Solutions to Lyapunov stability problems of sets: nonlinear systems with differentiable motions

Abstract

Time-invariant nonlinear systems with differentiable motions are considered. The algorithmic necessary and sufficient conditions are established in various forms for one-shot construction of a Lyapunov function, for asymptotic stability of a compact invariant set and for the exact determination of the asymptotic stability domain of the invariant set.The classical conditions are expressed in terms of existence of a system Lyapunov functions. The conditions of theorems presented herein are expressed via properties of the solution ? to ??=-p, or of the solution ? to ??=-(1-?)p, for arbitrarily selected p?P(S;f) or p?P1(S;f), where families P(S;f) and P1(S;f) are well defined. The equation ??=-p, or its equivalent ??=-(1-?)p, should be solved only for one selection of the function p.