Multi-objective and hierarchical control for coupled stochastic parabolic systems

TL;DR

This paper investigates the complex control problem of coupled stochastic parabolic systems with multiple leaders and followers, establishing existence, uniqueness, and new Carleman estimates for null controllability, pioneering work in stochastic coupled systems.

Contribution

It introduces a novel framework for multi-objective hierarchical control of stochastic coupled systems, including new Carleman estimates and equilibrium characterizations.

Findings

Existence and uniqueness of Nash equilibrium for the control system.

Reformulation as a null controllability problem for stochastic systems.

Development of new Carleman estimates for stochastic adjoint systems.

Abstract

We study the Stackelberg-Nash null controllability of a coupled system governed by two linear forward stochastic parabolic equations. The system includes one leader control localized in a subset of the domain, two additional leader controls in the diffusion terms, and \( m \) follower controls, where \( m \geq 2 \). We consider two different scenarios for the followers: first, when the followers minimize a functional involving both components of the system's state, and second, when they minimize a functional involving only the second component of the state. For fixed leader controls, we first establish the existence and uniqueness of the Nash equilibrium in both scenarios and provide its characterization. As a byproduct, the problem is reformulated as a classical null controllability issue for the associated coupled forward-backward stochastic parabolic system. To address this, we derive new Carleman estimates for the adjoint stochastic systems. As far as we know, this problem is among the first to be discussed for stochastic coupled systems.