A Maximum Principle for Optimal Control of Stochastic Evolution Equations

Abstract

A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable and linear unbounded operators act in both drift and diffusion terms, and the control set need not be convex. Abstract. A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable, as well as linear unbounded operators, acts in both drift and diffusion terms, and the control set need not be convex.

Necessary conditions for stochastic optimal control problems in infinite dimensions

“…One has a similar meaning for auu[t] v(t), v(t) and so on. 4 Note that the definition of S(t, x, u, y1, z1, y2, ω) in (5.7) is different from that in [5, the equality (4.4), p. 3708]. The main reason for this is due to the fact that the characterization of Q(·) in Theorem 2.2 is much weaker than the one in the finite dimensions.…”

“…For the general case when the diffusion term of the stochastic control system depends on the control variable and the control region lacks convexity, stimulated by the work [16] addressing the same problems but in finite dimensions (i.e., dim H < ∞), because presently dim H = ∞, one has to handle a difficult problem of the wellposedness of the second order adjoint equation, which is an operator-valued backward stochastic evolution equation (2.9) (see the next section). This problem was solved at almost the same time in [4,6,13], in which the Pontryagin-type maximum principles were established for optimal controls of general infinite dimensional nonlinear stochastic systems. Nevertheless, the techniques used in [13] and [4,6] are quite different.…”

“…This problem was solved at almost the same time in [4,6,13], in which the Pontryagin-type maximum principles were established for optimal controls of general infinite dimensional nonlinear stochastic systems. Nevertheless, the techniques used in [13] and [4,6] are quite different. Indeed, since the most difficult part, i.e., the correction term "Q 2 (t)" in (2.9) does not appear in the final formulation of the Pontryagin-type maximum principle, the strategy in both [4] and [6] is to ignore completely Q 2 (t) in the well-posedness result for (2.9).…”

“…Nevertheless, the techniques used in [13] and [4,6] are quite different. Indeed, since the most difficult part, i.e., the correction term "Q 2 (t)" in (2.9) does not appear in the final formulation of the Pontryagin-type maximum principle, the strategy in both [4] and [6] is to ignore completely Q 2 (t) in the well-posedness result for (2.9). By contrast, [13] characterized the above mentioned Q 2 (t) because it was anticipated that this term should be useful somewhere else.…”

First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints

“…Nevertheless, for a long time, almost all of the works on the necessary conditions for optimal controls of infinite dimensional SEEs addressed only the case that the diffusion term does NOT depend on the control variable (i.e., the function b(·, ·, ·) in (1.1) is independent of u). As far as we know, the stochastic maximum principle for general infinite dimensional nonlinear stochastic systems with control-dependent diffusion coefficients and possibly nonconvex control domains had been a longstanding problem till the very recent papers ( [10,18,29,30,31]). In these papers first order necessary optimality conditions for controlled SEEs are established by several authors with no constraint on the state.…”