On Robust Fixed-Time Stabilization of the Cauchy Problem in Hilbert Spaces

TL;DR

This paper extends finite-time and fixed-time stabilization theories to infinite-dimensional Hilbert space systems, providing well-posedness, settling time estimates, and partial stabilization with practical applications like heat equations with memory.

Contribution

It introduces new stabilization results for inhomogeneous evolution problems in Hilbert spaces, generalizing finite-dimensional theories and addressing partial state stabilization.

Findings

Established well-posedness for strong and weak solutions.

Derived upper bounds for settling times.

Demonstrated partial stabilization in heat equations with memory.

Abstract

This paper presents finite-time and fixed-time stabilization results for inhomogeneous abstract evolution problems, extending existing theories. We prove well-posedness for strong and weak solutions, and estimate upper bounds for settling times for both homogeneous and inhomogeneous systems. We generalize finite-dimensional results to infinite-dimensional systems and demonstrate partial state stabilization with actuation on a subset of the domain. The interest of these results are illustrated through an application of a heat equation with memory term.