On weak-invariance principles for nonlinear switched systems

In this technical note, we develop two weak-invariance principles for nonlinear switched systems. We first present a union weak-invariance principle for switched systems which includes as a special case the integral invariance principle. It is shown that the switched solution approaches the largest weakly invariant set of the combined zero loci of the output functions. Then, we extend the union weak-invariance principle to an intersection weak-invariance principle, which greatly reduces the convergence region. Unlike the existing results, in which the constructions of Lyapunov functions are inevitable, our principles do not require the existence of Lyapunov functions. Numerical examples are presented to demonstrate the feasibility of our principles, as well as applications to multi-agent consensus problems.