Piecewise quadratic stability of consensus in heterogeneous opinion dynamics

A piecewise quadratic Lyapunov function approach for the consensus in heterogeneous opinion dynamics is proposed. The connections among the agents are influenced by different thresholds which characterize the heterogeneity of the network. The continuous-time opinion dynamics model is represented as a piecewise affine system with the state space partitioned into convex polyhedra defined by the agents influence functions. Conditions on piecewise quadratic functions for their sign in the polyhedra and continuity over the common boundaries are given. A sufficient condition for the local asymptotic stability, i.e., the consensus, is formulated as a set of LMIs whose solution provides a continuous piecewise quadratic Lyapunov function. Numerical results show the effectiveness of the proposed approach.