Complete characterization of the set of Lyapunov functions for the autonomous system ẋ=−x

In this paper, we give the complete characterization of all possible Lyapunov functions for the specific autonomous system ẋ=-x. This characterization is in terms of strict quasi-convexity of a Lyapunov candidate at the equilibrium point, and it is in a way independent of system dynamics. From Lyapunov theory it is well known that, a continuously differentiable function V : ℝⁿ → ℝ is a Lyapunov function to conclude the global asymptotic stability of the equilibrium point of ẋ=-x if and only if V is positive definite, radially unbounded, and V̇(x)=-(∇V(x))^Tx<;0, ∀x≠0. For the autonomous system ẋ=-x, we give a necessary and sufficient condition, in terms of strict quasi-convexity at the equilibrium point 0, for V̇ (·) to be negative definite.