Well-posedness of the Optimal Control Problem Related to Degenerate Chemo-attraction Models-Reference-Cited by-同舟云学术

Well-posedness of the Optimal Control Problem Related to Degenerate Chemo-attraction Models

Published:2024-07-16 Issue: Volume:19 Page:200-208
ISSN:2224-2856
Container-title:WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL
language:en
Short-container-title:

Author:

Serhal Sarah¹,Chamoun Georges²,Saad Mazen¹,Sayah Toni³

Affiliation:

1. Ecole Centrale de Nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1 rue de la Noé, 44321 Nantes FRANCE

2. Faculty of engineering (ESIB), Saint Joseph University of Beirut LEBANON

3. Laboratoire de Mathématiques et Applications, URMM, CAR, Faculté des Sciences, Université Saint-Joseph de Beyrouth, B.P 11-514 Riad El Solh, Beyrouth 1107 2050 LEBANON

Abstract

This paper delves into the mathematical analysis of optimal control for a nonlinear degenerate chemotaxis model with volume-filling effects. The control is applied in a bilinear form specifically within the chemical equation. We establish the well-posedness (existence and uniqueness) of the weak solution for the direct problem using the Faedo Galerkin method (for existence), and the duality method (for uniqueness). Additionally, we demonstrate the existence of minimizers and establish first-order necessary conditions for the adjoint problem. The main novelty of this work concerns the degeneracy of the diffusive term and the presence of control over the concentration in our nonlinear degenerate chemotaxis model. Furthermore, the state, consisting of cell density and chemical concentration, remains in a weak setting, which is uncommon in the literature for solving optimal control problems involving chemotaxis models.

Publisher

World Scientific and Engineering Academy and Society (WSEAS)

Reference17 articles.

1. G. Chamoun, M. Saad, R. Talhouk, A coupled anisotropic chemotaxis–fluid model: the case of two-sidedly degenerate diffusion, Comput. Math. Appl. 68(9), 2014, 1052–1070.

2. P. Laurencot and D. Wrzosek, A Chemotaxis model with threshold density and degenerate diffusion, Nonlinear differential equations and their applications, vol. 64, Birkhauser, Boston, 2005, pp. 273-290,

3. C. Liu and Y. Yuan, Optimal Control of a Fully Parabolic Attraction-Repulsion Chemotaxis Model with Logistic Source in 2D, Applied Mathematics and Optimization; New York, Vol.85, Iss. 1, 2022.

4. F. Dai and B. Liu, Optimal control and pattern formation for a haptotaxis model of solid tumor invasion, J. Frankl. Inst. 356(16), 9364-9406, 2019.

5. F. Guillén-González, E. Mallea-Zepeda, É.J. Villamizar-Roa, On a bi-dimensional chemorepulsion model with nonlinear production and a related optimal control problem, Acta Appl. Math,2020, 170, 963–979.