The order of optimal control problems

TL;DR

This paper clarifies the concept of problem order in optimal control problems, showing it is generally non-integer for multi-input systems and addressing past misunderstandings related to this concept.

Contribution

It proves that the problem order is an integer only for single-input systems, providing a clearer understanding of the order in multi-input control systems.

Findings

Problem order equals an integer for single-input systems.

In multi-input systems, the problem order can be non-integer.

Clarifies the distinction between problem order and local order.

Abstract

The Pontryagin's Maximum Principle allows, in most cases, the design of optimal controls of affine nonlinear control systems by considering the sign of a smooth function. There are cases, although, where this function vanishes on a whole time interval and the Pontryagin's Maximum Principle alone does not give enough information to design the control. In these cases one considers the time derivatives of this function until a k-order derivative that explicitly depends on the control variable. The number q=k/2 is called problem order and it is the same to all the extremals. The local order is a related concept used in literature, but depending on each particular extremal. The confusion between these two concepts led to misunderstandings in past works, where the problem order was assumed to be an integer number. In this work we prove that this is true if the control system has a single input but, in general, it is not true if the control system has a multiple input.