Finite-Time Stabilization of Evolution Equations with Maximal Monotone Maps in Hilbert Space
Résumé
Building on the concept of exact solutions combined with the Cauchy problem and Lyapunov functions, we investigate a robust finite-time and fixed-time stability of abstract systems under nonlinear feedback control laws. First, we design a stabilizing feedback control based on a set-valued map, which precisely rejects matched perturbations and ensures finite-time convergence. Subsequently, an additional nonlinear power term is incorporated to achieve fixed-time convergence. For both cases, we analyze the well-posedness of the closed-loop system using the theory of maximal monotone operators. Furthermore, we extend the analysis to study partial finite-time stability for the same class of abstract systems. The theoretical findings are applied to derive robust partial finite-time stabilization results for the heat equation.
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