Consensus Control With Failure—Wait or Abandon?

This paper introduces and solves a decision-making problem under the context of consensus control with failure. We study an optimal consensus control problem in which
${n}$
autonomous agents try to arrive at a target at the same time. One of the agents suddenly fails and the rest
${n} - 1$
agents can either wait or abandon the failed agent. If they wait, they must slow down and delay the consensus time. If they abandon the failed agent, they can reach consensus earlier at the cost of losing one agent at consensus. This cost is an added delay to the consensus time. The decision problem is to decide whether to wait or abandon and, if abandon, when? To solve this problem, we derive analytical expressions and establish structural properties for target distance functions. We use numerical examples and simulation examples to demonstrate the applications of the derived formulas and results.