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Concept
Lyapunov function
Summary
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 of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, 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. There is no general technique for constructing Lyapunov functions for ODEs, however, depending on formulation type, a systematic method to construct Lyapunov functions for ordinary differential equations using their most general form in autonomous cases was given by Prof. Cem Civelek. Though, in many specific cases the construction of Lyapunov functions is known. For instance, according to a lot of applied mathematicians, for a dissipative gyroscopic system a Lyapunov function could not be constructed. However, using the method expressed in the publication above, even for such a system a Lyapunov function could be constructed as per related article by C. Civelek and Ö. Cihanbegendi. In addition, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems, and conservation laws can often be used to construct Lyapunov functions for physical systems.
A Lyapunov function for an autonomous dynamical system
with an equilibrium point at is a scalar function that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative is non positive (these conditions are required on some region containing the origin). The (stronger) condition that is strictly positive for is sometimes stated as is locally positive definite, or is locally negative definite.
Lyapunov functions arise in the study of equilibrium points of dynamical systems.
Official source
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