Input-to-state stability of exponentially stabilized semilinear control systems with inhomogeneous perturbations

In this paper we investigate the robustness of state feedback: stabilized semilinear systems subject to inhomogeneous perturbations in terms of input-to-state stability. We consider a general class of exponentially stabilizing feedback controls which covers sampled discrete feedbacks and discontinuous mappings as well as classical feedbacks and derive a necessary and sufficient condition for the corresponding closed-loop systems to be input-to-state stable with exponential decay and linear dependence on the perturbation. This condition is easy to check and admits a precise estimate for the constants involved in the input-to-state stability formulation. Applying this result to an optimal control based discrete feedback yields an equivalence between (open-loop) asymptotic null controllability and robust input-to-state (state feedback) stabilizability. (C) 1999 Elsevier Science B.V. All rights reserved.