Stochastic Reachability and Measurement Feedback under Control-Dependent Noise

Abstract

This paper deals with reachability under unknown disturbances and incomplete information on the state space variables. The unknown disturbances are described by a special type of vector-valued stochastic Brownian input noise which depends on the values of vector-valued control. The control may be either unbounded or bounded by hard bounds. The reachabilty sets introduced here are deterministic. They consist of all points whose mean-square deviations from a tube of given controlled trajectories are small. The “reach” sets are presented in terms of level sets to solutions of appropriate types of Hamilton-Jacobi-Bellman equation, which depend on the presence or absence of additional hard bounds on the controls. These allow explicit representation of the reach sets when the controls are unbounded and are presented in terms of solutions to some dual optimization problems when the controls are bounded. Accordingly, the reach sets are either ellipsoids or convex compact sets of more complicated structure. The last fact introduces significant changes with transition from scalar to vector-valued control-dependent noise. Finally the notions of reachability and control under incomplete feedback are introduced.