Finite-time stabilization of a class of unbounded parabolic bilinear systems in Hilbert space

Abstract

This paper investigates the finite-time stabilization of a class of unbounded bilinear systems in Hilbert spaces. Our methodology involves decomposing the original system into two interconnected subsystems: one that either inherently possesses finite-time stability or lacks it regardless of the feedback control applied, and another for which an appropriate feedback control must be designed to ensure the desired stability properties. By employing the theory of maximal monotone operators and Lyapunov-based methods, we establish sufficient conditions for finite-time stability without assuming the coercivity of the control operator. Applications are presented for heat and biharmonic heat equations, demonstrating the practical relevance of the results. This work extends existing frameworks and provides a broader theoretical foundation for the study of unbounded bilinear systems.