \sum_{\ell, L \in [1, r]} C_{\ell, L} := \sum_{\ell, L \in [1, r]} \alpha_{\ell} \alpha_{L} \sum_{j=1}^{+\infty} \left( c_{j}^{L, \ell} \omega_{j}^{k} \nu_{j}^{k-1} - c_{j}^{\ell, L} \omega_{j}^{k-1} \nu_{j}^{k} \right) = 0 - 0. $$

For all $1 \le \ell \le L \le r$, one has by definition,

$$ C_{\ell,L} = -\alpha_{\ell} \alpha_{L} \gamma_{2k-1}^{\ell,L}. $$

Moreover, if $1 \le L \le \ell \le r$, using Corollary A.5 with $p = k - 1$ and $\nu = 1$, one has

$$ C_{\ell,L} = \beta_0^1 \gamma_{2k-2}^{L,\ell} \omega_K - \beta_1^1 \gamma_{2k-1}^{L,\ell} = -\gamma_{2k-1}^{L,\ell}, $$

using $(\mathbf{H})_{\text{null}}$. Thus, we obtain $q(\alpha_1, \dots, \alpha_r) = 0$. Using $(\mathbf{H})_{\text{pos}}$, one has $(\alpha_1, \dots, \alpha_r) = 0$. Then, $((\langle \mu_\ell \varphi_1, \varphi_j \rangle)_{j \ge 1})_{\ell \in [1, r]}$ is a linearly independent family. Consequently, there exist $j_1, \dots, j_r \in \mathbb{N}^*$, pairwise distinct, such that $M := (\langle \mu_\ell \varphi_1, \varphi_{j_n} \rangle)_{(\ell,n) \in [1, r] \times [1, r]} \in GL_r(\mathbb{R})$. Then, using the remainder estimate (2.10), the expansion of the linearized system given by (2.3), an integration by parts and Cauchy-Schwarz's inequality, one finally gets, for all $j \ge 1$, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \begin{aligned} \langle \psi(T; u, \varphi_1) - \psi_1(T), \varphi_j e^{-i\lambda_1 T} \rangle = i \sum_{\ell \in [1, r]} \langle \mu_\ell \varphi_1, \varphi_j \rangle u_1^\ell(T) \\ + \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + |u_1(T)|^2 + \sqrt{T} \|u_1\|_{L^2(0,T)} \right). \end{aligned} $$
Focusing on $j \in \{j_1, \dots, j_{kr}\}$, we obtain

$$M(u_1^\ell(T)) = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + |u_1(T)|^2 + \sqrt{T} \|u_1\|_{L^2(0,T)} + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)} \right).$$

As $M$ is invertible and $\|u_1\|_{L^\infty(0,T)} \to 0$, one obtains the desired result. $\square$

Using (2.12) in (2.10) and (2.11), we finally obtain the following error estimates on the linear/quadratic expansion of the solution.

**Corollary 2.5.** For every $T > 0$, $u \in L^2((0,T), \mathbb{R})^r$, $\mu$ satisfying $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$, $(\mathbf{H})_{\text{null}}$, $(\mathbf{H})_{\text{pos}}$ and $p \in \mathbb{N}^*$, one has, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$\langle (\psi(\cdot; u, \varphi_1) - \psi_1 - \Psi)(T), \varphi_p \rangle = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2 \right), \quad (2.13)$$

$$\langle (\psi(\cdot; u, \varphi_1) - \psi_1 - \Psi - \xi)(T) \rangle = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^3 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^3 \right). \quad (2.14)$$

## 2.3 A new expression for the quadratic expansion

The purpose of this section is to show the following proposition, which has already been proved in [17, Proposition 5.1] for $\ell = L$ with some minor adaptations required here.

**Proposition 2.6.** Let $T > 0$, $1 \le \ell \le L \le r$ and $f, g \in L^2((0,T), \mathbb{R})$. If $(\mathbf{H})_{\text{conv}}$ and $(\mathbf{H})_{\text{null}}$ hold, then,

$$\begin{aligned} \Im \left( \mathcal{F}_T^{\ell,L}(f,g) + \mathcal{F}_T^{L,\ell}(g,f) \right) = & (-1)^{k+1} \gamma_{2k-1}^{\ell,L} \int_0^T f_k(t) g_k(t) \cos(\omega_K(t-T)) dt \\ & + \mathcal{O} \left( \sum_{p=1}^k \left( |f_p(T)|^2 + |g_p(T)|^2 \right) + T \| (f_k, g_k) \|_{L^2(0,T)}^2 \right). \end{aligned} \quad (2.15)$$

We first prove the following lemma.

