On reachability under uncertainty

The paper studies the problem of reachability for linear systems in the presence of uncertain ( unknown but bounded) input disturbances that may also be interpreted as the action of an adversary in a game-theoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between reachability under open-loop and closed-loop control. Solution schemes for calculating reachability sets are then indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) or nonanticipative (minmax) piecewise open-loop control with corrections and to the respective notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closed-loop reach sets under uncertainty may be found through a solution of the forward Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.