Norm and time optimal control problems of stochastic heat equations

TL;DR

This paper studies the control problems for stochastic heat equations, characterizing norm optimal controls, their relation to time optimal controls, and proving the bang-bang property, pioneering work in the stochastic setting.

Contribution

It introduces the first analysis of norm and time optimal control problems for stochastic heat equations, including their equivalence and bang-bang property.

Findings

Characterization of norm optimal control

Equivalence between norm and time optimal controls

Bang-bang property of time optimal control

Abstract

This paper investigates the norm and time optimal control problems for stochastic heat equations. We begin by presenting a characterization of the norm optimal control, followed by a discussion of its properties. We then explore the equivalence between the norm optimal control and time optimal control, and subsequently establish the bang-bang property of the time optimal control. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case.