# On the proof of Michel of the maximum Pontryagin Principle

## TL;DR

This paper improves the Pontryagin maximum principle for optimal control problems with final constraints, using advanced functional analysis tools and needlelike variations in a Banach space setting.

## Contribution

It introduces a novel proof of the maximum principle incorporating piecewise differentiability and recent multiplier rules, enhancing previous Michel's approach.

## Key findings

- Enhanced maximum principle with broader applicability
- Inclusion of piecewise differentiable state functions
- Use of functional analysis tools in proof structure

## Abstract

We provide an improvment of the maximum principle of Pon-tryagin of the optimal control problems, for a system governed by an ordinary differential equation, in presence of final constraints, in the setting of the piece-wise differentiable state functions (valued in a Banach space) and of piecewise continuous control functions (valued in a metric space). As Michel we use the needlelike variations, but we introduce tools of functional analysis and a recent multiplier rule of the static optimization to make our proofs. Mathematical Subject Classification 2010: 49K15, 47H10