Necessary condition on Lyapunov functions corresponding to the globally asymptotically stable equilibrium point

Abstract: It is well known that, the existence of a Lyapunov function is a sufficient condition for stability, asymptotic stability, or global asymptotic stability of an equilibrium point of an autonomous system $\dot{\mathbf{x}} = f(\mathbf{x})$. In variants of Lyapunov theorems, the condition for a Lyapunov candidate $V$ (continuously differentiable and positive definite function) to be a Lyapunov function is that its time derivative along system trajectories must be negative semi-definite or negative definite. Numerically checking positive definiteness of $V$ is very difficult; checking negative definiteness of $\dot{V}(\cdot)=\langle \nabla V(\cdot), f(\cdot) \rangle$ is even more difficult, because it involves dynamics of the system. We give a necessary condition independent of the system dynamics, for every Lyapunov function corresponding to the globally asymptotically stable equilibrium point of $\dot{\mathbf{x}} = f(\mathbf{x})$. This necessary condition is numerically easier to check than checking positive definiteness of a function. Therefore, it can be used as a first level test to check whether a given continuously differentiable function is a Lyapunov function candidate or not. We also propose a method, which we call a generalized steepest descent method, to check this condition numerically. Generalized steepest descent method can be used for ruling out Lyapunov candidates corresponding to the globally asymptotically stable equilibrium point of $\dot{\mathbf{x}} = f(\mathbf{x})$. It can also be used as a heuristic to check the local positive definiteness of a function, which is a necessary condition for a Lyapunov function corresponding to a stable and/or asymptotically stable equilibrium point of an autonomous system.