Linear Quadratic Optimal Control Problems

Abstract

In this chapter, we are concerned with linear quadratic optimal control problems (LQ problems for short) for stochastic evolution equations, in which the diffusion terms depend on the control variables and the coefficients are stochastic. In such a general setting, one has to introduce suitable operator-valued backward stochastic evolution equations (to characterize the optimal controls in the form of Pontryagin-type maximum principle or in the feedback forms), served as the second order adjoint equations or the Riccati type equations. As in the previous chapter, it is very difficult to show the existence of solutions to these equations. We shall use the stochastic transposition method to overcome this difficulty.