A Lyapunov function proves stability without solving the differential equation. It is an energy-like scalar whose derivative along trajectories has a controlled sign.
Direct stability theorem
Theorem (Lyapunov). Let \(x^*=0\), and suppose \(V\in C^1\) near zero satisfies \[ V(0)=0,\qquad V(x)>0\ (x\ne0),\qquad \dot V(x)=\nabla V(x)^\top f(x)\le0. \] Then zero is stable. If \(\dot V(x)<0\) for \(x\ne0\), zero is asymptotically stable.
Proof. Given a small sphere of radius \(\varepsilon\), compactness makes \[ m_\varepsilon=\min_{\|x\|=\varepsilon}V(x)>0. \] Choose \(\delta\) so \(V(x_0)<m_\varepsilon\) for \(\|x_0\|<\delta\). Since \(V\) is nonincreasing, the trajectory cannot cross the sphere, proving stability. If \(\dot V<0\), a precompact trajectory has omega-limit set on which \(V\) is constant; strict negativity forces that set to be \(\{0\}\), giving convergence.
Exponential estimates
Proposition. If locally \[ c_1\|x\|^2\le V(x)\le c_2\|x\|^2, \qquad \dot V(x)\le-c_3\|x\|^2, \] then \[ \|x(t)\| \le\sqrt{\frac{c_2}{c_1}} e^{-c_3t/(2c_2)}\|x_0\|. \]
Proof. \(\dot V\le-(c_3/c_2)V\), so \(V(t)\le e^{-c_3t/c_2}V(0)\). Apply the quadratic bounds and take square roots.
LaSalle invariance principle
Theorem (LaSalle). Let \(K\) be compact and positively invariant, and let \(\dot V\le0\) on \(K\). Every trajectory in \(K\) approaches the largest invariant subset of \[ E=\{x\in K:\dot V(x)=0\}. \]
Proof. \(V(\phi_t(x))\) is nonincreasing and bounded below, hence converges. On the omega-limit set it must be constant by invariance and continuity, so \(\dot V=0\) there. The omega-limit set is invariant and lies in \(E\), proving approach to its largest invariant subset.
Quadratic functions for linear systems
Theorem. A matrix \(A\) is Hurwitz if and only if for every \(Q\succ0\) there is a unique \(P\succ0\) satisfying \[ A^\top P+PA=-Q. \]
Proof. If \(A\) is Hurwitz, define \[ P=\int_0^\infty e^{tA^\top}Qe^{tA}\,\mathrm{d}t. \] The integral converges, is positive definite, and differentiation under the integral gives the equation. Uniqueness follows by integrating the homogeneous identity. Conversely, \(V=x^\top Px\) has \(\dot V=-x^\top Qx<0\), giving exponential stability and hence Hurwitz spectrum.
Constructing Lyapunov functions
Mechanical energy, entropy, relative entropy, and distance to an invariant set are common candidates. Sum-of-squares programming searches polynomial certificates but supplies sufficient, not necessary, conditions at fixed degree. Converse Lyapunov theorems show asymptotic stability implies existence of a Lyapunov function under regularity, though it may not have a simple closed form.