Asymptotic Controllability and Exponential Stabilization of Nonlinear Control Systems at Singular Points

Lars Grüne

doi:10.1137/s0363012997315919

Abstract

We discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points. Using a discounted optimal control approach, we construct discrete feedback laws minimizing the Lyapunov exponent of the linearization. Thus we obtain an equivalence result between uniform exponential controllability and uniform exponential stabilizability by means of a discrete feedback law.

Highlights

In this paper we will present a technique for the exponential stabilization of nonlinear control systems with constrained control range at singular points
In particular we address the relation between asymptotic controllability and exponential stabilization and will derive an equivalence theorem
This theorem states that the Lyapunov exponent that gives the characteristic number for null controllability can be approximated by the value function of a discounted optimal control problem

Summary

Introduction

In this paper we will present a technique for the exponential stabilization of nonlinear control systems with constrained control range at singular points. Using a similar feedback concept a result on the relation between asymptotic null controllability and practical stabilization for nonlinear systems has been developed in 5] using Lyapunov functions. The construction of the stabilizing discrete feedback | following the outline of 11] | is based on the minimization of the Lyapunov exponent This is related to minimizing (3.2) which forms an average time optimal control problem, for which the construction of optimal feedback controls is still an unsolved problem. This theorem states that the Lyapunov exponent that gives the characteristic number for null controllability can be approximated by the value function of a discounted optimal control problem. We will just give the idea of the construction and omit the proofs except for the concluding theorem

We approximate U by

Conclusions

Thus we can estimate