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

Lars Grüne

doi:10.1016/s0167-6911(99)00044-4

Abstract

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.

Highlights

An important issue in the analysis of feedback stabilization is the robustness of the resulting closed loop system with respect to exterior perturbations
The kind of feedbacks discussed in 6] and 8] emerge from discounted optimal control problems and are typically discontinuous; continuous dependence on the initial value will in general not hold for the closed loop system
In this paper we have shown that the closed loop system (3.2) with inhomogeneous perturbations satis es the input-to-state stability property with exponential decay and linear dependence on the perturbation if the exponentially stabilizing feedback for the associated semilinear system (3.1) satis es some robustness property with respect to small perturbations which can be checked on nite time intervals

Summary

Introduction

An important issue in the analysis of feedback stabilization is the robustness of the resulting closed loop system with respect to exterior perturbations. The question arises, whether the input-to-state stability property with linear dependence on initial value and perturbation and with exponential decay holds for the resulting closed loop system This system, will in general be nonlinear, the usual techniques for linear systems are no longer available. The kind of feedbacks discussed in 6] and 8] emerge from discounted optimal control problems and are typically discontinuous; continuous dependence on the initial value will in general not hold for the closed loop system It is necessary | and the aim of this paper | to nd a suitable condition for possibly discrete and possibly discontinuous exponentially stabilizing feedback laws which is easy to check and ensures input-to-state stability with respect to inhomogeneous perturbations.

Problem setup

The small-perturbation-robustness condition

Linear-Exponential Input-to-State Stability

An optimal control based feedback

Conclusions and Outlook