# Chronological calculus

Chronological calculus is a formalism for the analysis of flows of non-autonomous dynamical systems. It was introduced by A. Agrachev and R. Gamkrelidze in the late 1970s. The scope of the formalism is to provide suitable tools to deal with non-commutative vector fields and represent their flows as infinite Volterra series. These series, at first introduced as purely formal expansions, are then shown to converge under some suitable assumptions.

## Operator representation of points, vector fields and diffeomorphisms

Let $M$ be a finite-dimensional smooth manifold.
Chronological calculus works by replacing a non-linear finite-dimensional object, the manifold $M$, with a linear infinite-dimensional one, the commutative algebra $C^{\infty }(M)$. This leads to the following identifications:
* Points $q\in M$ are identified with nontrivial algebra homomorphisms
    ${\hat {q}}:C^{\infty }(M)\to \mathbb {R}$ defined by ${\hat {q}}(a)=a(q)$.
* Diffeomorphisms $P:M\to M$ are identified with $C^{\infty }(M)$-automorphisms ${\hat {P}}:C^{\infty }(M)\to C^{\infty }(M)$ defined by $({\hat {P}}a)(q)=a(P(q))$.
* Tangent vectors $v\in T_{q}M$ are identified with linear functionals ${\hat {v}}:C^{\infty }(M)\to \mathbb {R}$ satisfying the Leibnitz rule ${\hat {v}}(ab)={\hat {v}}(a)b(q)+a(q){\hat {v}}(b)$ at $q$.
* Smooth vector fields $V\in {\text{Vec}}(M)$ are identified with linear operators ${\hat {V}}:C^{\infty }(M)\to C^{\infty }(M)$
satisfying the Leibnitz rule ${\hat {V}}(ab)={\hat {V}}(a)b+a{\hat {V}}(b)$.
In this formalism, the tangent vector $V(P(q))$ is identified with the operator $q\circ P\circ V$.
We consider on $C^{\infty }(M)$ the Whitney topology, defined by the family of seminorms
    $\displaystyle {\|a\|_{s,K}:=\sup \left\{\left|{\frac {\partial ^{\alpha }a}{\partial q^{\alpha }}}(q)\right|\,\mid q\in K,\,\alpha \in \mathbb {N} ^{{\rm {dim}}(M)},\;|\alpha |\leq s\right\},}$
Regularity properties of families of operators on $C^{\infty }(M)$ can be defined in the weak sense as follows: $t\to A_{t}$ satisfies a certain regularity property if the family $t\to A_{t}(a)$ satisfies the same property, for every $a\in C^{\infty }(M)$. A weak notion of convergence of operators on $C^{\infty }(M)$ can be defined similarly.