**Lemma 2.7.** Let $n \in \mathbb{N}$ and $H \in C^{2n}(\mathbb{R}^2, \mathbb{C})$. There exists a quadratic form $S_n$ on $\mathbb{C}^{3n}$ such that for all $T > 0$ and $f, g \in L^1(0,T)$,

$$\begin{aligned} \int_0^T f(t) \left( \int_0^t g(\tau) H(t, \tau) d\tau \right) dt = & - \sum_{p=1}^n \int_0^T f_p(t) g_{p-1}(t) \partial_1^{p-1} \partial_2^{p-1} H(t, t) dt \\ & - \sum_{p=1}^n \int_0^T f_p(t) g_p(t) \partial_1^p \partial_2^{p-1} H(t, t) dt + \int_0^T f_n(t) \left( \int_0^t g_n(\tau) \partial_1^n \partial_2^n H(t, \tau) d\tau \right) dt \\ & + S_n (f_1(T), \dots, f_n(T), g_1(T), \dots, g_n(T), C_0^n(g), \dots, C_{n-1}^n(g)), \end{aligned}$$

$$\text{where } C_p^n(g) := \int_0^T g_n(\tau) \partial_1^p \partial_2^n H(t, \tau) d\tau.$$

*Proof.* We prove this lemma by induction on $n \in \mathbb{N}$. It holds for $n = 0$ with $S_0 = 0$. Assume that the result is true for $n \in \mathbb{N}$, fixed. Let $H \in C^{2(n+1)}(\mathbb{R}^2, \mathbb{C})$, $f, g \in L^1(0,T)$. With an integration by parts,

$$\begin{aligned} \int_0^T f_n(t) \left( \int_0^t g_n(\tau) \partial_1^n \partial_2^n H(t, \tau) d\tau \right) dt = & f_{n+1}(T) \int_0^T g_n(\tau) \partial_1^n \partial_2^n H(T, \tau) d\tau \\ & - \int_0^T f_{n+1}(t) \left( g_n(t) \partial_1^n \partial_2^n H(t, t) + \int_0^t g_n(\tau) \partial_1^{n+1} \partial_2^n H(t, \tau) d\tau \right) dt. \end{aligned}$$

Another integration by parts finally leads to

$$\begin{aligned} \int_0^T f_n(t) \left( \int_0^t g_n(\tau) \partial_1^n \partial_2^n H(t, \tau) d\tau \right) dt = & f_{n+1}(T) C_n^n(g) - \int_0^T f_{n+1}(t) g_n(t) \partial_1^n \partial_2^n H(t, t) dt \\ & - \int_0^T f_{n+1}(t) g_{n+1}(t) \partial_1^{n+1} \partial_2^n H(t, t) dt + \int_0^T f_{n+1}(t) \left( \int_0^t g_{n+1}(\tau) \partial_1^{n+1} \partial_2^{n+1} H(t, \tau) d\tau \right) dt. \end{aligned}$$
A final integration by parts finally gives $C_p^n(g) = g_{n+1}(T)\partial_1^p \partial_2^n H(T, T) - C_p^{n+1}(g)$, for all $p \in [0, n]$. Using the induction hypothesis, we obtain the result. $\square$

Thanks to this lemma, we are now able to prove Proposition 2.6.

**Proof of Proposition 2.6.** Let $T > 0$ and $1 \le \ell \le L \le r$. Using $(\mathbf{H})_{\text{conv}}$, one has $H_{\ell,L}, H_{L,\ell} \in C^{2k}(\mathbb{R}^2, \mathbb{C})$. First, note that for all $p \in [1, k]$, for all $t \in [0, T]$,

$$ \begin{aligned} \partial_1^{p-1} \partial_2^{p-1} (H_{L,\ell} - H_{\ell,L}) (t,t) &= (-1)^{p-1} e^{i\omega_{\kappa}(t-T)} \gamma_{2p-2}^{\ell,L}, \\ \partial_1^{p-1} \partial_2^p H_{L,\ell}(t,t) - \partial_1^p \partial_2^{p-1} H_{\ell,L}(t,t) &= i(-1)^{p-1} e^{i\omega_{\kappa}(t-T)} \gamma_{2p-1}^{\ell,L}. \end{aligned} \quad (2.16) $$

Let $f, g \in L^2((0, T), \mathbb{R})$ and $p \in [1, k]$. With an integration by parts,

$$ \begin{aligned} & \int_0^T f_{p-1}(t) g_p(t) \partial_1^{p-1} \partial_2^{p-1} H_{L,\ell}(t,t) dt = f_p(T) g_p(T) \partial_1^{p-1} \partial_2^{p-1} H_{L,\ell}(T,T) \\ & - \int_0^T f_p(t) g_{p-1}(t) \partial_1^{p-1} \partial_2^{p-1} H_{L,\ell}(t,t) dt - \int_0^T f_p(t) g_p(t) \left( \partial_1^p \partial_2^{p-1} + \partial_1^{p-1} \partial_2^p \right) H_{L,\ell}(t,t) dt. \end{aligned} $$

Then, applying Lemma 2.7 with $f \to g$, $g \to f$, $H_{L,\ell} \to H$, $k \to n$ and using the last equality, one gets

$$ \begin{aligned} & \int_0^T g(t) \left( \int_0^t f(\tau) H_{L,\ell}(t, \tau) d\tau \right) dt = \sum_{p=1}^k \int_0^T f_p(t) g_{p-1}(t) \partial_1^{p-1} \partial_2^{p-1} H_{L,\ell}(t,t) dt \\ & + \sum_{p=1}^k \int_0^T f_p(t) g_p(t) \partial_1^{p-1} \partial_2^p H_{L,\ell}(t,t) dt + \int_0^T g_k(t) \left( \int_0^t f_k(\tau) \partial_1^k \partial_2^k H_{L,\ell}(t,\tau) d\tau \right) dt \\ & + \mathcal{O} \left( \sum_{p=1}^k \left( |f_p(T)|^2 + |g_p(T)|^2 \right) + T \|f_k\|_{L^2(0,T)}^2 \right). \end{aligned} \quad (2.17) $$

