# Maximum principle for an optimal control problem associated to a SPDE with nonlinear boundary conditions

## TL;DR

This paper develops a maximum principle for an optimal control problem involving a stochastic PDE with nonlinear boundary conditions, where control acts on the boundary and influences the boundary dynamics.

## Contribution

It introduces necessary and sufficient optimality conditions for boundary-controlled stochastic PDEs with nonlinear boundary conditions, extending classical maximum principles.

## Key findings

- Derived maximum principle for boundary control of SPDEs
- Proved existence of optimal control in linear boundary control case
- Established conditions for optimality in nonlinear boundary settings

## Abstract

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.