Sub-optimal consensus protocol design for double integrator agents using krotov conditions

This work considers the problem of optimal consensus protocol design for the double integrator agents. The problem is tackled in the Krotov framework. The optimal consensus protocol design problem usually turns out to be non-convex due to the desired distributed nature of the protocol to be designed, a requirement imposed by the interaction topology of the agents, and hence the solution is not trivial to compute. This work employs the Krotov sufficient conditions to compute sub-optimal consensus protocol for the considered problem. These conditions transform the optimal control problem into another optimization problem which provides the sufficient conditions, called Krotov conditions, for the existence of optimal control laws. The latter optimization problem is obtained via a rather ad hoc selection of the so called Krotov function. In this work, these conditions are used to solve the consensus protocol design problem for double integrator multi-agent system by choosing the Krotov function such that the solution of the equivalent optimization problem has the desired distributed structure. The proposed method is demonstrated by a numerical example.