Stochastic Control on Hilbert Space for Linear Evolution Equations with Random Operator-Valued Coefficients

Abstract

We consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and Bensoussan and Voit [SIAM J. Control Optim., 13 (1975), pp. 904–926].