Optimal Trajectory Generation for Establishing Connectivity in Proximity Networks

We examine the problem of designing optimal trajectories to establish connectivity in a network of initially 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 (SDP) relaxation to efficiently approximate a solution of the resulting combinatorial optimization problem. Simulation results for double integrator agent dynamics are first provided to demonstrate feasibility for both approaches, and the results are compared with those obtained from exhaustive global search and random sampling. Additional simulation is performed for a specific spacecraft formation problem requiring the design of a stable connected network between a collection of fractionated spacecraft modules to illustrate the practicability and indicate the range of applications of the proposed approaches.