Linear Openness and Feedback Stabilization of Nonlinear Control Systems

TL;DR

This paper introduces the concept of linear openness in nonlinear control systems and develops variational analysis techniques to establish sufficient conditions for local exponential stabilization using continuous feedback laws.

Contribution

It replaces the openness condition with linear openness, providing new criteria for feedback stabilization in nonlinear control systems.

Findings

Derived sufficient conditions for local exponential stabilization.

Established necessary conditions for stabilization in continuous and discrete systems.

Extended results to nonlinear discrete-time control systems.

Abstract

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and smooth stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.