Existence of optimal controls for stochastic partial differential equations with fully local monotone coefficients

TL;DR

This paper proves the existence of optimal feedback controls for a class of controlled stochastic partial differential equations with fully monotone coefficients, using approximation and tightness arguments.

Contribution

It establishes the existence of stochastic optimal feedback controls for SPDEs with fully monotone coefficients via Faedo-Galerkin approximations and tightness methods.

Findings

Existence of optimal feedback controls for SPDEs with fully monotone coefficients.

Well-posedness of controlled SPDEs with these coefficients.

Applicability to controlled stochastic convection diffusion equations.

Abstract

This paper deals with a stochastic optimal feedback control problem for the controlled stochastic partial differential equations. More precisely, we establish the existence of stochastic optimal feedback control for the controlled stochastic partial differential equations with fully monotone coefficients by a minimizing sequence for the control problem. Using the Faedo-Galerkin approximations, the uniform estimates and the tightness in some appropriate space for the Faedo-Galerkin approximating solution can be obtain to prove the well-posedness of the controlled stochastic partial differential equations with fully monotone coefficients. The results obtained in the present paper may be applied to various types of controlled stochastic partial differential equations, such as the controlled stochastic convection diffusion equation.