Multi-objective and hierarchical control for coupled stochastic parabolic systems
Abdellatif Elgrou, Omar Oukdach
TL;DR
The work addresses Stackelberg-Nash null controllability for a coupled system of two forward stochastic parabolic equations with three leader controls and $m\ge 2$ followers. The authors develop Nash equilibrium existence, uniqueness, and a precise adjoint-based characterization for two follower objective types, and reformulate the control problem as a null controllability problem for a forward-backward stochastic system. Central to the analysis are new Carleman estimates and derived observability inequalities, which yield existence and energy bounds for leader controls that drive the state to zero at final time almost surely. The results advance hierarchical control of stochastic PDEs with diffusion coupling, providing a rigorous framework for multi-objective steering under stochastic dynamics and Dirichlet boundary conditions.
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.