Reachability analysis for controlled systems subjected to stochastic disturbances

Abstract

Among the recent problems of control of special interest are those of reachability under disturbances and resets of system dynamics. The present paper deals with the problem of reachability under stochastic disturbances. Considered is a linear system subjected to perturbations generated by Brownian noise which may or may not be dependent on the values of the control. The latter in its turn may be either unbounded or bounded by hard bounds. The reachabilty sets introduced here are deterministic. They consist of all points whose mean -square deviation from some controlled trajectory is small. These sets are presented in terms of level sets to solutions of certain types of the Hamilton-Jacobi-Bellman equation. The respective controls are explicitly calculated when the controls are unbounded and are presented in terms of solutions to some dual optimization problems when the controls are bounded.