Control problems with random and progressively known targets (Q1095105)

The situation considered is of optimally controlling a deterministic system from a given state to an initially unknown target y in a fixed time interval \([\tau _ 0,\tau]\). The target will be certainly known at a random time \(\tau\) in \([\tau _ 0,\tau]\). The controller knows the distributions of y and \(\tau\). We derive the Bellman equation for the problem, prove a verification theorem for it, and demonstrate how the distribution \(\tau\) influences the optimal control. We show that, in the linear-quadratic case, the optimal control is given by a feedback law which does not depend on the distribution of \(\tau\).