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Construction of Lyapunov functions with linear optimization

Construction of Lyapunov functions with linear optimization. Sigurður F. Hafstein, Reykjavík University. What can we do to get information about the solution ?. Analytical solution ( almost never possible )

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Construction of Lyapunov functions with linear optimization

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  1. Construction of Lyapunov functionswith linear optimization Sigurður F. Hafstein, Reykjavík University

  2. Whatcanwedoto get informationaboutthesolution? • Analyticalsolution (almostneverpossible) • Numericalsolution (not applicable for thegeneralsolution, badapproximation for largetimes for specialsolutions) • Search for trapsinthephase-space ( trap = forwardinvariant set)

  3. Let be the solution to the idealized closed physical system Dynamical systems Then, by conservation of energy, we have or equivalently

  4. Let be the solution to the non-idealized closed physical system Dynamical systems Then, by dissipation of energy, we have or equivalently

  5. Energy vs. Lyapunov-functions Real physical systems end up in a state where the energy of the system is at a local minimum. Such a state is called a stable equilibrium If we have a differential equation that does not possess an energy, can we do something similar? Answer by Lyapunov 1892: if similar to energy Kurzweil/Massera 1950‘s: such an energy exists YES !

  6. Example: where Partition of the domain of V:

  7. Example: where Grid :

  8. Example: where Values for , that fulfill the constraints

  9. Example: where Convex interpolation delivers a Lyapunov-function

  10. Example: where Region of attraction:

  11. Generated Lyapunov-function

  12. Generated Lyapunov-function

  13. Generated Lyapunov-function

  14. Generated common Lyapunov-function

  15. Arbitrary switched systems right-continuous and the discontinuity points form a discrete set common Lyapunov function asymptotically stable under arbitrary switching

  16. Variable structure system

  17. Variable structure system (sliding modes) Weallowthesystemtoswitch arbitiarybetweenthedynamicson a thinstripoverlappingtheboundaries

  18. Variable structure system (sliding modes) Weallowthesystemtoswitch arbitiarybetweenthedynamicson a thinstripoverlappingtheboundaries

  19. Triangle-Fan Lyapunov function (with Peter Giesl Uni Sussex)

  20. Extension of the region of attraction alsowith Peter Wemakeadditionallinearconstraintsthatsecure Thentheregion of attractionsecuredbythe Lyapunovfunctionmustcontainthegreen box

  21. Extension of the region of attraction withoutoptimization withoptimization

  22. Extension of the region of attraction

  23. Differential inclusions and Filippov solutions (with L. Grüne and R. Baier Uni Bayreuth) is convex and compact is a Filippov solution iff a.e. and one allows evil right-hand sides, but demands high regularity of the solutions

  24. Differential inclusions and Filippov solutions is convex and compact for where is upper semicontinuous

  25. Differential inclusions and Filippov solutions Clarke, Ledyaev, Stern 1998 is strongly (everysolution) asymptoticallystable possesses a smooth Lyapunov function The algorithm can generate a Lyapunov function for the differential inclusion, if one exists. One just has to demand LC4 for faces of the simplices if necessary

  26. THANKS FOR LISTENING!

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