The solution of a partially observed stochastic optimal control problem in terms of predicted miss

A stochastic adaptive control system whose parameters vary according to a finite-state jump Markov process was considered earlier by P. E. Caines and H. F. Chen (IEEE Trans. Automatic Control, vol. AC-30, 1985). They recognized, by using the nonlinear filtering equations for the conditional probabilities of the parameter states, that the control problem can be converted into a completely observed control problem. They then gave a verification theorem for checking that a control is optimal. However, they did not solve any examples and it appears that there have not been any previously solved examples of this type of `adaptive´ control system. The purpose of this study is to provide an explicit solution for a linear quadratic (LQ) problem of this type. The explicit solution of a partially observed LQ-problem driven by a combination of a Wiener process and an unobserved finite-state jump Markov process is given