Solving nonconvex optimal control problems by convex optimization

Motivated by aerospace applications, this paper presents a methodology to use second-order cone programming to solve nonconvex optimal control problems. The nonconvexity arises from the presence of concave state inequality constraints and nonlinear terminal equality constraints. The development relies on a solution paradigm, in which the concave inequality constraints are approximated by successive linearization. Analysis is performed to establish the guaranteed satisfaction of the original inequality constraints, the existence of the successive solutions, and the equivalence of the solution of the original problem to the converged successive solution. These results lead to a rigorous proof of the convergence of the successive solutions under appropriate conditions as well as nonconservativeness of the converged solution. The nonlinear equality constraints are treated in a two-step procedure in which the constraints are first approximated by first-order expansions, then compensated by second-order corrections in each of the successive problems for enhanced convergence robustness. Applications in highly constrained spacecraft rendezvous and proximity operations, finite-thrust orbital transfers, and optimal launch ascent are provided to demonstrate the effectiveness of the methodology.