Relative controllability of delay multi-agent systems

This article considers the controllability of delayed linear and nonlinear multi-agent systems, respectively, with leader-follower architecture and fixed communication topology. For the linear multi-agent systems, a relative protocol is designed to realize the interactions among agents and explicit solutions of the controlled agreement system are constructed in two cases, respectively, involving two kinds of delayed exponential matrix functions and the properties of the Kronecker product. Further Gramian and rank criteria for relative controllability are established, respectively. For the nonlinear ones, the control problem is transformed into the existence of fixed points which is tackled via. Krasnoselskii's fixed point theorem. Numerical examples of linear and nonlinear systems are given to verify the theoretical results.