LAUSR

In this article, we consider the exponential consensus of coupled inertial (double-integrator) agents, particularly with the general setting of the damping and stiffness control gains. Each agent has one damping gain and one stiffness gain. Here, the damping and stiffness control gains of all agents can be both fully heterogeneous (FH) and fully variable (FV), which are called the FH-FV gains for convenience of reference. Specifically, the FH gains are defined as follows: 1) the damping gains of all agents are heterogeneous; 2) the stiffness gains of all agents are heterogeneous; and 3) the set of the damping gains and the set of the stiffness gains are distinct without dependence. Otherwise, the control gains are said partially heterogeneous (PH). The FV or partially variable (PV) aspect of control gains is defined similarly. The FH-FV gains setting is novel and generalizes the specially PH settings of constant gains in previous papers. We also consider the general FH-PV gains and the PH-PV gains. Then, we provide the series of conditions that ensure exponential convergence to consensus, for the agents with the FH-FV gains, the general FH-PV gains, and the PH-PV gains, respectively. The series of the conditions for each type of control gains has particular meaning for characterizing heterogeneity of the gains, especially, when the digraph of the agents is far-from-balanced.