Necessary and Sufficient Conditions for Input-Output Finite-Time Stability of Linear Time-Varying Systems F. Amato § , G. Carannante † , G. De Tommasi † , A. Pironti † Abstract The problem of Input-output finite-time stabilization is tackled in this paper. A system is said to be input-output finite-time stable (IO-FTS) if, given a class of norm bounded input signals over a specified time interval T , the outputs of the system do not exceed an assigned threshold during T . IO-FTS constraints permit to specify quantitative bounds on the controlled variables to be fulfilled during the transient response. By using an approach based on Reachability Gramian theory, we present a necessary and sufficient condition for IO-FTS of linear time-varying systems when the case of L 2 is considered. We also prove that this condition requires the solution of a feasibility problem involving differential linear matrix inequalities (DLMIs). We show that the condition based on the Reachability Gramian is computationally more efficient, however the DLMIs formulation permits to solve the problem of IO finite-time stabilization via output feedback. The effectiveness of the two approaches for the analysis and the synthesis, respectively, is shown in the proposed examples. Keywords: Linear systems; time-varying systems; Reachability Gramian; IO-FTS; LMIs; DLMIs. I. INTRODUCTION TBD Notation. Given a vector v ∈ R m we will denote with |v| q its q-norm. Given the set Ω=[t 0 ,t 0 + T ], with t 0 ∈ R and T> 0, the symbol L p,q (Ω) denotes the space of vector-valued signals for which v ∈L p,q (Ω) = ⇒ ∫ Ω |v(t)| p q dt < +∞ , § F. Amato is with the School of Computer Science and Biomedical Engineering, Universit` a degli Studi Magna Græcia di Catanzaro, Via Tommaso Campanella 115, 88100 Catanzaro, Italy. † G. Carannante, G. De Tommasi and A. Pironti are with the Dipartimento di Informatica e Sistemistica, Universit` a degli Studi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy.