Construction of a complete lyapunov function using quadratic programming

A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is split into the chain-recurrent set, where the function is constant, and the part characterizing the gradient-like flow, where the function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about the stability of connected components of the chain-recurrent set and the basin of attraction of attractors therein. In a previous method, a complete Lyapunov function was constructed by approximating the solution of the PDE V⁰(x) = −1, where ⁰ denotes the orbital derivative, by meshfree collocation. We propose a new method to compute a complete Lyapunov function: we only fix the orbital derivative V⁰(x₀) = −1 at one point, impose the constraints V⁰(x) ≤ 0 for all other collocation points and minimize the corresponding reproducing kernel Hilbert space norm. We show that the problem has a unique solution which can be computed as the solution of a quadratic programming problem. The new method is applied to examples which show an improvement compared to previous methods.