Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces

TL;DR

This paper develops an optimal relaxed control framework for dissipative semilinear stochastic PDEs in Banach spaces, utilizing advanced stochastic analysis tools to handle multiplicative noise and control via the drift term.

Contribution

It introduces a novel approach combining the factorization method and Young measures to analyze relaxed controls in Banach space-valued stochastic PDEs with dissipative nonlinearities.

Findings

Established existence of optimal relaxed controls under dissipativity conditions.

Demonstrated the applicability of the factorization method in UMD Banach spaces.

Provided compactness results for Young measures on control sets.

Abstract

We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.