Research projects

Project CEMAPRE internal

TitleControl of evolution equations and Markov processes
ParticipantsManuel Guerra (Principal Investigator), Andrey Sarychev
SummaryTraditional 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 (continuous-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.