Lyapunov Stability

Lyapunov theory judges stability from an energy-like function, applying to nonlinear systems without solving their equations.

Stability Without Solving the Equations

For nonlinear systems, computing the response to check stability is usually impossible. Lyapunov stability theory, from Aleksandr Lyapunov's 1892 work, sidesteps this by examining an energy-like scalar function of the state. If that function always decreases along the system's trajectories, the system settles toward equilibrium, and this conclusion follows without ever solving the differential equations.

Notions of stability

• Stable in the sense of Lyapunov: trajectories starting near an equilibrium stay near it.
• Asymptotically stable: trajectories starting near the equilibrium not only stay near but converge to it.
• Globally asymptotically stable: every trajectory, from any starting point, converges to the equilibrium.
• Exponentially stable: convergence occurs at least as fast as a decaying exponential.

The Lyapunov function

A Lyapunov function V(x) is a scalar function that is zero at the equilibrium and positive everywhere else, an abstract measure of how far the state is from equilibrium, generalizing energy. It need not be an actual physical energy, only share the property of shrinking as the system approaches rest.

The direct method

Lyapunov's direct method states the core test: if a positive-definite V can be found whose time derivative along the system's trajectories is negative, then the equilibrium is stable, and if the derivative is strictly negative, asymptotically stable. Finding such a V proves stability; failing to find one proves nothing, which is the method's main practical difficulty.

Why it dominates nonlinear analysis

Because it works directly on the nonlinear equations and requires no solution, Lyapunov theory is the primary tool for nonlinear stability. It also underpins controller design: many nonlinear and adaptive controllers are constructed by first choosing a Lyapunov function and then designing the control law to force its derivative negative, a strategy that builds stability into the design from the start.

For linear systems, the search for a Lyapunov function reduces to solving the Lyapunov matrix equation, a tractable linear-algebra problem that recovers the familiar eigenvalue stability test.