Lossless convexification of non-convex optimal control problems with disjoint semi-continuous inputs

TL;DR

This paper introduces a novel convex relaxation technique for a class of non-convex optimal control problems with semi-continuous inputs, enabling globally optimal solutions to be efficiently computed.

Contribution

It presents the first lossless convexification method for mixed-integer non-convex optimal control problems, solving them via second-order cone programming with guarantees.

Findings

Achieves globally optimal solutions for a class of non-convex problems

Significantly reduces computation time compared to traditional mixed-integer methods

Validates approach with a spacecraft docking example

Abstract

This paper presents a convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems that are non-convex in the input norm, which is a semi-continuous variable that can be zero or lower- and upper-bounded. Using lossless convexification, the non-convex problem is relaxed to a convex problem whose optimal solution is proved to be optimal almost everywhere for the original problem. The relaxed problem can be solved using second-order cone programming, which is a subclass of convex optimization for which there exist numerically reliable solvers with convergence guarantees and polynomial time complexity. This is the first lossless convexification result for mixed-integer optimization problems. An example of spacecraft docking with a rotating space station corroborates the effectiveness of the approach and features a computation time almost three orders of magnitude shorter than a mixed-integer programming formulation.