Criteria for the response of nonlinear systems to be l-asymptotically periodic

We consider the behavior of a general type of system governed by an input-output operator G that maps each excitation x into a corresponding response r. Here excitations and responses are Rn-valued functions defined on a set T. To accommodate both continuous time and discrete time cases, T is allowed to be either [0, ∞) or {0, 1, 2, …}. We address the following question. Under what conditions on G and x is it true that the response r is L-asymptotically periodic in the sense that r = p + q, where p is periodic with a given period τ, and q has finite energy (i.e., is square summable)? This type of question arises naturally in many applications. The main results given (which include a necessary and sufficient condition) are basically “tool theorems.” To illustrate how they can be used, an example is discussed involving an integral equation that is often encountered in the theory of feedback systems.