Lossless Convexification of Optimal Control Problems with Semi-continuous Inputs

TL;DR

This paper introduces a lossless convexification method for solving a broad class of mixed-integer non-convex optimal control problems with semi-continuous inputs, enabling reliable polynomial-time solutions.

Contribution

It extends lossless convexification to overlapping input sets with different bounds, integral costs, and discrete state constraints, a first in the field.

Findings

Convex relaxation yields globally optimal solutions almost everywhere.

Method solves problems with binary variables reliably and efficiently.

Rocket landing example demonstrates practical effectiveness.

Abstract

This paper presents a novel convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the input norms to be either zero or lower- and upper-bounded. The non-convex problem is relaxed to a convex one whose optimal solution is proved to be optimal almost everywhere for the original problem, a procedure known as lossless convexification. This paper is the first to allow individual input sets to overlap and to have different norm bounds, integral input and state costs, and convex state constraints that can be activated at discrete time instances. The solution relies on second-order cone programming and demonstrates that a meaningful class of optimal control problems with binary variables can be solved reliably and in polynomial time. A rocket landing example with a coupled thrust-gimbal constraint corroborates the effectiveness of the approach.