Lyapunov functions for trichotomies with growth rates

Shatter functions with polynomial growth rates

We study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer’s Lemma.

Nondecreasing Lyapunov functions

We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switche...

On quadratic Lyapunov functions

Control Lyapunov Functions :

A control design method for nonlinear systems based on control Lyapunov functions and inverse optimality is analyzed. This method is shown to recover the LQ optimal control when applied to linear systems. More generally, it is shown to recover the optimal control whenever the level sets of the control Lyapunov function match those of the optimal value function. The method can be readily applied...

Nonsmooth Control-Lyapunov Functions

It is shown that the existence of a continuous controlLyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finitedimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative that has been studied in set-valued analysis and the theory of differential inclusions with various names such as “upper conting...