Lyapunov function: Difference between revisions

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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 (CORRECTION: 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 in 'Civelek, C. (2018). Archives of Control Sciences, volume 28 (LXIV), No. 2, pages 201–222 Doi:10.24425/1234562') 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 given in "Civelek, C.; Cihanbeğendi, Ö. (2020). Frontiers of Information Technology & Electronic Engineering, volume 21, pages 629–634, https://doi.org/10.1631/FITEE.1900014'In addion, [[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.

== Definition ==

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* {{cite book |first=Joseph |last=La Salle |first2=Solomon |last2=Lefschetz |title=Stability by Liapunov's Direct Method: With Applications |___location=New York |publisher=Academic Press |year=1961 }}

*{{PlanetMath attribution|id=4386|title=Lyapunov function}}

* Civelek, C. (2018). Archives of Control Sciences, volume 28 (LXIV), No. 2, pages 201–222 Doi:10.24425/123456

* Civelek, C.; Cihanbeğendi, Ö. (2020). Frontiers of Information Technology & Electronic Engineering, volume 21, pages 629–634. https://doi.org/10.1631/FITEE.1900014

==External links==

* [https://web.archive.org/web/20110926230621/http://www.exampleproblems.com/wiki/index.php/ODELF1 Example] of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function