To estimate $C_p^k(f)$, we used the boundness of $H_{L,\ell}$ and the Cauchy–Schwarz’s inequality. Applying Lemma 2.7 again with $H_{\ell,L} \to H$, $k \to n$, using (2.16), (2.17) and the assumption $(\mathbf{H})_{\text{null}}$, one gets

$$ \begin{aligned} & (-1)^{k+1} e^{i\omega_{\kappa}T} \left( \mathcal{F}_T^{\ell,L}(f,g) + \mathcal{F}_T^{L,\ell}(g,f) \right) = i\gamma_{2k-1}^{\ell,L} \int_0^T f_k(t) g_k(t) e^{i\omega_{\kappa}t} dt \\ & + \sum_{j=1}^{+\infty} \omega_j^k \nu_j^k \int_0^T e^{i\nu_j t} \left( c_j^{\ell,L} f_k(t) \left( \int_0^t g_k(\tau) e^{i\omega_j\tau} d\tau \right) + c_j^{L,\ell} g_k(t) \left( \int_0^t f_k(\tau) e^{i\omega_j\tau} d\tau \right) \right) dt \\ & + \mathcal{O} \left( \sum_{p=1}^k \left( |f_p(T)|^2 + |g_p(T)|^2 \right) + T \|(f_k,g_k)\|_{L^2(0,T)}^2 \right). \end{aligned} $$

Using the Cauchy–Schwarz’s inequality and taking the imaginary part, we obtain the result. $\square$

## 2.4 Vectorial relations

The equations (2.6) and (2.15) give a quadratic expansion of the solution to (1.1). We then estimate the boundary terms $|u_p^\ell(T)|^2$ that appear. To do that, we need the following lemma, which is an equivalent in the infinite-dimensional case of [34, Lemma 3.12].

**Lemma 2.8.** Assume that the hypotheses $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$ hold. Then, the following family is $\mathbb{R}$-linearly independent in $\mathbb{C}^{N^*}$

$$ \left( \langle (\mu_{\ell} \varphi_1, \varphi_j)(-i\omega_j)^p \rangle_{j \ge 1} \right)_{(p, \ell) \in [0, k-1] \times [1, r]} $$
Proof. We use a proof technique similar to Proposition 2.4. By contradiction, assume that there exist $(\alpha_{p,l})_{(p,l)\in[0,k-1]\times[1,r]} \in \mathbb{R}^{k \times r} \setminus \{0\}$ s.t.

$$ \forall j \ge 1, \quad \sum_{p \in [0,k-1]} \sum_{l \in [1,r]} \alpha_{p,l} \langle \mu_l \varphi_1, \varphi_j \rangle (-i\omega_j)^p = 0. $$

Let $l_0 := \max\{p \in [0, k-1]; (\alpha_{p,1}, \dots, \alpha_{p,r}) \neq 0\}$. By hypothesis, the following quantity is zero

$$ \sum_{\ell,L \in [1,r]} \sum_{p,q \in [0,l_0]} \alpha_{p,\ell} \alpha_{q,L} (-i)^{p+q} \sum_{j=1}^{+\infty} \left( c_j^{L,\ell} \omega_j^{p+k-l_0} \nu_j^{q+k-l_0-1} - c_j^{\ell,L} \omega_j^{q+k-l_0-1} \nu_j^{p+k-l_0} \right) = 0. $$

We will denote this quantity by $\sum_{\ell,L \in [1,r]} C_{\ell,L}$. Let $v := (v_j)_{j \ge 1}$ be a sequence of real numbers and $m, n \in \mathbb{N}$. Subject to convergence, we define

$$ R_1^{m,n}(v) := \sum_{j=1}^{+\infty} v_j \omega_j^{m+k-l_0} \nu_j^{n+k-l_0-1} \quad \text{and} \quad R_2^{m,n}(v) := \sum_{j=1}^{+\infty} v_j \omega_j^{m+k-l_0-1} \nu_j^{n+k-l_0}. $$

By definition of $R_1$ and $R_2$, one has, for all $1 \le \ell \le L \le r$,

$$ C_{\ell,L} = \sum_{p=0}^{l_0} \alpha_{p,\ell} \alpha_{p,L} (-1)^{p+1} \gamma_{2(p+k-l_0)-1}^{\ell,L} + \sum_{\substack{p,q=0, \\ p \ne q}}^{l_0} \alpha_{p,\ell} \alpha_{q,L} (-i)^{p+q} (R_1^{p,q}(c^{L,\ell}) - R_2^{q,p}(c^{\ell,L})). $$

