On Lyapunov-Krasovskii Functionals for Switched Nonlinear Systems with Delay

We present a set of results concerning the existence of Lyapunov-Krasovskii functionals for classes of nonlinear switched systems with time-delay. In particular, we first present a result for positive systems that relaxes conditions recently described in \cite{SunWang} for the existence of L-K functionals. We also provide related conditions for positive coupled differential-difference positive systems and for systems of neutral type that are not necessarily positive. Finally, corresponding results for discrete-time systems are described.Comment: 19 Page

Functions of perturbed operators

We prove that if 0<\a<1 and $f$ is in the H\"older class \L_\a(\R), then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and \|f(A)-f(B)\|\le\const\|A-B\|^\a. We prove a similar result for functions $f$ of the Zygmund class \L_1(\R): \|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|, where $A$ and $K$ are self-adjoint operators. Similar results also hold for all H\"older-Zygmund classes \L_\a(\R), \a>0. We also study properties of the operators $f(A)-f(B)$ for f\in\L_\a(\R) and self-adjoint operators $A$ and $B$ such that $A-B$ belongs to the Schatten--von Neumann class \bS_p. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.Comment: 6 page