International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 1, Pages 103-112
doi:10.1155/S0161171294000141

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 ν to ν˙=−p, or of the solution ω to ω˙=−(1−ω)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 ν˙=−p, or its equivalent ω˙=−(1−ω)p, should be solved only for one selection of the function p.