For all $p \in [0, l_0 - 1]$, for all $q \in [p+1, l_0]$, Corollary A.5 with $q-p-1 \to \nu$ and $p+k-l_0 \to p$ gives

$$ R_1^{p,q}(c^{L,\ell}) - R_2^{q,p}(c^{\ell,L}) \in \operatorname{Span} \left( \gamma_{2(p+k-l_0)+r}^{\ell,L}, r \in [0, q-p-1] \right) \subset \mathbb{R}. $$

Moreover, $2(p + k - l_0) + r \le 2k - 2$. Using $(\mathbf{H})_{\text{null}}$, this sum is zero. Finally, for all $p \in [1, l_0]$, for all $q \in [0, p-1]$, Corollary A.5 with $p-q+1 \to \nu$ and $q+k-l_0-1 \to p$ gives

$$ R_1^{p,q}(c^{L,\ell}) - R_2^{q,p}(c^{\ell,L}) \in \operatorname{Span} \left( \gamma_{2(q+k-l_0-1)+r}^{\ell,L}, r \in [0, p-q+1] \right). $$

Similarly, $2(q + k - l_0 - 1) + r \le 2k - 2$ so this sum is zero by $(\mathbf{H})_{\text{null}}$. Finally,

$$ C_{\ell,L} = \alpha_{l_0,\ell} \alpha_{l_0,L} (-1)^{l_0+1} \gamma_{2k-1}^{\ell,L}. \qquad (2.18) $$

With similar computations, we prove that for all $1 \le L \le \ell \le r$,

$$ C_{\ell,L} = \alpha_{l_0,\ell} \alpha_{l_0,L} (-1)^{l_0+1} \gamma_{2k-1}^{L,\ell}. \qquad (2.19) $$

As $\sum_{\ell,L \in [1,r]} C_{\ell,L} = 0$, (2.18) and (2.19) lead to

$$ \sum_{\ell=1}^{r} \gamma_{2k-1}^{\ell} \frac{\alpha_{l_0,\ell}^2}{2} + \sum_{1 \le \ell < L \le r} \gamma_{2k-1}^{\ell,L} \alpha_{l_0,\ell} \alpha_{l_0,L} = 0, \quad \text{i.e.} \quad q(\alpha_{l_0,1}, \dots, \alpha_{l_0,r}) = 0. $$

Using $(\mathbf{H})_{\text{pos}}$, $(\alpha_{l_0,\ell})_{\ell \in [1,r]} = 0$. This is a contradiction with the choice of $l_0$. We then obtain the result. $\square$

**Remark 2.9.** The proof of Lemma 2.8 may seem a little mysterious at first sight, but there is a strict analogy with the result proved in the finite-dimensional case – see [34, Lemma 3.12]. Note that, if $\mu \in C_c^\infty((0,1), \mathbb{R})^r$, then using [35, Lemma A.9],

$$ \forall 1 \le \ell \le r, \quad 0 \le p \le k-1, \quad (\langle \mu_\ell \varphi_1, \varphi_j \rangle (-i\omega_j)^p)_{j \ge 1} = i^p (\langle \underline{ad}_A^p(\mu_\ell) \varphi_1, \varphi_j \rangle)_{j \ge 1}. $$

Consequently, we proved that the family $(\underline{ad}_A^p(\mu_\ell)\varphi_1)_{(p,\ell)\in[0,k-1]\times[1,r]}$ is linearly independent. This is the same family as in the finite-dimensional case. In [34], we consider $(\alpha_{p,l})_{(p,l)\in[0,k-1]\times[1,r]} \in \mathbb{R}^{k \times r}$ scalars, not all zero, such that

$$ B\varphi_1 := \sum_{p \in [0,k-1]} \sum_{l \in [1,r]} \alpha_{p,l} i^p \underline{ad}_A^p(\mu_l)\varphi_1 = 0. $$
Then, we define $l_0 := \max\{p \in [0, k-1]; (\alpha_{p,1}, \dots, \alpha_{p,r}) \neq \{0\}\}$. As $B\varphi_1 = 0$, one has $[\underline{ad}_A^{k-l_0-1}(B), \underline{ad}_A^{k-l_0}(B)] \varphi_1 = 0$, and we expand this term. Actually, this is exactly what we did in the proof of Lemma 2.8, in a disguised way, by not assuming regularity on $\mu_\ell$ since

$$ \sum_{\ell, L \in [1, r]} C_{\ell, L} = - \left[ \underline{ad}_A^{k-l_0-1}(B), \underline{ad}_A^{k-l_0}(B) \right] \varphi_1. $$

## 2.5 Closed-loop estimates

Using Lemma 2.8, we can now estimate the boundary terms, as stated in the following proposition.

