International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 1, Pages 103-112

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

Ljubomir T. Grujić

Department of Electrical Engineering, University of Natal, Rm. 1-05, Elec. Eng. Bldg., King George V Avenue, Durban 4001, South Africa

Received 23 January 1991; Revised 28 April 1993

Copyright © 1994 Ljubomir T. Grujić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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˙=(1ω)p, for arbitrarily selected pP(S;f) or pP1(S;f), where families P(S;f) and P1(S;f) are well defined. The equation ν˙=p, or its equivalent ω˙=(1ω)p, should be solved only for one selection of the function p.