Lossless convexification of a class of non-convex optimal control problems for linear systems

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem. This lossless convexification approach enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. We demonstrate this solution approach with a number of planetary soft landing problems.