Optimal time-varying topology for network systems

This paper treats an optimization problem of time-varying network topology for continuous-time linear network systems. Although the topology from controllers to nodes has been widely investigated in the literature, this paper alternatively focuses on the topology among nodes. For the cost function, we employ the trace of the controllability Gramian to reduce the required control energy. For the constraint, we introduce an upper bound of the L
⁰
norm of the data transmission among nodes to find an effective communication schedule. Since the formulated optimization problem is combinatorial, we introduce a convex relaxation for its computational tractability. After reformulating the problem into an optimal control problem where Pontryagin's maximum principle can be applied, we give a sufficient condition under which the relaxation problem gives a solution of the original problem. We finally provide a numerical algorithm based on a derived optimality condition and illustrate the proposed method with an example.