Pinning dynamic systems of networks with Markovian switching couplings and controller-node set

TL;DR

This paper investigates the stabilization of networks with dynamically changing couplings and controller nodes governed by Markovian switching, providing conditions for stability under different switching speeds.

Contribution

It introduces new stability criteria for networks with stochastic switching couplings and controller sets, applicable to mobile agent systems and systems with slow or fast switching.

Findings

Stability achieved with slow Markovian switching if subsystems are individually stabilizable.

Stability achieved with fast switching if average system parameters are stabilizable.

Numerical examples validate theoretical stability conditions for mobile agents and switching systems.

Abstract

In this paper, we study pinning control problem of coupled dynamical systems with stochastically switching couplings and stochastically selected controller-node set. Here, the coupling matrices and the controller-node sets change with time, induced by a continuous-time Markovian chain. By constructing Lyapunov functions, we establish tractable sufficient conditions for exponentially stability of the coupled system. Two scenarios are considered here. First, we prove that if each subsystem in the switching system, i.e. with the fixed coupling, can be stabilized by the fixed pinning controller-node set, and in addition, the Markovian switching is sufficiently slow, then the time-varying dynamical system is stabilized. Second, in particular, for the problem of spatial pinning control of network with mobile agents, we conclude that if the system with the average coupling and pinning gains can be stabilized and the switching is sufficiently fast, the time-varying system is stabilized. Two numerical examples are provided to demonstrate the validity of these theoretical results, including a switching dynamical system between several stable sub-systems, and a dynamical system with mobile nodes and spatial pinning control towards the nodes when these nodes are being in a pre-designed region.