Distributed optimal control design for the identical continuous-time systems

This paper studies the distributed optimal control design problem for the continuous-time systems. We consider an identical decoupled dynamical equation of the multi-agent system and a coupled performance index which is described through the Laplacian matrix. Using the LQR method, the optimal feedback gain matrix is got. It is proved that the optimal feedback matrix corresponds to a complete undirected graph for the Laplacian matrix in the cost function. We aim to get the distributed control that only uses the information of its state and the states of its neighbors. Then the existence conditions of the distributed optimal control for special cost function are given and the asymptotic stability of the system is discussed. It is also showed that optimal feedback matrix corresponds to a complete undirected graph when the conditions don´t satisfy. Examples are given to show the efficiency of the proposed results.