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

This paper presents lossless convexification for a class of finite horizon optimal control problems with non-convex control constraints and quadratic state constraints. Some special cases where the state at most touches the state constraint have been addressed previously in the literature. In this paper, the convexification results are generalized to allow optimal trajectories with boundary arcs. There are a number of practical examples that belong to the class of problems studied here. The optimal control problems considered have convex cost, specialized linear dynamics, quadratic state constraints, and non-convex control constraints. Hence, the control constraints are the single source of non-convexity. The control set is relaxed to a convex set by introducing a scalar slack variable. It is shown that optimal solutions of the relaxed problem are also optimal solutions of the original problem, hence the term lossless convexification. The main contribution of this paper is to extend the lossless convexification to the problem with quadratic state constraints. The proof uses a maximum principle with state variable inequality constraints and requires an assumption on the bounds of an external disturbance. A numerical example is presented to illustrate the approach. Because the numerical problem is a second order cone problem, convergence to the global minimum is guaranteed in a deterministic, finite number of steps.