Stackelberg-Nash null controllability for stochastic parabolic equations

TL;DR

This paper investigates hierarchical control of stochastic parabolic equations using a Stackelberg-Nash framework, establishing null controllability through Carleman estimates and observability inequalities.

Contribution

It introduces a novel hierarchical control approach for stochastic parabolic equations with multiple leaders and followers, and proves null controllability via new Carleman estimates.

Findings

Established null controllability for stochastic parabolic equations.

Developed a new observability inequality using Carleman estimates.

Formulated a Stackelberg-Nash control strategy for complex stochastic systems.

Abstract

We study a hierarchical control problem for stochastic parabolic equations involving gradient terms. We employ the Stackelberg-Nash strategy with two leaders and two followers. The leaders are responsible for selecting the policy targeting null controllability, while the followers solve a bi-objective optimal control problem which consists of maintaining the solution process close to prefixed targets. Once the Nash equilibrium is determined, the problem reduces to achieving null controllability of a coupled forward-backward stochastic system. To solve this problem, via Carleman estimates, we establish a suitable observability inequality. Subsequently, we achieve the desired controllability result.