A Decomposition Method for Finite-Time Stabilization of Bilinear Systems with Applications to Parabolic and Hyperbolic Equations

TL;DR

This paper introduces a decomposition-based method for finite-time stabilization of bilinear systems, focusing on infinite-dimensional parabolic and hyperbolic equations, by splitting the system into stable and controllable parts.

Contribution

A novel decomposition approach that simplifies finite-time stabilization analysis for bilinear systems, applicable to parabolic and hyperbolic equations, with established conditions for well-posedness.

Findings

Successfully stabilizes infinite-dimensional systems within finite time

Applicable to both parabolic and hyperbolic equations

Provides sufficient conditions for system and control operator well-posedness

Abstract

In this work, we address the problem of finite-time stabilization for a class of bilinear system. We propose a decomposition-based approach in which the nominal system is split into two subsystems, one of which is inherently finite-time stable without control. This allows the stabilization analysis to focus solely on the remaining subsystem. To ensure the well-posedness of the closed-loop system, we establish sufficient conditions on the system and control operators. The stabilization results are then derived using a suitable Lyapunov function and an observation condition. The effectiveness of the proposed approach is demonstrated through examples involving both parabolic and hyperbolic infinite-dimensional systems.