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 ofa Lyapunov function, for asymptotic stability of a compact invariant set and for the exact determination ofthe 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 v to-p, or of the solution w to-(1 w)p, for arbitrarily selected p C. P(S;f) or p C. Pt(S;f), where families P(S;f) and Pt(S;f) are well defined. The equation-p, or its equivalent /,-(1 w)p, should be solved only for one selection of the function p