Maximum principle for optimal control of stochastic partial differential equations

TL;DR

This paper develops a stochastic maximum principle for optimal control of stochastic partial differential equations driven by martingales, without requiring convex control domains and allowing control in the martingale part.

Contribution

It derives necessary optimality conditions for stochastic PDE control problems with non-convex control sets and controls affecting the martingale component.

Findings

Established maximum principle without convexity assumptions.

Included controls in the martingale part of the SPDE.

Extended the theory to unbounded linear operators.

Abstract

We shall consider a stochastic maximum principle of optimal control for a control problem associated with a stochastic partial differential equations of the following type: d x(t) = (A(t) x(t) + a (t, u(t)) x(t) + b(t, u(t)) dt + [<\sigma(t, u(t)), x(t)>_K + g (t, u(t))] dM(t), x(0) = x_0 \in K, with some given predictable mappings $a, b, \sigma, g$ and a continuous martingale $M$ taking its values in a Hilbert space $K,$ while $u(\cdot)$ represents a control. The equation is also driven by a random unbounded linear operator $A(t,w), \; t \in [0,T ], $ on $K .$ We shall derive necessary conditions of optimality for this control problem without a convexity assumption on the control domain, where $u(\cdot)$ lives, and also when this control variable is allowed to enter in the martingale part of the equation.