## Project CEMAPRE internal

Title | Control of evolution equations and Markov processes |

Participants | Manuel Guerra (Principal Investigator), Andrey Sarychev |

Summary | Traditional approaches to control of Markov processes focus on controlling sample trajectories. An alternative approach concentrates on control of conditional distributions: one intends to study controlled evolution equations (such as Kolmogorov or Fokker-Planck equations) in the space of probability measures. In general, these are infinite-dimensional evolution equations of a distinct type which, up to now, lacked attention in the control-theoretic literature. In many interesting cases the equations are nonlinear. Our research will focus on controlled Ito processes. For these processes, the control enters nonlinearly in the infinitesimal generator of the respective evolution equation. We will study the accessibility and controllability properties of such evolution equations in full state space, in projections on the space of moments, and in truncated functional series. We intend to proceed by Lie algebraic methods of geometric control, which relate the controllability properties to the structure of the Lie algebra generated by the operators which enter the evolution equation. Markov processes taking values in separable spaces are approximable in probabilistic sense by finite-state (continuos-time) Markov processes. The Kolmogorov and Fokker-Planck equations of these processes reduce to finite-dimensional systems of ODE's and in control version, to bilinear control systems. We start by studying the accessibility and controllability properties in the space of distributions of finite-state continuous-time Markov systems with n states. We believe that results in this direction will provide useful insights into the properties of the full infinite-dimensional systems. |