## Volterra expansion and right-chronological exponential

Consider a complete non-autonomous vector field $(t,q)\mapsto X_{t}(q)$ on $M$, smooth with respect to $q$ and measurable with respect to $t$. Solutions to ${\dot {q}}(t)=X_{t}(q(t))$, which in the operator formalism reads
| ${\frac {d}{dt}}q(t)=q(t)\circ X_{t},$ | (1) |
define the flow of $X_{t}$, i.e., a family of diffeomorphisms $t\to P^{t}$, $P^{0}=\mathrm {Id}$. The flow satisfies the equation
| ${\frac {d}{dt}}P^{t}=P^{t}\circ X_{t},\;P^{0}=\mathrm {Id} .$ | (2) |
Rewrite 2 as a Volterra integral equation $P^{t}=\mathrm {Id} +\int _{0}^{t}P^{\tau _{1}}\circ X_{\tau _{1}}d\tau _{1}$.
Iterating one more time the procedure, we arrive to
    ${\begin{aligned}P^{t}&=\mathrm {Id} +\int _{0}^{t}P^{\tau _{1}}\circ X_{\tau _{1}}d\tau _{1}=\mathrm {Id} +\int _{0}^{t}\left(\mathrm {Id} +\int _{0}^{\tau _{1}}P^{\tau _{2}}\circ X_{\tau _{2}}d\tau _{2}\right)\circ X_{\tau _{1}}d\tau _{1}\\&=\mathrm {Id} +\int _{0}^{t}X_{\tau _{1}}d\tau _{1}+\int _{0}^{t}\int _{0}^{\tau _{1}}P^{\tau _{2}}\circ X_{\tau _{2}}\circ X_{\tau _{1}}d\tau _{2}d\tau _{1}.\end{aligned}}$
In this way we justify the notation, at least on the formal level, for the right chronological exponential
| $P^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau =\mathrm {Id} +\sum _{n=1}^{+\infty }\int \dots \int _{\Delta _{n}(t)}\ X_{\tau _{n}}\circ \dots \circ X_{\tau _{1}}d\tau _{n}\dots d\tau _{1},$ | (3) |
where $\Delta _{n}(t)=\{(\tau _{1},\dots ,\tau _{n})\in \mathbb {R} ^{n}\,\mid \,0\leq \tau _{1}\leq \dots \leq \tau _{n}\leq t\}$ denotes the standard $n$-dimensional simplex.
Unfortunately, this series never converges on $C^{\infty }(M)$; indeed, as a consequence of Borel's lemma, there always exists a smooth function $a\in C^{\infty }(M)$ on which it diverges. Nonetheless, the partial sum
    $\displaystyle {S_{m}(t)=\mathrm {Id} +\sum _{n=1}^{m}\int \dots \int _{\Delta _{n}(t)}\ X_{\tau _{n}}\circ \dots \circ X_{\tau _{1}}d\tau _{n}\dots d\tau _{1}}$
can be used to obtain the asymptotics of the right chronological exponential: indeed it can be proved that, for every $a\in C^{\infty }(M)$, $s\geq 0$ and $K\subset M$ compact, we have
| $\left\|\left({\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau -S_{m}(t)\right)(a)\right\|_{s,K}\leq Ce^{C\int _{0}^{t}\|X_{\tau }\|_{s,K'}d\tau }{\frac {1}{m!}}\left(\int _{0}^{t}\|X_{\tau }\|_{s+m-1,K'}d\tau \right)^{m}\|a\|_{s+m,K'},$ | (4) |
for some $C>0$, where $K'=\cup _{s\in [0,t]}P^{s}(K)$. Also, it can be proven that the asymptotic series $S_{m}(t)$ converges, as $m\to +\infty$, on any normed subspace $L\subset C^{\infty }(M)$ on which $X_{t}$ is well-defined and bounded, i.e.,
    $\displaystyle {X_{t}(L)\subset L,\quad \|X_{t}\|_{L}:=\sup \left\{\|X_{t}(a)\|_{L}\,\mid \,a\in L,\,\|a\|_{L}\leq 1\right\}<+\infty .}$
Finally, it is worth remarking that an analogous discussion can be developed for the left chronological exponential $Q^{t}$, satisfying the differential equation
    $\displaystyle {{\frac {d}{dt}}Q^{t}=X_{t}\circ Q^{t},\;Q^{0}=\mathrm {Id} .}$

