Establishing connectivity in proximity networks

We examine the problem of designing the optimal paths to establish connectivity in a network of intially scattered dynamic agents, specifically minimizing the squared integral of the total control effort. The network edges are modeled by proximity relationships between endpoint agents, leading to a dynamic, state-dependent network topology. We formulate an optimal control problem with specified initial states, linear dynamics, and a connectivity constraint on the final induced topology. Our approach utilizes the Hamiltonian and resultant Euler-Lagrange equations to restructure the optimal control formulation as a parameter optimization problem based on final agent states. We provide both a heuristic approach and an iterative semidefinite programming relaxation to efficiently approximate a solution of the resulting combinatorial optimization problem. Simulation results are provided (for double integrator agent dynamics) to demonstrate feasibility for both approaches, and the results are compared with those obtained from exhaustive global search and random sampling.