Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case

TL;DR

This paper derives new necessary conditions involving Lie brackets for impulsive control problems where the system's vector fields depend linearly on control derivatives, focusing on the commutative case.

Contribution

It introduces novel necessary conditions for impulsive control systems with commutative vector fields, expanding the theoretical framework for such problems.

Findings

Derived new conditions involving Lie brackets and adjoint states

Focused on systems with commutative vector fields

Enhanced understanding of impulsive control optimality criteria

Abstract

In this article we study control problems with systems that are governed by ordinary differential equations whose vector fields depend linearly in the time derivatives of some components of the control. The remaining components are considered as classical controls. This kind of system is called `impulsive system'. We assume that the vector fields multiplying the derivatives of each component of the control are commutative. We derive new necessary conditions in terms of the adjoint state and the Lie brackets of the data functions.