Behavior of a stable nonlinear infinite-dimensional system under the influence of a nonlinear exosystem

This paper considers a nonlinear infinite-dimensional system Σ^N obtained by the feedback interconnection of a well-posed linear system Σ^p with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for Σ^N, for any initial state in its state space X and any input signal of class L²_loc. Then we investigate the behavior of Σ^N when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that Σ^P is exponentially stable, denoting the state space of the exosystem by W, we find that there exists a continuous map Π: W → X such that regardless of initial states lim_t→∞ ||Πw(t) - x(t)|| = 0, where w(t) is the state of the exosystem and x(t) is the state of Σ^N. In particular, when w is T-periodic, then the state of the interconnection tends to a T-periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.