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A smooth Lyapunov function from a class-$$\mathcal{KL}$$ estimate involving two positive semidefinite functions. (English) Zbl 0953.34042
Summary: The authors consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-$${\mathcal K\mathcal L}$$ estimate in terms of time and a second positive semidefinite function of the initial condition. They show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-$$\mathcal K\mathcal L$$ estimate, exists if and only if the class-$$\mathcal K\mathcal L$$ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains the open question whether all class-$$\mathcal K\mathcal L$$ estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature.

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 34A60 Ordinary differential inclusions
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##### References:
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