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Practical Stability and Lyapunov Functions

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Practical Stability and Lyapunov Functions

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Title: Practical Stability and Lyapunov Functions
Author: Lakshmikantham, V.; Bernfeld, Stephen R.
Abstract: The notion of "practical stability" was discussed in the monograph by LaSalle and Lefschetz [6] in which they point out that stability investigations may not assure "practical stability" and vice versa. For example an aircraft may oscillate around a mathematically unstable path, yet its performance may be acceptable. Motivated by this, Weiss and Infante introduced the concept of finite time stability [7]. They were interested in the behavior of systems contained within specified bounds during a fixed time interval. Many problems fall into this category including the travel of a space vehicle between two points and the problem, in a chemical process, of keeping the temperature within certain bounds. In particular, Weiss and Infante [7] provided sufficient conditions for finite time stability in terms of Lyapunov functions. Moreover, Weiss [9] provided necessary and sufficient conditions for uniform finite time stability and exponential contractive stability. These results were extended by Kayande [3] who obtained necessary and sufficient conditions for contractive stability (without requiring the exponential behavior assumed in [9]). The sufficiency part of the above results were extended by Kayande and Wong [4], and Gunderson [1], who applied the comparison principle. Moreover Hallam and Komkov [2] generalized the concept of the finite time stability of the zero solution to that of arbitrary closed sets.
URI: http://hdl.handle.net/10106/2293
Date: 1979-10

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