# Asymptotic Stability and Smooth Lyapunov Functions

@article{Clarke1998AsymptoticSA, title={Asymptotic Stability and Smooth Lyapunov Functions}, author={Frank H. Clarke and Yu. S. Ledyaev and Ronald J. Stern}, journal={Journal of Differential Equations}, year={1998}, volume={149}, pages={69-114} }

Abstract We establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth Lyapunov function. Since well-known concepts of generalized solutions of differential equations with discontinuous right-hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions, this result gives a Lyapunov characterization of asymptotic…

## 253 Citations

### Smooth Lyapunov Functions for Discontinuous Stable Systems

- Mathematics
- 1999

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper…

### Asymptotic Stability and Smooth Lyapunov Functions for a Class of Abstract Dynamical Systems

- Mathematics
- 2015

This paper deals with asymptotic stability of a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot…

### Results on existence of smooth Lyapunov functions for (pre-)asymptotically stable hybrid systems with non-open basins of attraction

- Mathematics2007 American Control Conference
- 2007

This work demonstrates several applications of the converse Lyapunov theorem by establishing robustness of pre-asymptotic stability to various types of perturbations and achieving input-to-state stabilization for hybrid-feedback-control systems.

### Non-coercive Lyapunov functions for infinite-dimensional systems

- MathematicsJournal of Differential Equations
- 2019

### Smooth Lyapunov Functions for Hybrid Systems Part II: (Pre)Asymptotically Stable Compact Sets

- MathematicsIEEE Transactions on Automatic Control
- 2008

The converse Lyapunov theorems are used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, and to the insertion of jumps according to an ldquoaverage dwell-timerdquo rule.

### Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness

- MathematicsIEEE Transactions on Automatic Control
- 2007

The equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth Lyapunov function is established.

### On the topological structure of attraction basins for differential inclusions

- MathematicsSyst. Control. Lett.
- 2011

### Results on converse Lyapunov functions from class-KL estimates

- MathematicsProceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304)
- 1999

We state results on converse Lyapunov functions for differential inclusions where a positive semidefinite function of the solutions satisfies a class-KL estimate in terms of time and a second…

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