On a Class of Stochastic Bang-Bang Control Problems

Abstract

To begin with, let us recall a well known separation result for a stochastic linear regulator problem [6],[9],[I]. Let (Ω,P) be the underlying probability space and consider the following system:

$$\begin{gathered} dx(t,w) = [A(t)x(t,w) + B(t)u(t,w)]dt + F(t)dW(t,w), \hfill \\ x(0,w) = {x_0}(w), \hfill \\ dy(t,w) = C(t)x(t,w)dt + G(t)dW(t,w),y(0,w) = 0, \hfill \\ 0 \leqslant t \leqslant {t_1}, \hfill \\ \end{gathered} $$

((1.1))

where x, y, u correspond to state, observation and control processes respectively. W(t,ω) is a Wiener process, the matrices A, B,... are, say, continuous in t and of appropriate dimensions, X ₀ is a r.v. independent of W(t,ω), t ≥ 0.