On Converse Lyapunov Theorem for Fixed-Time Input-to-State Stability
Denis Efimov, Andrey Polyakov
Abstract
Input-to-state stability is one of the most utilizable robust stability properties for nonlinear dynamical systems, while (nearly) fixed-time convergence is a kind of decay for trajectories of disturbance-free systems that is independent in initial conditions. The presence of both these features for a system can be checked by existence of a proper Lyapunov function. The objective of this work is to provide the conditions for a converse result that (nearly) fixed-time input-to-state stable systems admit a respective Lyapunov function. Similar auxiliary results for uniform finite-time stability and uniform (nearly) fixed-time stability are obtained.
Topics & Concepts
MathematicsLyapunov functionConverseControl theory (sociology)Stability (learning theory)Lyapunov redesignLyapunov equationFixed pointStability theoryLyapunov exponentState (computer science)Function (biology)Control-Lyapunov functionApplied mathematicsNonlinear systemMathematical analysisControl (management)Computer scienceAlgorithmEvolutionary biologyMachine learningArtificial intelligenceGeometryQuantum mechanicsPhysicsBiologyAdaptive Control of Nonlinear SystemsStability and Control of Uncertain SystemsControl and Stability of Dynamical Systems