It is well know that to demonstrate stability of internal dynamics in nonlinear systems, specially after applying input-output linearization, some conditions need to be attend, Lyapunov function for internal stability must be upper and lower bounded by class K functions and its first derivative (of lyapunov function) must be less or equal to another class K function Had someone constructed such kappa functions? how can I do it? If possible, share some paper where this procedure had been done!
Thanks in Advanced!
Dear Yohan Díaz,
If the Lyapunov function does not depend on time explicitely, is positive definite in a neighbourhood of $0$ having negative definite derivative along the system trajectories the lower bounds for $V$ and $-\dot{V}$ can be constructed by minimizing them on a sphere of a given radius of $r$, and maximizing to get an upper bounds. On this way you may obtain naturally $K$-ffunctions. Combining them you can deduce EXS. This to some extend concerns time-dependent Lyapunov functions.
Regards,
Piotr Grabowski
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