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

Abstract

This article 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 article 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 article are applied to Mars landing problems, demonstrating that under glide-slope constraints, the exact convex can be realized.

Problem Addressed

Exact convex relaxation of general nonconvex optimal control problems remains difficult, with existing sufficient conditions restricted to particular problem structures and not directly applicable to broader formulations.

Key Contributions

• Two unified sufficient conditions guaranteeing exactness of convex relaxation for general nonconvex optimal control problems
• Novel proof technique based on a locally constructed subproblem
• Transformation into classical controllability, normality, and strong observability conditions for specific constraint set forms
• Approximation methods modifying cost function, control constraints, and system dynamics to satisfy the conditions

Conclusions

The proposed conditions unify and extend prior exact convex relaxation theory, and the approximation methods ensure the sufficient conditions hold in specific scenarios. Applied to Mars landing problems, the theoretical results are confirmed numerically, demonstrating that under glide-slope constraints the nonconvex problem can be converted into an exactly equivalent convex one.