State-constrained semilinear elliptic optimization problems with unrestricted sparse controls

Casas Eduardo; Tröltzsch Fredi

doi:10.3934/mcrf.2020009

Article Contents

2020, Volume 10, Issue 3: 527-546. Doi: 10.3934/mcrf.2020009

This issue Previous Article Optimality conditions in variational form for non-linear constrained stochastic control problems Next Article Optimal periodic control for scalar dynamics under integral constraint on the input

State-constrained semilinear elliptic optimization problems with unrestricted sparse controls

Casas Eduardo^1, and
Tröltzsch Fredi^2, ,

1.
Departamento de Matemática Aplicada y Ciencias de la Computació, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain
2.
Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany

^* Corresponding author: Eduardo Casas
^* Corresponding author: Eduardo Casas

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

Received: January 31, 2018

Revised: November 30, 2018

Early access: December 2019

Published: July 2020

The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

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Abstract

In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.

Keywords:

Optimal control,
semilinear elliptic equations,
first and second order optimality conditions,
sparse controls.

Mathematics Subject Classification: Primary: 35J61, 49K20; Secondary: 49J20.

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References

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