Stackelberg-Nash Controllability for Abstract Stochastic Evolution Equations and Applications

TL;DR

This paper develops a framework for controllability in stochastic evolution equations with leader-follower dynamics, establishing Nash equilibrium existence, duality principles, and applying results to stochastic heat equations with new Carleman estimates.

Contribution

It introduces a novel controllability concept in stochastic Stackelberg-Nash settings and characterizes Nash equilibria for abstract stochastic evolution equations.

Findings

Existence and uniqueness of Nash equilibrium proven.

Duality between controllability and observability established.

Application to stochastic heat equations with new Carleman estimates.

Abstract

This paper presents the concepts of exact, null, and approximate controllability in the Stackelberg-Nash sense for abstract forward and backward stochastic evolution equations, involving two types of controls: leaders and followers. We begin by proving the existence and uniqueness of the Nash equilibrium, as well as its characterization for fixed leader controls. We then establish a duality between these controllability concepts and the corresponding observability properties. Finally, we apply our theoretical results to the forward and backward stochastic heat equations. The results for the backward heat equation are obtained by deriving a new Carleman estimate.