A note on interrupted stochastic control processes

Consider a discrete stochastic control process in which the state of the system at time n is specified by the state vector χn, the control vector is yn, and the change of state is determined by the relation xn+1=g(xn,yn,rn),x0=c, where rn is a sequence of independent random vectors with common known distribution. If we ask that feedback control, as represented by the yz, be applied in such a way as to minimize the expected value of some prescribed function of the terminal state χN, a straight-forward dynamic programming treatment yields an algorithm for the solution.Suppose we now consider the foregoing process with the added feature that there is a chance at any particular stage that the true state of the system will not be known to the decision maker. We shall call this an interrupted stochastic control process.The problem of determining optimal control in a situation of this type was posed to us by J. Craig of The RAND Corporation. As we shall see, processes of this type possess some interesting and novel aspects and present some complex analytic and computational questions