Lyapunov methods for input-to-state stability of time-varying evolution equations
Rahma Heni, Andrii Mironchenko, Fabian Wirth, Hanen Damak, Mohamed Ali Hammami
TL;DR
This work extends input-to-state stability theory to time-varying infinite-dimensional systems by proving that (L)ISS and iISS follow from the existence of Lyapunov functions in the setting of evolution families $W(t,s)$. It develops Lyapunov criteria for both linear and semilinear systems, including non-coercive LISS and iISS constructions, and applies them to parabolic PDEs such as the Kuramoto–Sivashinsky and heat equations. The results unify ISS/LISS for time-varying systems with unbounded $A(t)$ and bounded/unbounded input operators, and provide explicit Lyapunov function forms, e.g., $V(t,x)=\int_t^{\infty}\|W(\tau,t)x\|^2 d\tau$ and $Z(t,x)=\ln(1+V(t,x))$, to certify stability.
Abstract
We prove that (local) input-to-state stability ((L)ISS) and integral input-to-state stability (iISS) of time-varying infinite-dimensional systems in abstract spaces follows from the existence of a {corresponding} Lyapunov function. In particular, input-to-state stability of linear time-varying control systems in Hilbert spaces with bounded input operators is discussed. Methods for the construction of non-coercive LISS/iISS Lyapunov functions are presented for a certain class of time-varying semi-linear evolution equations. Two examples are given to illustrate the effectiveness of the results.