This paper analyzes a class of finite horizon optimal control problems with mixed non-convex and convex control constraints and linear state constraints. A convex relaxation of the problem is proposed, and it is proved that a solution of the relaxed problem is also a solution of the original problem. This process is called lossless convexification, and its generalization for problems with state constraints is the primary contribution of the paper. Doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non-convex problem. The approach is also demonstrated on example problems. (C) 2014 Elsevier Ltd. All rights reserved.