Finite-Time Stabilizability, Detectability, and Dynamic Output Feedback Finite-Time Stabilization of Linear Systems

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In this note, we deal with linear systems and we extend to the finite-time setting the concepts of stabilizability and detectability. It will be shown that, similarly to what happens in the classical Lyapunov framework, even in the finite-time context, stabilizability and detectability play a role into the existence of stabilizing dynamical controllers. We prove that a dynamic output feedback controller, which finite-time stabilizes the overall closed-loop system, exists if and only if the open-loop system is finite-time detectable and stabilizable plus a further linear matrix inequality coupling condition. We also show that, in the finite-time context, the equivalence between stabilizability via output feedback and stabilizability via observer-based controllers is no longer true.