Practical Stability and Lyapunov Functions
Bernfeld, Stephen R.
MetadataShow full item record
The notion of "practical stability" was discussed in the monograph by LaSalle and Lefschetz  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 . 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  provided sufficient conditions for finite time stability in terms of Lyapunov functions. Moreover, Weiss  provided necessary and sufficient conditions for uniform finite time stability and exponential contractive stability. These results were extended by Kayande  who obtained necessary and sufficient conditions for contractive stability (without requiring the exponential behavior assumed in ). The sufficiency part of the above results were extended by Kayande and Wong , and Gunderson , who applied the comparison principle. Moreover Hallam and Komkov  generalized the concept of the finite time stability of the zero solution to that of arbitrary closed sets.