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$h$-stability for linear dynamic equations on time scales. (English) Zbl 1112.34031

This paper extends the concept of $h$-stability introduced e.g., by M. Pinto [Appl. Anal. 43, 1–20 (1992; Zbl 0748.34029)] to dynamic equations on time scales. This stability notion is quite flexible, since it includes the classical theories of uniform or uniform exponential stability within one common framework.

The authors show that $h$-stability of linear systems is preserved under Lyapunov transformations (cf B. Aulbach and the reviewer [J. Comput. Appl. Math. 141, No. 1-2, 101–115 (2002; Zbl 1032.39008)]). A Gronwall argument yields robustness w.r.t linear perturbations. Using a Bihari-type inequality, similar results are also derived for nonlinear perturbations. Finally, quadratic Lyapunov functions enable the authors to obtain sufficient conditions for $h$-stability of linear systems.

An example and various special cases illustrate the paper.

##### MSC:
 34D20 Stability of ODE 34D05 Asymptotic stability of ODE 34D10 Stability perturbations of ODE 39A14 Partial difference equations 39A13 Difference equations, scaling ($q$-differences)