Unified Sufficient Conditions for Exact Convex Relaxation of Nonconvex Optimal Control Problems

This paper focuses on achieving exact convex relaxation of optimal control problems characterized by nonconvex control constraints and convex state constraints. By employing the convex hull of the original nonconvex control constraint set, the constraints are relaxed, thereby transforming the original problem into a convex problem. The paper introduces two unified sufficient conditions to ensure the relaxation's exactness, guaranteeing that the solution derived from the relaxed problem remains globally optimal for the original nonconvex problem. Although one of the proposed sufficient conditions is abstract and nontrivial, we prove that it can be transformed into standard controllability, normality, and strong observability conditions proposed by the previous work about exact convex relaxation, if the control and state constraint sets have certain properties. Furthermore, approximation methods are developed to modify the cost function, control constraints, and system dynamics to ensure that the sufficient conditions are satisfied in certain scenarios. The results of this paper are applied to Mars landing problems, demonstrating that under glide-slope constraints, the exact convex can be realized.