Minimum time consensus for damped second order agents using Gröbner basis

A problem of achieving minimum time consensus for a set of $N$ second-order LTI system agents with bounded inputs and fuel constraints is considered. Unlike our other works, here the damping effect in agent dynamics is included. First, the attainable set for each agent with fuel budget constraints is characterized, and its boundary equations are derived. Then, using the convexity property, the minimum time at which attainable sets of all agents have a non-empty intersection is computed. By applying Helly's theorem, the computation reduces to finding the minimum time to consensus and the corresponding consensus point for each of the triplets separately.