On Asymptotic and Finite-Time Stabilization of Bilinear Systems

Abstract

This manuscript addresses the analysis and design of feedback laws for the stabilization of bilinear control systems in infinite-dimensional spaces. It first examines weak, strong, and polynomial stabilization within a Hilbert space framework, emphasizing the role of observability conditions. It then studies exponential stabilization in Banach spaces, highlighting the additional challenges arising from the lack of a Hilbertian structure. Finally, it introduces finite-time stabilization, presenting recent results and open problems within the broader context of nonlinear infinite-dimensional control theory. Several applications are also discussed.

This manuscript contains 56 pages and was, for the most part, delivered as a course at the Spring School on Control of Dynamical Systems, Corsica, March 16--20, 2026