Generalized Semistability and Stochastic Semistability for Switched Nonlinear Systems Using Fixed Point Theory1

This paper generalizes our previous results on semistability and stochastic semistability for switched nonlinear systems published in the Proceedings of 2021 Modeling, Estimation and Control Conference. The paper also provides the results on semistability in mean square for switched nonlinear discrete-time systems. The theoretical result involves generalized sufficient conditions for (stochastic) semistability and semistability in mean square of discrete-time nonlinear dynamical systems under time-varying or random (arbitrary) switching by means of Fixed Point Theory. An advantage of these results is to overcome fundamental challenges arising from using existing methods such as Lyapunov and LaSalle methods. As an application of the theoretical results presented, a constrained distributed consensus problem over random multi-agent networks is considered for which a generalized asynchronous and totally asynchronous iterative algorithm is derived. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a numerical example is given in which there is a distribution dependency among communication graphs to demonstrate the results.