# Duality and upper bounds in optimal stochastic control governed by partial differential equations

## TL;DR

This paper introduces a dual control framework for stochastic PDE-governed systems, establishing duality theorems that help bound and identify the optimal control values.

## Contribution

It develops a dual control problem for stochastic PDE systems and proves duality theorems linking the original and dual problems, aiding in optimal value estimation.

## Key findings

- Duality theorems relate original and dual problem values.
- Dual problem provides upper bounds for the original control problem.
- The approach helps in estimating and achieving optimal control values.

## Abstract

A dual control problem is presented for the optimal stochastic control of a system governed by partial differential equations. Relationships between the optimal values of the original and the dual problems are investigated and two duality theorems are proved. The dual problem serves to provide upper bounds for the optimal and maximum value of the original one or even to give the optimal value.