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Construction of Lyapunov functions with linear optimization

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Construction of Lyapunov functionswith linear optimization

Sigurður F. Hafstein, Reykjavík University

Whatcanwedoto get informationaboutthesolution?

- Analyticalsolution (almostneverpossible)
- Numericalsolution (not applicable for thegeneralsolution, badapproximation for largetimes for specialsolutions)
- Search for trapsinthephase-space ( trap = forwardinvariant set)

Let be the solution to the idealized closed physical system

Dynamical systemsThen, by conservation of energy, we have

or equivalently

Let be the solution to the non-idealized closed physical system

Dynamical systemsThen, by dissipation of energy, we have

or equivalently

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 !

Arbitrary switched systems

right-continuous and the discontinuity

points form a discrete set

common Lyapunov function

asymptotically stable under arbitrary switching

Variable structure system (sliding modes)

Weallowthesystemtoswitch

arbitiarybetweenthedynamicson a thinstripoverlappingtheboundaries

Variable structure system (sliding modes)

Weallowthesystemtoswitch

arbitiarybetweenthedynamicson a thinstripoverlappingtheboundaries

Triangle-Fan Lyapunov function

(with Peter Giesl Uni Sussex)

Extension of the region of attraction

alsowith

Peter

Wemakeadditionallinearconstraintsthatsecure

Thentheregion of attractionsecuredbythe

Lyapunovfunctionmustcontainthegreen box

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

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

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