Input–to–state stability finite–time lyapunov functions for continuous–time systems

Abstract

In this paper we propose an input-to-state stability (ISS) criterion for continuous–time systems based on a finite–time decrease condition for a positive definite function of the norm of the state. This yields a so–called ISS finite–time Lyapunov function, which allows for eas-ier choice of candidate functions compared to standard ISS Lyapunov functions. An alternative converse ISS theorem in terms of ISS finite– time Lyapunov functions is also provided. Moreover, we prove that ISS finite–time Lyapunov functions are equivalent with standard ISS Lya-punov functions using a Massera–type construction. The developed ISS framework can be utilized in combination with Sontag’s “univer-sal” stabilisation formula to develop input–to–state stabilizing control laws for continuous–time nonlinear systems that are affine in the control and disturbance inputs, respectively.