Lossless convexification for a class of optimal control problems with linear state constraints

This paper presents lossless convexification for a class of finite horizon optimal control problems with convex cost, linear dynamics, linear state constraints, and non-convex control constraints. There are a number of examples that belong to this class of problems since many practical problems are constrained to evolve in a bounded state space. The control set is relaxed to a convex set by introducing a scalar slack variable, and it is proved that optimal solutions of the relaxed problem are optimal solutions of the original problem, hence the term lossless convexification. Extending the proof to problems with state boundary arcs is the main theoretical contribution of this paper. The practical implication is that the non-convex optimization problem can be solved as a convex problem with guaranteed convergence properties. A numerical example is presented to illustrate the approach.