Lyapunov characterizations on input-to-state stability of nonlinear systems with infinite delays

This paper addresses Lyapunov characterizations on input-to-state stability (ISS) of time-varying nonlinear systems with infinite delays. With novel ISS definitions in the case of nonlinear systems with infinite delays, we present several results on their ISS Lyapunov characterizations in the form of both ISS Lyapunov theorems and converse ISS Lyapunov theorems. It is shown that an infinite-delayed system is (locally) ISS if it has a (local) ISS Lyapunov functional, and conversely, there exists a (local) ISS Lyapunov functional if it is (locally) ISS. To prove the converse ISS Lyapunov theorems, we establish a key technical lemma bridging ISS/LISS and robust asymptotic stability of systems with infinite delays and two converse Lyapunov theorems concerning robust asymptotic stability of systems with infinite delays. Two distinctive advantages of this work are that a large class of infinite dimensional spaces are allowed and the results are established based on a more general Lipschitz condition, i.e., the right hand side Lipschitz (RS-L) condition. An example is provided for illustration of the obtained results.