**Proposition 2.10.** Assume that $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$, $(\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$ hold. Then, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \sum_{p \in [1, k]} \sum_{\ell \in [1, r]} |u_p^\ell(T)| = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + \sqrt{T} \|u_k\|_{L^2(0,T)} + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)} \right). $$

*Proof.* By Lemma 2.8, there exist $j_1, \dots, j_{kr} \in \mathbb{N}^*$, pairwise distinct, such that

$$ M := \left( \langle \mu_\ell \varphi_1, \varphi_{j_n} \rangle (-i\omega_{j_n})^p \right)_{((p,\ell),n) \in [0,k-1] \times [1,r] \times [1,rk]} \in GL_{kr}(\mathbb{C}). $$

Then, using the sharp remainder estimate (2.13) and the expansion of the linearized system given by (2.3), one has for all $j \ge 1$, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \langle \psi(T; u, \varphi_1) - \psi_1(T), \varphi_j e^{-i\lambda_1 T} \rangle = i \sum_{\ell=1}^{r} \langle \mu_\ell \varphi_1, \varphi_j \rangle \int_0^T u^\ell(t) e^{i\omega_j(t-T)} dt + \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2 \right). $$

Using integrations by parts and Cauchy-Schwarz's inequality, one gets, for all $j \ge 1$,

$$ \langle \psi(T; u, \varphi_1) - \psi_1(T), \varphi_j e^{-i\lambda_1 T} \rangle = i \sum_{p \in [0, k-1]} \sum_{\ell \in [1, r]} \langle \mu_\ell \varphi_1, \varphi_j \rangle (-i\omega_j)^p u_{p+1}^\ell(T) + \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + \sqrt{T} \|u_k\|_{L^2(0,T)} + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2 \right). $$

Focusing on $j \in \{j_1, \dots, j_{kr}\}$, we finally obtain

$$ M(u_p^\ell(T))_{p \in [1, k]} = \mathcal{O} \left( \|u_1\|_{L^2(0,T)}^2 + \sqrt{T} \|u_k\|_{L^2(0,T)} + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)} \right). $$

As $M$ is invertible, one obtains the desired result. $\square$

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

## 2.6 Interpolation inequality

In this paper, we expand the solution to the Schrödinger equation (1.1) to the second-order term. The remainder term is given by (2.14) and is estimated as $\mathcal{O}(\|u_1\|_{L^2(0,T)}^3)$. The purpose of the following lemma is to compare this error term with the drift size $\|u_k\|_{L^2(0,T)}^2$.

**Lemma 2.11.** Assume that $k \ge 2$. There exists $C > 0$ such that, for all $T > 0$, $f \in H^{2k-3}((0,T), \mathbb{R})$,

$$ \|f_1\|_{L^2(0,T)}^3 \le C (1 + T^{-2k+3}) \|f\|_{H^{2k-3}(0,T)} \|f_k\|_{L^2(0,T)}^2. $$

*Proof.* We apply the Gagliardo-Nirenberg interpolation inequalities – see [33, 43] – with $j = k-1$, $l = 3k-3$, $\alpha = \frac{1}{3}$, $p = q = r = s = 2$ et $\varphi = f_k$ to obtain

$$ \|f_1\|_{L^2(0,T)}^3 \le C \left\| f^{(2k-3)} \right\|_{L^2(0,T)} \|f_k\|_{L^2(0,T)}^2 + CT^{-3(k-1)} \|u_k\|_{L^2(0,T)}^3. $$

Moreover,

$$ \|u_k\|_{L^2(0,T)} \le T^k \|u\|_{L^2(0,T)} \le t^k \|u\|_{H^{2k-3}(0,T)}. $$

Thus, we obtain the desired result. $\square$

<figure><img src="image_3.png" /></figure>
## 2.7 Proof of the drift

We prove Theorem 1.10 as a consequence of the following more precise statement.

**Theorem 2.12.** Let $k, K, r \in \mathbb{N}^*$, $\mu$ be functions satisfying $(\mathbf{H})_{\text{conv}}$, $(\mathbf{H})_{\text{reg}}$, $(\mathbf{H})_{\text{lin}}$, $(\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$. If $k \ge 2$ (resp. $k=1$), there exist $C, T^* > 0$ such that for all $T \in (0, T^*)$, there exists $\eta > 0$ such that for all $u \in H^{2k-3}((0,T), \mathbb{R})^r$ (resp. $u \in L^2((0,T), \mathbb{R})^r$) with $\|u\|_{H^{2k-3}(0,T)} \le \eta$ (resp. $\|u_1\|_{L^\infty(0,T)} \le \eta$),

$$ (-1)^{k+1} \operatorname{sgn}(\gamma_{2k-1}^1) \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle \ge C \|u_k\|_{L^2}^2 - C \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2}^2. \quad (2.20) $$

**Remark 2.13.** Theorem 2.12 shows that there exists $0 < R < 1$ such that the following targets cannot be reached by the solution to (1.1)

$$ \forall \delta \in (0, R), \quad \psi_f := \left( \sqrt{1 - \delta^2} \varphi_1 + i(-1)^k \operatorname{sgn}(\gamma_{2k-1}^1) \delta \varphi_K \right) e^{-i\lambda_1 T}. $$

Indeed, if there exist controls $u$ such that $\psi(T; u, \varphi_1) = \psi_f$, the equation (2.20) leads to

$$ -\delta \ge K \|u_k\|_{L^2(0,T)}^2 - 2K(1 - \sqrt{1 - \delta^2}) \ge -2K\delta^2. $$

This is impossible when $\delta \to 0$. Thus, we obtain Theorem 1.10.

