Abstract
The results of the paper concern a broad family of time‐varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a function p(⋅) from a defined functional family to determine a Lyapunov function v(⋅), [v(⋅)], by solving v′(⋅) = −p(⋅) {or equivalently, v′(⋅) = −p(⋅)[1 − v(⋅)]}, respectively. Illstrative examples are worked out.
Highlights
Theorems established for time-varying nonlinear systems have been expressed in terms of existence of a Lyapunov function v(.)Iv(-)] without clarifying how to determine it, that is without clarifying how to choose p(-) in v’ (-) p(-) {or equivalently, in v’ p(-)[1 v(-)}, and what are, with respect to a selected p(.), the necessary and sufficient conditions for a solution v(-) Iv(-)], respectively, to guarantee uniform asymptotic stability of x 0 and/or to determine accurately its lomain of uniform asymptotic stability
NOVEL SOLUTIONS TO TIIE CLASSICAL STABILITY PROBLEMS For the sake of clearness we emphasize that the notions of a positive definite function and of a decrescent function will be used in the usual sense (c.f Hahn 15], Zubov [20]), that is that a function v(-) /L x Rn R
For the function p(.) (7.3) we find that @1(’) and @2(’), l(X) (x’ + 2-1x22) and ’2(x) (32xI + x), satisfy
Summary
The well known fundamental advantageous feature of the Lyapunov method consists in the use of both a positive definite function and its total derivative along system motions without knowing the motions themselves in order to investigate qualitative properties of the system behavior, among which there are various asymptotic stability properties. Theorems established for time-varying nonlinear systems have been expressed in terms of existence of a Lyapunov function v(.)Iv(-)] without clarifying how to determine it, that is without clarifying how to choose p(-) in v’ (-) p(-) {or equivalently, in v’ p(-)[1 v(-)}, and what are, with respect to a selected p(.), the necessary and sufficient conditions for a solution v(-) Iv(-)], respectively, to guarantee uniform asymptotic stability of x 0 and/or to determine accurately its lomain of uniform asymptotic stability Such a crucial incompleteness ofthe existing Lyapunov stability theory has been an inherent obstacle to broader and more effective applications of the theory than have been realized. It was overcome in [2]-[12] for different classes of time-invariant systems by proposing three distinct approaches. This paper is aimed to establish analogous solutions for a broad family of time-varying nonlinear systems
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