Lyapunov function

A Lyapunov function proves stability by assigning a scalar measure of deviation from an equilibrium. Consider an autonomous system ẋ = f(x) with equilibrium x* and a continuously differentiable function V defined near x*. If V(x*) = 0, V(x) is positive for nearby x ≠ x*, and its derivative along solutions satisfies V̇(x) = ∇V(x)·f(x) ≤ 0, then x* is stable in the sense of Lyapunov. The nonincrease of V prevents trajectories that start sufficiently close from escaping a prescribed neighborhood.

If V̇ is strictly negative away from x*, the equilibrium is typically asymptotically stable: trajectories approach the equilibrium as time tends to infinity. The function need not represent physical energy; it may be an algebraically constructed certificate.

The distinction between nonincrease and strict decrease determines what a Lyapunov argument can establish. A positive-definite V with V̇ ≤ 0 generally proves stability, but not necessarily convergence, because a trajectory can remain on a set where V̇ = 0. LaSalle's invariance principle strengthens the test by identifying the largest invariant subset of that zero-derivative set; if it contains only the equilibrium, asymptotic stability follows.¹

Global conclusions require global hypotheses. If V is positive definite and radially unbounded, while V̇ is negative definite throughout the state space, the equilibrium is globally asymptotically stable under standard smoothness assumptions. For a nonautonomous or forced system, the function may also depend explicitly on time, and practical stability may replace convergence to a single point.

Constructing a useful Lyapunov function is often the hardest part of the analysis. For linear systems ẋ = Ax, a quadratic form V(x) = xᵀPx is obtained by choosing a positive-definite matrix P that solves a Lyapunov equation AᵀP + PA = −Q for a positive-definite Q; this gives a direct certificate when A is Hurwitz.¹

For nonlinear systems, common candidates combine physical energy, weighted norms, polynomial expressions, or terms suggested by the dynamics. In feedback control, a controller and a Lyapunov function are frequently designed together: the control law makes V̇ negative, yielding a certificate of stabilization and sometimes robustness. Sum-of-squares optimization, barrier functions, and control-Lyapunov functions extend the approach to computational verification and safety constraints.²

A Lyapunov function is a certificate, not usually a unique physical quantity. Many different functions can prove the same stability property, and a poor choice may establish only local stability even when the system is globally stable. Conversely, converse Lyapunov theorems show that under broad conditions, an asymptotically stable equilibrium admits a suitable Lyapunov function, although finding it explicitly may be difficult.¹

The method also covers discrete-time maps, switched systems, hybrid systems, and stochastic processes, with the derivative replaced by a difference, mode-dependent condition, or expected generator inequality. In applications, Lyapunov functions can certify bounds and input-to-state stability rather than merely equilibrium behavior. These extensions make the method useful for systems with disturbances, jumps, and changing dynamics.²

Positive definite

A function that is zero at the equilibrium and strictly positive at every other point in the domain under consideration.

Radially unbounded

A function whose value tends to infinity as the norm of the state tends to infinity.

Lyapunov derivative

The rate of change of the function along system trajectories, commonly written V̇ = ∇V·f.

Control-Lyapunov function

A function for which an admissible control input can be chosen to produce a required decrease condition.