# Lyapunov function

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## Lyapunov function

[lē′ap·ə‚nȯf ‚fəŋk·shən]
(mathematics)
A function of a vector and of time which is positive-definite and has a negative-definite derivative with respect to time for nonzero vectors, is identically zero for the zero vector, and approaches infinity as the norm of the vector approaches infinity; used in determining the stability of control systems. Also spelled Liapunov function.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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It plays a very important role in the system stability proof and the controller design, and Lyapunov functions are needed when using this method.
But in our paper, by using the quadratic Lyapunov functions, the aforementioned problem is solved.
Then, design partial Lyapunov functions and intermediate virtual control for each subsystem until back to the entire system.
System (32) is globally asymptotically stable with respect to the Lyapunov functions (18) and (26).
What is more, by using suitable Lyapunov functions, we demonstrate the global stability of equilibria.
Up to now, searching the common Lyapunov functions or variations of the same framework is the main tool for studying dynamic characteristics of a system.
By using a common or multiple Lyapunov functions method, stability issues for switched fuzzy systems were considered in [35-39].
Kurak, "A computational method for determining quadratic Lyapunov functions for non-linear systems," Automatica, vol.
Osorio, "Strict Lyapunov functions for the super-twisting algorithm," IEEE Transactions on Automatic Control, vol.
Parallel to Lemma 3.1 in , we state the comparison lemma in terms of Lyapunov functions. As the proof is similar to that of Theorem 2.1 in , we omit it.
In this section, the uniform (asymptotic) stability of system (3.1) is investigated using Lyapunov functions in the spirit of Razumikhin.

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