Maximum principle for optimal control of stochastic partial differential equations

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 + [<σ(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.

References in corpus (1)

• Differentiability of quadratic BSDEs generated by continuous martingales