Finite-Time Input-To-State Stability for Infinite-Dimensional Systems

In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite-dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS-LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS-LF for a class of abstract infinite-dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in-domain disturbances. Since the non-linear terms of the corresponding abstract system are not Lipschitz continuous, the well-posedness is proved based on the application of compact semigroup theory, and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.