Abstract
We extend the metric proof of the converse Lyapunov Theorem,given in [13] for continuous multivalued dynamics, bymeans of tools issued from weak KAM theory, to the case where theset-valued vector field is just upper semicontinuous. Thisgenerality is justified especially in view of application todiscontinuous ordinary differential equations. The more relevantnew point is that we introduce, to compensate the lack ofcontinuity, a family of perturbed dynamics, obtained throughinternal approximation of the original one, and perform somestability analysis of it.
Highlights
This is the prosecution of our previous work [13], where we have proposed a proof of the inverse Lyapunov Theorem for continuous multivalued dynamics, enjoying suitable stability and attractiveness properties with respect to an attractor A, based on the introduction of a Lipschitz–continuous intrinsic metric suitably related to the set–valued vector field and the exploitation of some ideas borrowed from weak KAM theory
As declared in the title, the step forward here is that we consider dynamics just upper semicontinuous
We use [13, Lemma 6.1] to construct from V , Ψ a smooth Lyapunov pair (V, Ψ) satisfying the statement
Summary
This is the prosecution of our previous work [13], where we have proposed a proof of the inverse Lyapunov Theorem for continuous multivalued dynamics, enjoying suitable stability and attractiveness properties with respect to an attractor A, based on the introduction of a Lipschitz–continuous intrinsic metric suitably related to the set–valued vector field and the exploitation of some ideas borrowed from weak KAM theory. To show that a Lipschitz–continuous Lyapunov function for F , say v, constructed using the intrinsic distance, can be uniformly approximated locally around any point y ∈/ A by a sequence of C∞ functions enjoying the infinitesimal (strict) decrease property with respect to F in some neighborhood of y This is a crucial step for proving the final regularization theorem. In the continuous setting this is performed by directly smoothing v through mollifiers with suitably small support In doing it and in checking the validity of the decrease property for the regularized functions, one has to cope, after using Jensen inequality, with pairings between vectors of F (y), and differentials of v at points close to y.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
