Robust Stackelberg Controllability for the Kuramoto-Sivashinsky Equation

TL;DR

This paper investigates a hierarchical robust control framework for the nonlinear Kuramoto-Sivashinsky equation, combining Stackelberg and saddle point strategies, and develops iterative algorithms for control approximation.

Contribution

It introduces a novel robust Stackelberg controllability approach for a nonlinear fourth-order PDE, with algorithms for practical implementation.

Findings

Successful formulation of a hierarchical control problem for the Kuramoto-Sivashinsky equation.

Development of iterative algorithms for robust control and Stackelberg strategy.

Verification of controllability to trajectories under the proposed framework.

Abstract

In this article the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely, the Kuramoto-Sivashinsky equation. When three external sources are acting into the system, the RSC problem consists essentially in combining two subproblems: the first one is a saddle point problem among two sources. Such an sources are called the "follower control" and its associated "disturbance signal". This procedure corresponds to a robust control problem. The second one is a hierarchic control problem (Stackelberg strategy), which involves the third force, so-called leader control. The RSC problem establishes a simultaneous game for these forces in the sense that, the leader control has as objective to verify a controllability property, while the follower control and perturbation solve a robust control problem. In this paper the leader control obeys to the exact controllability to the trajectories. Additionally, iterative algorithms to approximate the robust control problem as well as the robust Stackelberg strategy for the nonlinear Kuramoto-Sivashinsky equation are developed and implemented.