**Proof of Theorem 2.12.** Using $(\mathbf{H})_{\text{lin}}$ and the remainder estimate (2.14), the quadratic expansion of the solution gives, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle = \Im \langle \xi(T), \varphi_K e^{-i\lambda_1 T} \rangle + \mathcal{O} \left( \|u_1\|_{L^2}^3 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2}^3 \right). $$

Using (2.6) and Proposition 2.6, one gets

$$ \begin{aligned} \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle = & (-1)^{k+1} \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \\ & + \mathcal{O} \left( \sum_{p \in [1,k]} \sum_{\ell \in [1,r]} |u_p^\ell(T)|^2 + T \|u_k\|_{L^2(0,T)}^2 + \|u_1\|_{L^2(0,T)}^3 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^3 \right). \end{aligned} $$

We use Proposition 2.10 to estimate the boundary terms. We obtain, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \begin{aligned} \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle = & (-1)^{k+1} \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \\ & + \mathcal{O} \left( (T \|u_k\|_{L^2(0,T)}^2 + \|u_1\|_{L^2(0,T)}^3 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2) \right). \end{aligned} \quad (2.21) $$

Let $T^* := \frac{\pi}{3\omega_K}$ if $K \ne 1$, $T^* = +\infty$ else. Let $T \in (0, T^*)$. By $(\mathbf{H})_{\text{pos}}$ and Remark 1.8, there exists $C_2 > 0$ such that $\operatorname{sgn}(\gamma_{2k-1}^1)q(a) \ge 4C_2 \|a\|^2$, for every $a \in \mathbb{R}^r$. Thus,

$$ \operatorname{sgn}(\gamma_{2k-1}^1) \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \ge 2C_2 \|u_k\|_{L^2(0,T)}^2. \quad (2.22) $$

First obstruction – $k=1$: using (2.21), we immediately obtain

$$ \begin{aligned} \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle = & \int_0^T q(u_1(t)) \cos(\omega_K(t-T))dt \\ & + \mathcal{O} \left( (T + \|u_1\|_{L^\infty(0,T)} \|u_1\|_{L^2(0,T)}^2 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2) \right). \end{aligned} $$

Then, there exist $C_1, T_1 > 0$ such that, for all $T \in (0, T_1)$, there exists $\eta_1 > 0$ such that, for all $u \in L^2((0,T), \mathbb{R})^r$ satisfying $\|u_1\|_{L^\infty(0,T)} < \eta_1$,

$$ \left| \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle - \int_0^T q(u_1(t)) \cos(\omega_K(t-T))dt \right| \le C_1 (T + \|u_1\|_{L^\infty(0,T)} \|u_1\|_{L^2(0,T)}^2 + C_1 \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2. $$
We fix $T_f := \min(T_1, T^*, \frac{C_2}{2C_1})$ and $\eta := \min(\eta_1, \frac{C_2}{2C_1})$. This equation together with (2.22) conclude the proof.

Other obstructions - $k \ge 2$: using the interpolation inequality proved in Lemma 2.11 in (2.21), one gets, as $\|u_1\|_{L^\infty(0,T)} \to 0$,

$$ \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle = (-1)^{k+1} \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \\ + \mathcal{O} \left( (T + (1 + T^{-2k+3}) \|u\|_{H^{2k-3}(0,T)} \|u_k\|_{L^2(0,T)}^2 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2 \right). $$

Then, there exist $C_1, T_1 > 0$ such that, for all $T \in (0, T_1)$, there exists $\eta_1 > 0$ such that, for all $u \in H^{2k-3}((0,T), \mathbb{R})^r$ satisfying $\|u_1\|_{L^\infty(0,T)} < \eta_1$,

$$ \left| \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle - (-1)^{k+1} \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \right| \\ \le C_1 \left( (T + (1 + T^{-2k+3}) \|u\|_{H^{2k-3}(0,T)} \|u_k\|_{L^2(0,T)}^2 + \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2 \right). \quad (2.23) $$

Let $T_f := \min(T_1, T^*, \frac{C_2}{3C_1})$. For all $T \in (0, T_f)$, we define $\eta := \min(\eta_1, \frac{C_2}{3C_1}, \frac{C_2}{3C_1}T^{2k-3})$. Then, for all $u \in H^{2k-3}((0,T), \mathbb{R})^r$ with $\|u\|_{H^{2k-3}(0,T)} \le \eta$, the estimate (2.23) leads to

$$ \left| \Im \langle \psi(T; u, \varphi_1), \varphi_K e^{-i\lambda_1 T} \rangle - (-1)^{k+1} \int_0^T q(u_k(t)) \cos(\omega_K(t-T))dt \right| \\ \le C_2 \|u_k\|_{L^2(0,T)}^2 + C_1 \|\psi(T; u, \varphi_1) - \psi_1(T)\|_{L^2(0,1)}^2. $$

This equation together with (2.22) lead to the desired result. $\square$

# A Postponed proofs

## A.1 Existence of $\mu_1, \dots, \mu_r$ verifying the hypotheses

**Theorem A.1.** Let $k, K, r \in \mathbb{N}^*$. There exist $\mu_1, \dots, \mu_r$ satisfying $(\mathbf{H})_{\text{reg}}, (\mathbf{H})_{\text{conv}}, (\mathbf{H})_{\text{lin}}, (\mathbf{H})_{\text{null}}$ and $(\mathbf{H})_{\text{pos}}$.

