Multi-objective optimal formation reconfiguration with scalarization and stochastic boundaries

A new method is proposed for multi-objective optimal control of satellite formation reconfiguration. First, necessary optimality conditions of single-objective fuel-optimal reconfiguration are studied. Problem of initial guess of the costate and control discontinuity are addressed by using a new developed stochastic based method. Terminal conditions of Two Point Boundary Value Problem (TPBVP) of the optimal control equations are relaxed using augmented Gaussian. Variances or bandwidths of the terminal conditions are set to decision variables to be optimized. Responses of varied initial costate are modeled and propagated using Riccati equation. Using quadratic convolution of relaxed terminal conditions, objective function of the fuel-optimal problem is reformulated into a quadratic equation, and a gradient based optimizer is used to search the optimal initial guess. In the second part of the paper, optimal control of format flying reconfiguration with two competitive objectives:fuel and time of flight (ToF) is studied. With Pascoletti-Serafini scalarization, the Multi-objective Optimal Control Problem (MOCP) is scalarized into a set of weighted Single-objective Optimal Control Problems (SOCP). Hamiltonian and switch function for the weighted SOCP are developed and constructed. New developed stochastic based approach is then used to search the optimal Pareto solutions. Verification and performances of proposed algorithms are demonstrated through numerical simulations of single-objective and multi-objective optimal control of low-thrust elliptical orbit formation reconfiguration. Potential applications of the algorithms to some other preliminary space mission designs are also discussed.