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Some remarks on relative stability

Published online by Cambridge University Press:  17 February 2009

Peter E. Kloeden
Peter E. Kloeden
Affiliation:
Department of Mathematics, University of Queensland
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In 1962 Lakshmikantham ([1], [2]) extended the concept of extreme stability (e.g. [4]) of a system described by an ordinary differential equation, not necessarily with uniqueness, to relative stability of two such systems. Here we show the restrictiveness of his definition of relative stability in that it implies not only are the solutions of two systems unique for each initial condition, they are in fact identical. We then introduce and give an example of a weaker version of relative stability which is of some interest for control systems. For greater simplicity and generality we use Roxin's attainability set defined General Control Systems [3] to describe the dynamics of our systems, as they subsume both ordinary differential equations without uniqueness and ordinary differential control equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 1 , June 1975 , pp. 112 - 115
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Lakshmikantham, V., Differential systems and extension of Lyapunov's method, Michigan Math. J., 9 (1962), 161–166.CrossRefGoogle Scholar
[2]Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities, Volume 1, Academic Press, New York (1969); pp. 241–243.Google Scholar
[3]Roxin, E. O., Stability in general control systems, J. DifferentialEquations, 1 (1965), 115–150.Google Scholar
[4]Yoshizawa, T., Stability Theory by Lyapunov's Second Method, Math. Soc. Japan, Tokyo (1967).Google Scholar