Lyapunov function: Difference between revisions

Latest revision as of 15:14, 10 July 2026

A Lyapunov function is a scalar function defined on the phase space of a dynamical system that decreases monotonically along trajectories and attains its minimum at an equilibrium point. Unlike Lyapunov exponents, which quantify instability through linearization, a Lyapunov function proves stability globally without requiring the system to be close to equilibrium. The existence of a Lyapunov function is sufficient for asymptotic stability but not necessary; conversely, the absence of positive Lyapunov exponents is necessary for stability but not sufficient. The two concepts — Lyapunov function and Lyapunov exponent — are complementary pillars of stability theory: one gives global, nonlinear proofs of order, the other gives local, linear measures of chaos. Lyapunov functions are central to control theory, where they are used to design stabilizing feedback, and to the theory of dissipative systems, where they represent the system's free energy or entropy production. \n\nIn control theory, the stronger notion of a Control Lyapunov function is used to design feedback laws that guarantee stabilization of nonlinear systems.

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-A '''Lyapunov function''' is a scalar function defined on the state space of a [[dynamical system]] that enables the analysis of stability without solving the system's equations of motion. Named after Aleksandr Lyapunov, it generalizes the intuitive notion of energy: it is positive everywhere except at an equilibrium point, and its rate of change along system trajectories is negative. The existence of such a function guarantees [[Lyapunov stability]] or asymptotic stability; its non-existence tells us only that the energy-landscape method does not apply, not that the system is unstable.
+A '''Lyapunov function''' is a scalar function defined on the phase space of a dynamical system that decreases monotonically along trajectories and attains its minimum at an equilibrium point. Unlike [[Lyapunov exponents]], which quantify instability through linearization, a Lyapunov function proves stability globally without requiring the system to be close to equilibrium. The existence of a Lyapunov function is sufficient for asymptotic stability but not necessary; conversely, the absence of positive Lyapunov exponents is necessary for stability but not sufficient. The two concepts — Lyapunov function and Lyapunov exponent — are complementary pillars of stability theory: one gives global, nonlinear proofs of order, the other gives local, linear measures of chaos. Lyapunov functions are central to control theory, where they are used to design stabilizing feedback, and to the theory of [[Dissipative Systems|dissipative systems]], where they represent the system's free energy or entropy production.
-The construction of Lyapunov functions for nonlinear systems remains an art rather than an algorithm. For linear systems, quadratic forms suffice; for mechanical systems, total energy often works; for general nonlinear systems, one may need to search through classes of candidate functions using sum-of-squares optimization or machine learning approaches. A [[control-Lyapunov function]] is a Lyapunov function for which an explicit stabilizing control law can be derived, forming the bridge between stability analysis and controller design.
+[[Category:Systems]] [[Category:Mathematics]]\n\nIn control theory, the stronger notion of a [[Control Lyapunov function]] is used to design feedback laws that guarantee stabilization of nonlinear systems.
-[[Category:Mathematics]]
-[[Category:Systems]]