A Hierarchical Robust Control Strategy for Stochastic Kuramoto--Sivashinsky--Korteweg--de Vries Equations

TL;DR

This paper develops a hierarchical control strategy for a stochastic KS-KdV equation, addressing null controllability under worst-case disturbances using advanced Carleman estimates.

Contribution

It introduces a novel Stackelberg game framework for stochastic KS-KdV equations and establishes null controllability via new Carleman estimates.

Findings

Existence of a saddle point characterizing the robust control problem.

Reduction to null controllability of a coupled stochastic system.

Development of new Carleman estimates for stochastic fourth-order equations.

Abstract

We investigate the robust Stackelberg null controllability of a one-dimensional forward linear stochastic Kuramoto--Sivashinsky--Korteweg--de Vries (KS--KdV) equation. The control framework is formulated as a hierarchical Stackelberg game involving two leaders, one follower, and worst-case disturbances acting in both the drift and diffusion terms. The first leader acts to drive the system to rest, while the second leader is introduced to overcome analytical difficulties arising from the stochastic setting. The follower, by reducing the effect of the disturbances, addresses a tracking-type control problem aimed at keeping the system state and its first and second spatial derivatives close to prescribed target trajectories. First, the robust control problem is characterized by the existence of a saddle point. Then, the analysis is reduced to the null controllability of a strongly coupled forward--backward stochastic KS--KdV system. The problem is addressed by combining a duality technique with new Carleman estimates for forward and backward stochastic fourth-order parabolic equations.