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Lyapunov’s second method for nonautonomous differential equations. (English) Zbl 1128.37010

Converse Lyapunov theorems are important, at least theoretically, for deducing existence of Lyapunov functions from stability of invariant sets. The authors prove converse Lyapunov theorems for the pullback, forward, and uniform attractors concentrating attention on obtaining Lyapunov functions that recover certain attraction rates in terms of given comparison functions from the classes \(\mathcal{K}\) and \(\mathcal{KL}\). Recall that \(\mathcal{K}\) contains all continuous functions \(\gamma:\mathbb{R}_+\to\mathbb{R}_+\) such that \(\gamma(0)=0\) and \(\gamma\) is strictly increasing, whereas \(\mathcal{KL}\) contains all continuous functions \(\beta:\mathbb{R}_+^2\to\mathbb{R}_+\) that belong to class \(\mathcal{K}\) in the first argument and decrease monotonically to zero in the second argument. It is shown how the different notions of stability and attractivity can be characterized in terms of attraction rates provided by comparison functions.
First the notions of the pullback, forward, and uniform attractors are introduced with respect to the attraction of arbitrary compact sets, which implies stability properties too. Lyapunov functions are defined, and it is shown that if the base space of the skew product flow is compact, then only the maximal invariant set can posses Lyapunov functions. It is proved that a skew product flow satisfies a decay condition expressed in terms of comparison functions if and only if there exists a Lyapunov function that characterizes this type of decay. Then it is demonstrated how different notions of stability and attractivity may be equivalently expressed in terms of nonautonomous comparison functions. Finally, Lyapunov and converse Lyapunov theorems for different stability notions are established.

MSC:

37B55 Topological dynamics of nonautonomous systems
34D20 Stability of solutions to ordinary differential equations
37B25 Stability of topological dynamical systems
93D30 Lyapunov and storage functions
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