Editing Lyapunov function (section)

{{Short description|Concept in the analysis of dynamical systems}}{{No footnotes|date=October 2023}}

In the theory of [[ordinary differential equations]] (ODEs), '''Lyapunov functions''', named after [[Aleksandr Lyapunov]], are scalar functions that may be used to prove the stability of an [[equilibrium point|equilibrium]] of an ODE.  Lyapunov functions (also called Lyapunov’s second method for stability) are important to [[stability theory]] of [[dynamical system]]s and [[control theory]]. A similar concept appears in the theory of general state-space [[Markov chain]]s usually under the name Foster–Lyapunov functions.

For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, [[quadratic function|quadratic]] functions suffice for systems with one state, the solution of a particular [[linear matrix inequality]] provides Lyapunov functions for linear systems, and [[Conservation law (physics)|conservation law]]s can often be used to construct Lyapunov functions for [[physical system]]s.