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It is well known from the seminal Brockett's theorem that the opennessproperty of the mapping on the right-hand side of a given nonlinear ODE controlsystem is a necessary condition for the existence of locally asymptoticallystabilizing continuous stationary feedback laws. However, this condition failsto be sufficient for such a feedback stabilization. In this paper we develop anapproach of variational analysis to continuous feedback stabilization ofnonlinear control systems with replacing openness by the linear opennessproperty, which has been well understood and characterized in variationaltheory. It allows us, in particular, to obtain efficient conditions via thesystem data supporting the sufficiency in Brockett's theorem and ensuring localexponential stabilization by means of continuous stationary feedback laws.Furthermore, we derive new necessary conditions for local exponential andasymptotic stabilization of continuous-time control systems by using bothcontinuous and smooth stationary feedback laws and establish also somecounterparts of the obtained sufficient conditions for local asymptoticstabilization by continuous stationary feedback laws in the case of nonlineardiscrete-time control systems.

Linear Openness and Feedback Stabilization of Nonlinear Control Systems