ALEXANDER MIHAILOVICH LYAPUNOV

founded 1892 the general stability theory

of dynamical systems.

Lyapunov established two different methodologies for studying stability properties of dynamical systems:

One is valid for time-invariant linear systems. It provides the necessary and sufficient conditions for (global) asymptotic stability of the equilibrium state and for the exact construction of a Lyapunov function for a given linear dynamical system. It starts with negative definitness of the total time- derivative of a Lyapunov function accepted in the form of an arbitrarily chosen negative definite quadratic form and results in a positive definite quadratic form if and only if the equilibrium is (globally) asymptotically stable. The result follows from a single application of the methodology.

Another one is for nonlinear dynamical systems. It provides only sufficient conditions for the asymtotic stability and for the construction of a system Lyapunov function. It enables only an estimate of the domain of asympttic stability of the equilibrium. It permits an infinite number of trials in order to get a final result. It is inconsistent with his methodology for linear dynamical systems.

The consistent Lyapunov methodology is valid for linear and nonlinear, time-invariant and time-varying, dynamical systems. It starts with an arbitrary negative definite function p(.) that plays the role of a total time derivative of a Lyapunov function v(.) to be found. The unique solution of the diffrential equation in v(.), on the right hand side of which is the chosen function p(.), is positive definite if and only if the equilibrium state is asymptotically stable. The determination of the Lyapunov function v(.) finishes in a single application of the methodology. Besides, the methodology establishes simultaneously necessary and sufficient conditions for a set to be the exact domain of the asymptotic stability.

The following monograph on various stability domains contains the complete theoretical solutions to all three unsolved fundamental problems of Lyapunov stability theory:

The necessary and sufficient conditions for asymptotic stability of the equilbrium state ofa nonlinear dynamical system, which are not expressed in terms of the existence of a system Lyapunov function.

The direct, one-shot, construction of a Lyapunov function for a nonlinear dynamical system.

The direct determination for the exact domain of asymptotic stability of the equilbrium state of a nonlinear dynamical system.

Published 2004

Published 1987