Low-gain integral control of well-posed linear infinite-dimensional systems with input and output nonlinearities

Time-varying low-gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable, well-posed, linear, infinite-dimensional, single-input–single-output, systems—subject to globally Lipschitz, nondecreasing input and output nonlinearities. It is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense, provided that (a) the steady-state gain of the linear part of the system is positive, (b) the reference value r is feasible in an entirely natural sense, and (c) the positive gain function t ↦ k(t) is ultimately sufficiently small and not of class L1. Under a weak restriction on the initial data it is shown that (a), (b), and (c) ensure asymptotic tracking. If, additionally, the impulse response of the linear part of the system is a finite signed Borel measure, the global Lipschitz assumption on the output nonlinearity may be considerably relaxed.