We use arguments very similar to those developed in [18, 17, 35]. That is why we will only give a proof skeleton for $r = 2$.

**Ideas of proof.** We prove more precisely the existence of $\mu_1, \mu_2 \in C_c^\infty(0,1)$ such that the previous assumptions are satisfied. If $\mu_1, \mu_2 \in C_c^\infty(0,1)$, $(\mathbf{H})_{\text{reg}}$ is already verified. This is also the case for $(\mathbf{H})_{\text{conv}}$ by Remark 1.5. In [35], using a trick that divides function supports into two parts, we have explained how to guarantee the existence of functions $\mu_1, \mu_2 \in C_c^\infty(0,1)$ so that the following equalities hold

$$ \langle \mu_1 \varphi_1, \varphi_K \rangle = \langle \mu_2 \varphi_1, \varphi_K \rangle = 0, \quad (A.1) $$

$$ \forall 1 \le p \le 2k-2, \forall 1 \le \ell \le L \le 2, \quad \gamma_{p,\ell,L} = 0, \quad (A.2) $$

$$ \gamma_{2k-1}^{1,2} = 0 \quad (A.3) $$

As a consequence, $(\mathbf{H})_{\text{lin}}$ and $(\mathbf{H})_{\text{null}}$ are satisfied. Moreover using Remark 1.9, as $\gamma_{2k-1}^{1,2} = 0$, the assumption $(\mathbf{H})_{\text{pos}}$ becomes

$$ \gamma_{2k-1}^{1} \gamma_{2k-1}^{2} > 0. $$

Then, we want to guarantee that (A.1), (A.2), (A.3), $\gamma_{2k-1}^{1,2} > 0$ and $\gamma_{2k-1}^{2} > 0$ can be satisfied simultaneously to conclude the proof. This is what Bournissou did in [17, Theorem A.4]. By adapting her method to our setting, we obtain the claimed result. $\square$
## A.2 Some series expansions

We recall that $\omega_j$ and $\nu_j$ are defined in (2.1).

**Definition A.2.** Let $a := (a_j)_{j \ge 1}$, $b := (b_j)_{j \ge 1}$ be sequences of real numbers and $l \in \mathbb{N}$ s.t. $(a_j j^{2l})_{j \ge 1}$, $(b_j j^{2l})_{j \ge 1} \in \ell^1(\mathbb{N}^*)$. We define $\gamma_l(a, b) := \sum_{j=1}^{+\infty} (a_j \omega_j^{\lfloor \frac{l}{2} \rfloor} \nu_j^{\lfloor \frac{l+1}{2} \rfloor} - b_j \omega_j^{\lfloor \frac{l+1}{2} \rfloor} \nu_j^{\lfloor \frac{l}{2} \rfloor})$.

**Remark A.3.** We recall that $\gamma_p^{\ell,L}$ is specified in Definition 1.6. If $(\mathbf{H})_{\text{conv}}$ holds,

$$ \forall 0 \le p \le 2k-1, \quad 1 \le \ell \le L \le r, \qquad \gamma_p^{\ell,L} = \gamma_p (c^{\ell,L}, c^{L,\ell}). $$

**Lemma A.4.** Let $\nu \in \mathbb{N}$. There exist coefficients $(\beta_l^\nu)_{l \in [0, \nu]} \in \mathbb{Z}^{\nu+1}$ such that, for every $p \in \mathbb{N}$ and $a := (a_j)_{j \ge 1}$, $b := (b_j)_{j \ge 1} \in \mathbb{R}^{\mathbb{N}^*}$ verifying $\sum_{j=1}^{+\infty} (|a_j| + |b_j|) j^{2(2p+\nu)} < +\infty$,

$$ \sum_{j=1}^{+\infty} (a_j \omega_j^{p+\nu} \nu_j^p - b_j \omega_j^p \nu_j^{p+\nu}) = \sum_{l=0}^{\nu} \beta_l^{\nu} (-1)^l \gamma_{2p+l}(a, b) \omega_K^{\nu-l}. $$

*Proof.* We prove this statement by induction on $\nu \in \mathbb{N}$. *Initialisation.* For $\nu = 0$, the desired equality is true by definition with $\beta_0^0 = 1$. For $\nu = 1$, one can notice that $\omega_K - \nu_j = \omega_j$. Then,

$$ \sum_{j=1}^{+\infty} (a_j \omega_j^{p+1} \nu_j^p - b_j \omega_j^p \nu_j^{p+1}) = \sum_{j=1}^{+\infty} a_j \omega_j^p \nu_j^p (\omega_K - \nu_j) - \sum_{j=1}^{+\infty} b_j \omega_j^p \nu_j^p (\omega_K - \omega_j) \\ = \omega_K \gamma_{2p}(a, b) - \gamma_{2p+1}(a, b). $$

