Second order necessary conditions for optimal control problems of evolution equations involving final point equality constraints

Hélène Frankowska; Qi Lü

doi:10.1051/cocv/2021065

All issues

Volume 27 (2021)

ESAIM: COCV, 27 (2021) 71

Abstract

Special issue in honor of Enrique Zuazua's 60th birthday

Open Access

Issue		ESAIM: COCV Volume 27, 2021 Special issue in honor of Enrique Zuazua's 60th birthday


Article Number		71
Number of page(s)		38
DOI		https://doi.org/10.1051/cocv/2021065
Published online		06 July 2021

ESAIM: COCV 27 (2021) 71

Second order necessary conditions for optimal control problems of evolution equations involving final point equality constraints

Hélène Frankowska¹^*^,** and Qi Lü²^,***

¹ CNRS, IMJ-PRG, UMR 7586, Sorbonne Université, case 247, 4 place Jussieu, 75252 Paris, France.
² School of Mathematics, Sichuan University, Chengdu, 610064, PR China.

^* Corresponding author: helene.frankowska@imj-prg.fr

Received: 9 October 2020
Accepted: 12 June 2021

Abstract

We establish some second order necessary conditions for optimal control problems of evolution equations involving final point equality and inequality constraints. Compared with the existing works, the main difference is due to the presence of end-point equality constraints. With such constraints, we cannot simply use the variational techniques since perturbations of a given control may be no longer admissible. We also cannot use the Ekeland’s variational principle, which is a first order variational principle, to obtain second order necessary conditions. Instead, we combine some inverse mapping theorems on metric spaces and second order linearization of data to obtain our results.

Mathematics Subject Classification: 49K20

Key words: Optimal control / time evolution partial differential equation / second order necessary condition / local minimizer

^**

The research of this author is partially supported by the AFOSR grant FA 9550-18-1-0254 and the CNRS-NSFC PRC Project under grant 271392. She also benefited from the support of the FJMH Program PGMO and from the support to this program from EDF-THALES-ORANGE-CRITEO.

^***

The research of this author is supported by the NSF of China under grants 11971334, 12025105 and 11931011, by the Chang Jiang Scholars Program from the Chinese Education Ministry, and the CNRS-NSFC PRC Project under grant 11530142.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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