REACHABILITY ANALYSIS UNDER CONTROL-DEPENDENT STOCHASTIC NOISE

Abstract

The list of important problems of modern control theory includes those of reachability under under unknown disturbances, incomplete information on the system model and possible resets of system dynamics. The present paper deals special type of stochastic disturbances, when a linear system is subjected to perturbations generated by Brownian noise. The latter is assumed to depend on the values of the control, which in its turn may be either unbounded or bounded by hard bounds. The ”reach” sets introduced here are deterministic. They consist of all points whose mean-square deviation from some controlled trajectory is small. These are presented in terms of level sets to solutions of appropriate Hamilton-Jacobi-Bellman (HJB) equations. The solutions to the HJB equations then allow explicit representation when the controls are unbounded and are given here in terms of solutions to some dual optimization problems when the controls are bounded. Accordingly, the reach sets are either ellipsoids or Euclidean neighborhoods of the reach sets to the respective averaged system.