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**H. Damak**

University of Sfax Tunisia,

Faculty of Science of Sfax, Department of Mathematics

**M. A. Hammami**

University of Sfax Tunisia,

Faculty of Science of Sfax, Department of Mathematics

**A. Kicha**

University of Sfax Tunisia,

Faculty of Science of Sfax, Department of Mathematics

The question proposed in this paper is related to the study of the preservation of uniform h-stability and uniform boundedness of time-varying nonlinear differential equations with a perturbation using Gronwall's inequalities and Lyapunov's theory. Moreover, we show the linearization technique for the uniform h-stability of a nonlinear system and give necessary and sufficient conditions for the global boundedness of perturbed systems. The last part is devoted to the study of the problem of h-stabilization for certain classes of nonlinear systems. Some examples and simulations are given to illustrate the main results.

- Global uniform boundedness
- h-stabilization
- Linearization system
- Lyapunov function
- uniform h-stability

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