Extension of Chronological Calculus for dynamical systems on manifolds

• Extension of the classical Chronological Calculus for infinite-dimensional manifolds is developed.
• This extension is suitable for flows of vector fields which are measurable in time (control dynamical systems).
• The central product rule for such systems in distributions form is derived and it is applied for derivation of formulas of variation of parameters and etc. for operator differential equations.
• Theory of operator remainder terms is developed and formula for representation of brackets of flows in terms of brackets of vector fields is obtained.
• Infinite-dimensional variant of Chow–Rashevskii theorem is proven by using results of the paper.

We propose an extension of the Chronological Calculus, developed by Agrachev and Gamkrelidze for the case of C∞C∞-smooth dynamical systems on finite-dimensional C∞C∞-smooth manifolds, to the case of CmCm-smooth dynamical systems and infinite-dimensional CmCm-manifolds. Due to a relaxation in the underlying structure of the calculus, this extension provides a powerful computational tool without recourse to the theory of calculus in Fréchet spaces required by the classical Chronological Calculus. In addition, this extension accounts for flows of vector fields which are merely measurable in time. To demonstrate the utility of this extension, we prove a variant of Chow–Rashevskii theorem for infinite-dimensional manifolds.