# Necessary conditions involving Lie brackets for impulsive optimal control problems

## TL;DR

This paper develops higher order necessary conditions for impulsive optimal control problems involving nonlinear control-affine systems, incorporating Lie brackets and needle variations to handle impulsive trajectories.

## Contribution

It introduces a Higher Order Maximum Principle that includes Lie bracket conditions and extends classical needle variation methods to impulsive control scenarios.

## Key findings

- Higher order necessary conditions involve Lie brackets.
- Impulsive trajectories are characterized using a distributional approach.
- An example demonstrates the exclusion of certain extremals by the new conditions.

## Abstract

We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward "impulsive", namely discontinuous, trajectories. As is known, a distributional approach does not make sense in such a nonlinear setting, where instead a suitable embedding in the graph space is needed. We will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.