sliding mode control is a discontinuous control technique which has a surface to be reached. For ensuring that whether sliding modes exists on that pre-defined surface we often take time derivative. A fact is that derivative exists only if the function is continuous. Lipschitz condition or existance of Lipschitz for a function guarantee the continuity (locally or globally)

"A fact is that derivative exists only if the function is continuous."

Derivative exists iff a function is differentiable. Just continuity does not suffice.

Moreover, a Lipschitz continuous function does not need to be differentiable everywhere.

Going through some examples will be very useful for understanding the concept and application for your specific problem. Good examples can be found in following links:

https://www.tu-chemnitz.de/mathematik/part_dgl/publications/Alt_Griesse_Metla_R%C3%B6sch__Lipschitz_Stability_for_Elliptic_Optimal_Control_Problems_with_Mixed_Control-State_Constraints.pdf

http://ac.els-cdn.com/S0196885805000345/1-s2.0-S0196885805000345-main.pdf?_tid=f8107888-a664-11e4-964f-00000aacb360&acdnat=1422391511_e0507c90fec94bf73e767f59313bfb51

http://www.mathnet.or.kr/mathnet/kms_tex/984.pdf

http://page.math.tu-berlin.de/~troeltz/arbeiten/ppr749-02.pdf

Dear Sruthi Nair

In my lectures:

http://home.agh.edu.pl/~pgrab/grabowski_files/nonlinear/nonlinearse3.xml#x7-30003

you can find some indications how nonlinear (Lipschitz) semigroups can be applied

to a system with a slide regime. For a more general Lyapunov functionals you may

consult 2 old paper by Weissenberger present on the list of references.

Regards,

Piotr Grabowski