Convergence and synchronization in heterogeneous networks of smooth and   piecewise smooth systems
  This paper presents a framework for the study of convergence when the nodes' dynamics may be both piecewise smooth and/or nonidentical across the network. Specifically, we derive sufficient conditions for global convergence of all node trajectories towards the same bounded region of their state space. The analysis is based on the use of set-valued Lyapunov functions and bounds are derived on the minimum coupling strength required to make all nodes in the network converge towards each other. We also provide an estimate of the asymptotic bound $\epsilon$ on the mismatch between the node states at steady state. The analysis is performed both for linear and nonlinear coupling protocols. The theoretical analysis is extensively illustrated and validated via its application to a set of representative numerical examples. 
Pietro DeLellis, Mario di Bernardo, Davide Liuzza
math.DS