Yuri S. Ledyaev | Searcharxiv

arXiv2014

Pontryagin Maximum Principle for Control Systems on Infinite Dimensional Manifolds

We discuss a mathematical framework for analysis of optimal control problems on infinite-dimensional manifolds. Such problems arise in study of optimization for partial differential equations with some symmetry. It is shown that some nonsmooth analysis methods and Lagrangian charts techniques can be used for study of global variations of optimal trajectories of such control systems and derivation of maximum principle for them.

math.OC↗

arXiv2014

Extension of Chronological Calculus for Dynamical Systems on Manifolds

We propose an extension of the Chronological Calculus, developed by Agrachev and Gamkrelidze for the case of $C^\infty$-smooth dynamical systems on finite-dimensional $C^\infty$-smooth manifolds, to the case of $C^m$-smooth dynamical systems and infinite-dimensional $C^m$-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.

math.OC↗