# A Higher-order Maximum Principle for Impulsive Optimal Control Problems

## TL;DR

This paper develops higher-order necessary optimality conditions for impulsive control problems involving nonlinear systems with unbounded controls, using a graph-space embedding and Lie brackets, under weak regularity assumptions.

## Contribution

It introduces a novel higher-order maximum principle for impulsive optimal control problems with nonlinear dynamics and unbounded controls, avoiding constant rank conditions.

## Key findings

- Provides necessary conditions involving iterated Lie brackets.
- Handles impulsive controls with weak regularity assumptions.
- Extends optimality theory to nonlinear, unbounded control systems.

## Abstract

We consider a nonlinear system, affine with respect to an unbounded control $u$ which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to $u$. This lack of coercivity gives the problem an {\it impulsive} character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. 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 provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions.