Edinburgh Napier University

  We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay- and partial-differential equa-tions are known to belong to this class of infinite-dimensional systems. We present refinements ofrecent incremental input-to-state stability results [14] and use them to derive convergence results fortrajectories generated by Stepanov almost periodic inputs. In particular, we show that the incrementalstability conditions guarantee that for every Stepanov almost periodic input there exists a unique pairof state and output signals which are almost periodic and Stepanov almost periodic, respectively. Thealmost periods of the state and output signals are shown to be closely related to the almost periodsof the input, and a natural module containment result is established. All state and output signalsgenerated by the same Stepanov almost periodic input approach the almost periodic state and theStepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficientconditions guaranteeing incremental input-to-state stability and the existence of almost periodic stateand Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-knownabsolute stability criteria such as the complex Aizerman conjecture and the circle criterion.