Geometric homogeneity with applications to finite-time stability. (English) Zbl 1110.34033

Bhat, S. P.; Bernstein, D. S.

Math. Control Signals Syst. 17, No. 2, 101-127 (2005).

The authors study properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of their results, the authors consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.

Reviewer: Alexander O. Ignatyev (Donetsk)

MSC:

34D20	Stability of solutions to ordinary differential equations
34A26	Geometric methods in ordinary differential equations

Keywords:

homogeneous systems; stability; Lyapunov functions

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