We obtain the result with $\beta_0^1 = \beta_1^1 = 1$. *Induction step:* assume that the result holds for $\nu$ and $\nu + 1$. Let $p \in \mathbb{N}$, $(a_j)_{j \ge 1}$, $(b_j)_{j \ge 1} \in \mathbb{R}^{\mathbb{N}^*}$ be such that $\sum_{j=1}^{+\infty} (|a_j| + |b_j|) j^{2(2p+\nu+2)} < +\infty$. Using the same strategy,

$$ \sum_{j=1}^{+\infty} (a_j \omega_j^{p+\nu+2} \nu_j^p - b_j \omega_j^p \nu_j^{p+\nu+2}) = \omega_K \sum_{j=1}^{+\infty} (a_j \omega_j^{p+\nu+1} \nu_j^p - b_j \omega_j^p \nu_j^{p+\nu+1}) \\ - \sum_{j=1}^{+\infty} ((a_j \omega_j \nu_j) \omega_j^{p+\nu} \nu_j^p - (b_j \omega_j \nu_j) \omega_j^p \nu_j^{p+\nu}). $$

We use the equality $\gamma_{2p+l}((a_j \omega_j \nu_j)_{j \ge 1}, (b_j \omega_j \nu_j)_{j \ge 1}) = \gamma_{2(p+1)+l}(a, b)$ and the induction hypothesis to obtain the result, with $\beta_l^{\nu+2} = \beta_l^{\nu+1} - \beta_{l-2}^{\nu}$. $\square$

**Corollary A.5.** Assume that $(\mathbf{H})_{\text{conv}}$ holds. Let $\nu \in \mathbb{N}$, there exist $(\beta_l^\nu)_{l \in [0, \nu]}$, $(\delta_l^\nu)_{l \in [0, \nu]} \in \mathbb{Z}^{\nu+1}$ such that, for every $p \in \mathbb{N}$ satisfying $2p + \nu \le 2k - 1$, for every $1 \le \ell \le L \le r$,

$$ \sum_{j=1}^{+\infty} (c_j^{\ell,L} \omega_j^{p+\nu} \nu_j^p - c_j^{L,\ell} \omega_j^p \nu_j^{p+\nu}) = \sum_{l=0}^{\nu} \beta_l^{\nu} (-1)^l \gamma_{2p+l}^{\ell,L} \omega_K^{\nu-l}, $$

$$ \sum_{j=1}^{+\infty} (c_j^{L,\ell} \omega_j^{p+\nu} \nu_j^p - c_j^{\ell,L} \omega_j^p \nu_j^{p+\nu}) = \sum_{l=0}^{\nu} \delta_l^{\nu} (-1)^l \gamma_{2p+l}^{\ell,L} \omega_K^{\nu-l}. $$

Moreover, $\beta_0^1 = \beta_1^1 = 1$.

*Proof.* The first point is a direct consequence of Lemma A.4 with $a = c^{\ell,L}$ and $b = c^{L,\ell}$. The second one can be proved in the same way (induction). $\square$
**Remark A.6.** Assume that $\mu_1, \dots, \mu_r \in C_c^\infty(0,1)$ so that $(\mathbf{H})_{\text{conv}}$ holds by Remark 1.5. Then, for all $p, \nu \in \mathbb{N}$ with $2p + \nu \le 2k - 1$, for all $1 \le \ell \le L \le r$, a bracket computation gives

$$ \langle [\underline{ad}_A^p(\mu_\ell), \underline{ad}_A^{p+\nu}(\mu_L)]\varphi_1, \varphi_K \rangle = (-1)^\nu \sum_{j=1}^{+\infty} (c_j^{\ell,L} \omega_j^{p+\nu} \nu_j^p - c_j^{L,\ell} \omega_j^p \nu_j^{p+\nu}). \quad (A.4) $$

Using (A.4) and [35, Proposition A.11], the first expansion of Corollary A.5 can be written

$$ \langle [\underline{ad}_A^p(\mu_\ell), \underline{ad}_A^{p+\nu}(\mu_L)]\varphi_1, \varphi_K \rangle \\ = \sum_{l=0}^{\nu} \beta_l^{\nu} (-1)^l \langle \underline{ad}_A^{\nu-l} \left( \left[ \underline{ad}_A^{p+\left\lfloor \frac{l+1}{2} \right\rfloor}(\mu_\ell), \underline{ad}_A^{p+\left\lfloor \frac{l}{2} \right\rfloor}(\mu_L) \right] \right) \varphi_1, \varphi_K \rangle. $$

Thus, Corollary A.5 is the equivalent of [34, Lemma A.1] in the infinite-dimensional case. Moreover, the sequences defined in Corollary A.5 and [34, Lemma A.1] are the same.

## Acknowledgement

The author would like to sincerely thank Karine Beauchard and Frédéric Marbach for the many discussions that shaped this article. The author acknowledges support from grants ANR-20-CE40-0009 (Project TRECOS) and ANR-11-LABX-0020 (Labex Lebesgue), as well as from the Fondation Simone et Cino Del Duca – Institut de France.

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