Lyapunov

Aleksandr Mikhailovich Lyapunov (1857–1918) was a Russian mathematician who laid foundational work in the theory of stability for dynamical systems. His studies introduced the concepts of Lyapunov stability and the Lyapunov function, which have become central tools in nonlinear analysis, differential equations, and control theory.

Lyapunov stability refers to the behavior of solutions near an equilibrium of a dynamical system. For a system described by x' = f(x) with an equilibrium at x = 0, the system is stable in the sense of Lyapunov if, for every ε > 0, there exists a δ > 0 such that ||x(0)|| < δ implies ||x(t)|| < ε for all t ≥ 0. If x(t) → 0 as t → ∞ for all such initial conditions, the equilibrium is asymptotically stable. Exponential stability indicates decay at an exponential rate.

A Lyapunov function is a scalar function V: R^n → R that is positive definite (V(0) = 0 and V(x) > 0 for x ≠ 0) and has a nonpositive time derivative along system trajectories, V̇(x) ≤ 0. If V̇(x) < 0 for x ≠ 0, the equilibrium is asymptotically stable. In linear systems x' = A x, a common approach uses a positive definite P to form V(x) = xᵀ P x, with a condition AᵀP + P A = -Q for some Q > 0, which establishes stability.

Beyond Lyapunov stability, the term Lyapunov exponent—also named after Lyapunov—measures the average exponential rate of divergence of nearby trajectories and is a key concept in chaos theory. Lyapunov’s methods have wide applications in physics, engineering, and applied mathematics.