 Lyapunov function

A Lyapunov function near an equilibrium point $x_{\ast }$ is a continuously differentiable scalar $V$ with $V(x_{\ast })=0$, $V(x)>0$ for nearby $x\ne x_{\ast }$, and nonpositive orbital derivativeIts sublevel sets act as nested barriers: a trajectory cannot cross from a lower value to a higher one. The First Lyapunov theorem uses these barriers to prove Lyapunov stability without an explicit solution.

For $\dot{x}=-x$, choose $V=x^2/2$; then $\dot{V}=-x^2<0$ away from zero and trajectories converge to the equilibrium. For the planar rotation $\dot{x}=-y$, $\dot{y}=x$, the choice $V=(x^2+y^2)/2$ instead has $\dot{V}=0$. It proves stability, but trajectories remain on circles and do not converge.

Thus a nonpositive derivative alone does not establish asymptotic stability. Strict negativity, or an argument such as LaSalle's invariance principle controlling the invariant part of the zero-derivative set, supplies the missing information. A Lyapunov functional applies the same idea to a whole function-valued state.

A Lyapunov function is a mathematical construct used in the field of stability theory to analyze the stability of dynamic systems, particularly in the context of differential equations and control theory. It is a scalar function that helps in determining the stability of an equilibrium point of a dynamical system.