# Existence of optimal controls for SPDE with locally monotone coefficients

## TL;DR

This paper proves the existence of optimal feedback controls for a class of stochastic partial differential equations with locally monotone coefficients, extending previous results to various types of controlled SPDEs.

## Contribution

It adapts existing methods to establish the existence of optimal controls for locally monotone SPDEs, including nonlocal, semilinear, reaction diffusion, and linear equations.

## Key findings

- Existence of optimal controls for locally monotone SPDEs established.
- Results applicable to stochastic nonlocal, semilinear, reaction diffusion, and linear equations.
- Method extends previous work on stochastic Navier-Stokes equations.

## Abstract

The aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations (SPDEs) with locally monotone coefficients controlled by different external forces which are feedback controls. To attain our objective we adapt the argument of H. Lisei, (Existence of optimal and epsilon-optimal controls for the stochastic Navier - Stokes equation, Nonlinear Analysis, 51, (2002) 95-118) where the existence of optimal control to the stochastic Navier-Stokes equation was studied. The results obtained in the present paper may be applied to demonstrate the existence of optimal control to various types of controlled SPDEs such as: a stochastic nonlocal equation and stochastic semilinear equations which are locally monotone equations; we also apply the result to a monotone equation such as the stochastic reaction diffusion equation and to a stochastic linear equation.