A constructive converse Lyapunov theorem on asymptotic stability for nonlinear autonomous ordinary differential equations

Abstract

An ordinary differential equation's (ODE) equilibrium is asymptotically stable, if and only if the ODE possesses a Lyapunov function, that is, an energy-like function decreasing along any trajectory of the ODE and with exactly one local minimum. Theorems regarding the `only if' part are called converse theorems. Recently, the author presented a linear programming problem, of which every feasible solution parameterizes a Lyapunov function for the nonlinear autonomous ODE in question. In 2004 the author proved the first general constructive converse theorem by showing that if the equilibrium of the ODE is exponentially stable, then the linear programming problem possesses a feasible solution. In this paper we prove a constructive converse theorem on asymptotic stability for nonlinear autonomous ODEs and so improve the 2004 results. The only restriction on the ODE = f(x) is that f is a class function. Note, that these results imply that the algorithm presented by the author in 2002 is capable of constructing a Lyapunov function for all nonlinear systems, of which the equilibrium is asymptotically stable.