Geometric Conditions for Lossless Convexification in Linear Optimal Control with Discrete-Valued Inputs: Real-Time Implementation for Spacecraft Rendezvous

Optimal control problems with discrete-valued inputs are inherently challenging due to their mixed-integer nature, rendering them generally intractable for real-time, safety-critical aerospace applications. Lossless convexification offers a powerful alternative by reformulating these mixed-integer programs into computationally efficient convex programs. This paper develops a lossless convexification framework for the optimal control of linear time-varying systems with discrete-valued inputs. We extend existing theoretical results by demonstrating that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation. Furthermore, we establish that under simple geometric conditions on the input set, the solution to the relaxed convex problem strictly satisfies the original non-convex input constraints. This framework enables the real-time computation of optimal discrete-valued controls without resorting to mixed-integer optimization. The proposed algorithm is validated on a spacecraft rendezvous maneuver utilizing discrete-valued reaction thrusters in an elliptical orbit. Numerical results from Monte Carlo simulations confirm that the algorithm consistently yields exact discrete-valued control inputs with computational timelines compatible with safety-critical, on-board applications.