A converse theorem for practical \(h\)-stability of time-varying nonlinear systems. (English) Zbl 1477.34078
The article considers non-autonomous nonlinear ordinary differential equations and studies their \(h\)-stability, which consists in a bound on the size of solutions that depends linearly on the size of the initial condition and whose dependence with respect to time is given by the function \(h=h(t)\). The authors modify such definition, due to Pinto, by considering its practical stability version, obtained by adding a positive constant to the bound on trajectories. The results obtained by the authors are of three types: first, given a non-autonomous linear \(h\)-stable system, conditions are given on an additive perturbative dynamical term that guarantee practical \(h\)-stability of the perturbed system; then a direct and a converse Lyapunov results are obtained for a general non-autonomous nonlinear system; finally, given a \(h\)-stable system to which the converse Lyapunov result applies, conditions are given on an additive perturbative dynamical term so that the perturbed system is practically \(h\)-stable.
Reviewer: Mario Sigalotti (Paris)
MSC:
| 34D20 | Stability of solutions to ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
Keywords:
Lyapunov function; practical uniform \(h\)-stability; semi-global practical uniform \(h\)-stability; time-varying systemsReferences:
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