On Reachability Under Uncertainty

The paper studies the problem of reachability for linear systems in the presence of uncertain input disturbances that may also be interpreted as the action of an adversary in a game-theoretic setting. It emphasizes the differences between reachability under open-loop and closed-loop control. 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. As the number of corrections tends to infinity, one comes in both cases to a nonanticipative feedback control strategy that ensures reachability under uncertainty and may be found through a solution of the forward Hamilton-Jacobi-Bellman-Isaacs equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.