Lossless Convexification for Optimal Control Problems with Multiple Inputs and a Control Exclusivity Constraint

Lossless convexification is a method for numerically solving continuous time non-convex optimal control problems via a Cook reduction to a convex program. Previous research has shown the method to be applicable to a general class of state-constrained continuous optimal control problems with a single input constrained to a non-convex set. Here we extend the methodology to a class of mixed-integer optimal control problems with multiple inputs subject to the same non-convex constraint but only a subset of which can be simultaneously active. We present a relaxation of the original problem to a convex program and a solution algorithm composed of two second-order cone programs. Our main contribution is a proof that this algorithm solves the original problem to global optimality. Second-order cone programming is a subclass of convex optimization for which numerically reliable and efficient algorithms are available that have guaranteed convergence and run in polynomial time. Our result thus shows a subclass of $\mathcal{NP}$-hard optimal control problems to be $\mathcal{P}$. A satellite rendezvous and a supersonic aircraft center-of-gravity management example demonstrate the effectiveness of the approach over mixed-integer programming.