An Extension of Barbashin-Krasovski-LaSalle Theorem to a Class of Nonautonomous Systems

In this paper we give an extension of the Barbashin-Krasovski-LaSalle Theorem to a class of time-varying dynamical systems, namely the class of systems for which the restricted vector field to the zero-set of the time derivative of the Liapunov function is time invariant and this set includes some trajectories. Our goal is to improve the sufficient conditions for the case of uniform asymptotic stability of the equilibrium. We obtain an extension of an well-known linear result to the case of zero-state detectability (given (C,A) a detectable pair, if there exists a positive semidefinite matrix P>=0 such that: A^TP+PA+C^TC=0 then A is Hurwitz - i.e. it has all eigenvalues with negative real part) as well as a result about robust stabilizability of nonlinear affine control systems.