## Variation of constants formula

Consider the perturbed ODE
    $\displaystyle {{\frac {d}{dt}}P^{t}=P^{t}\circ (X_{t}+Y_{t}),\quad P^{0}=\mathrm {Id} .}$
We would like to represent the corresponding flow, $P^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}(X_{\tau }+Y_{\tau })d\tau$, as the composition of the original flow ${\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau$ with a suitable perturbation, that is, we would like to write an expression of the form
    $\displaystyle {P^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}(X_{\tau }+Y_{\tau })d\tau =R^{t}\circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau .}$
To this end, we notice that the action of a diffeomorphism $S$ on $M$ on a smooth vector field $W$, expressed as a derivation on $C^{\infty }(M)$, is given by the formula
    $\displaystyle {S_{*}W=S^{-1}\circ W\circ S=\mathrm {Ad} S^{-1}(W).}$
In particular, if $S^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}V_{\tau }d\tau$, we have
    ${\begin{aligned}{\frac {d}{dt}}(\mathrm {Ad} S^{t})W&={\frac {d}{dt}}S^{t}\circ W\circ S^{-t}=S^{t}\circ \left(V_{t}\circ W-W\circ V_{t}\right)\circ S^{-t}\\&=\mathrm {Ad} S^{t}[V_{t},W]=(\mathrm {Ad} S^{t})\mathrm {ad} V_{t}W.\end{aligned}}$
This justifies the notation
    $\displaystyle {\mathrm {Ad} S^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}\mathrm {ad} V_{\tau }d\tau .}$
Now we write
    $\displaystyle {{\frac {d}{dt}}P^{t}=P^{t}\circ (X_{t}+Y_{t})=P^{t}\circ X_{t}+R^{t}\circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau \circ Y_{t}}$
and
    $\displaystyle {{\frac {d}{dt}}P^{t}={\dot {R}}^{t}\circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau +R^{t}\circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau \circ X_{t}={\dot {R}}^{t}\circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau +P^{t}\circ X_{t}}$
which implies that
    $\displaystyle {{\dot {R}}^{t}=R^{t}\circ \left({\overrightarrow {\mathrm {exp} }}\int _{0}^{t}\mathrm {ad} X_{\tau }d\tau \right)Y_{t},\quad R^{0}=\mathrm {Id} .}$
Since this ODE has a unique solution, we can write
    $\displaystyle {R^{t}={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}\left({\overrightarrow {\mathrm {exp} }}\int _{0}^{\tau }\mathrm {ad} X_{\theta }d\theta \right)Y_{\tau }d\tau ,}$
and arrive to the final expression, called the variation of constants formula:
| ${\overrightarrow {\mathrm {exp} }}\int _{0}^{t}(X_{\tau }+Y_{\tau })d\tau ={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}\left({\overrightarrow {\mathrm {exp} }}\int _{0}^{\tau }\mathrm {ad} X_{\theta }d\theta \right)Y_{\tau }d\tau \circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau .$ | (5) |
Finally, by virtue of the equality $(\mathrm {Ad} P){\overrightarrow {\mathrm {exp} }}\int _{0}^{t}V_{\tau }d\tau ={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}(\mathrm {Ad} P)V_{\tau }d\tau$, we obtain a second version of the variation of constants formula, with the unperturbed flow ${\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau$ composed on the left, that is,
| ${\overrightarrow {\mathrm {exp} }}\int _{0}^{t}(X_{\tau }+Y_{\tau })d\tau ={\overrightarrow {\mathrm {exp} }}\int _{0}^{t}X_{\tau }d\tau \circ {\overrightarrow {\mathrm {exp} }}\int _{0}^{t}\left({\overrightarrow {\mathrm {exp} }}\int _{t}^{\tau }\mathrm {ad} X_{\theta }d\theta \right)Y_{\tau }d\tau .$ | (6) |

## Sources

* Agrachev, Andrei A.; Sachkov, Yuri L. (2004). "Elements of Chronological Calculus". Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences. Vol. 84. Springer. ISBN 9783662064047.
* Agrachev, Andrei A.; Gamkrelidze, Revaz V. (1978). "Exponential representation of flows and a chronological enumeration. (Russian)". Mat. Sb. New Series. 107 (149): 467–532, 639.
* Agrachev, Andrei A.; Gamkrelidze, Revaz V. (1980). "Chronological algebras and nonstationary vector fields. (Russian)". Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii. 11: 135–176.
* Kawski, Matthias; Sussmann, Héctor (1997). Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. European Consort. Math. Indust. Teubner, Stuttgart. pp. 111–128.
* Kawski, Matthias (2002). The combinatorics of nonlinear controllability and noncommuting flows. ICTP Lect. Notes, VIII. Abdus Salam Int. Cent. Theoret. Phys., Trieste. pp. 223–311.
* Sarychev, Andrey V. (2006). "Lie extensions of nonlinear control systems". Journal of Mathematical Sciences. 135 (4): 3195–3223. doi:10.1007/s10958-006-0152-4.