Some results on robust stability of general input-output systems

We consider general input-output systems governed by nonlinear operator equations that relate the system's input, state, and output. The systems under consideration need not be of a feedback type. Assuming that the governing equations depend on a parameterA in a linear space that is allowed to vary in a vicinity Nr(A0) of a “nominal” valueA0, we study conditions under which the system is stable for each A∈Nr (A0), i.e., when the system is robust. By stability we essentially mean that the input-output map is continuous. Depending on the type of continuity used, two concepts of robustness are introduced. The main theorem shows that a certain generalized monotonicity condition imposed on the nominal system combined with a Lipschitz-like condition imposed on the perturbed system guarantees robustness. Moreover, several particular cases of the governing equations are investigated. As examples, we consider (1) a singular system of nonlinear ordinary differential equations (a semistate equation), (2) a feedback system, and (3) a feedback, feedforward system. At the end of this paper some extensions and modifications of the presented theory are discussed.