Properties of the composite quadratic Lyapunov functions

A composite quadratic Lyapunov function introduced recently was shown to be very useful in the study of set invariance properties for linear systems with input and state constraints and for systems with a class of convex/concave nonlinearities. In this note, more properties about this function are revealed. In particular, we study the continuity of the optimal parameter involved in this function. This continuity is crucial in the construction of a continuous feedback law which makes the convex hull of a group of ellipsoids invariant.