On a distributed control problem for a coupled chemotaxis-fluid model

M. Ángeles Rodríguez-Bellido; Diego A. Rueda-Gómez; Élder J. Villamizar-Roa

doi:10.3934/dcdsb.2017208

Article Contents

2018, Volume 23, Issue 2: 557-571. Doi: 10.3934/dcdsb.2017208

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On a distributed control problem for a coupled chemotaxis-fluid model

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas and IMUS, Universidad de Sevilla, C/Tarfia, S/N, 41012 Sevilla, Spain
2.
Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia

* Corresponding author: M. Ángeles Rodríguez-Bellido
* Corresponding author: M. Ángeles Rodríguez-Bellido

Received: January 31, 2017

Revised: May 31, 2017

Early access: November 2017

Published: June 2018

The first author has been partially supported by MINECO grants MTM2012-32325 and MTM2015-69875-P (Ministerio de Economía y Competitividad, Spain) with the participation of FEDER. The second and third authors have been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842- 157-2016.

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Abstract

In this paper we analyze an optimal distributed control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations. We consider that the movement and the interaction of cells are occurring in a smooth bounded domain of $\mathbb{R}^n,n = 2,3,$ subject to homogeneous boundary conditions. We control the system through a distributed force and a coefficient of chemotactic sensitivity, leading the chemical concentration, the cell density, and the velocity field towards a given target concentration, density and velocity, respectively. In addition to the existence of optimal solution, we derive some optimality conditions.

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Mathematics Subject Classification: Primary: 35Q35, 49J20, 35Q92; Secondary: 76D05, 76D03.

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