arXiv subjects

M. A. Rodríguez-Bellido

Publications and source records attributed to M. A. Rodríguez-Bellido.

3 recordsLinked to original sources

arXiv2022

Bilinear optimal control for weak solutions of the Keller-Segel logistic model in $2D$ domains

An optimal control problem associated to the Keller-Segel with logistic reaction system will be studied in $2D$ domains. The control acts in a bilinear form only in the chemical equation. The existence of optimal control and a necessary optimality system are deduced. The main novelty is that control can be rather singular and the state (cell density $u$ and the chemical concentration $v$) remains only in a weak setting, which is not usual in the literature to solve optimal control problems subject to chemotaxis models (see e.g. Guillén-González et al. ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp).

math.OC↗

arXiv2020

Study of a chemo-repulsion model with quadratic production. Part I: Analysis of the continuous problem and time-discrete numerical schemes

We consider a chemo-repulsion model with quadratic production in a bounded domain. Firstly, we obtain global in time weak solutions, and give a regularity criterion (which is satisfied for $1D$ and $2D$ domains) to deduce uniqueness and global regularity. After, we study two cell-conservative and unconditionally energy-stable first-order time schemes: a (nonlinear and positive) Backward Euler scheme and a linearized coupled version, proving solvability, convergence towards weak solutions and error estimates. In particular, the linear scheme does not preserve positivity and the uniqueness of the nonlinear scheme is proved assuming small time step with respect to a strong norm of the discrete solution. This hypothesis is reduced to small time step in $nD$ domains ($n\le 2$) where global in time strong estimates are proved. Finally, we show the behavior of the schemes through some numerical simulations.

math.NA↗

arXiv2020

Study of a chemo-repulsion model with quadratic production. Part II: Analysis of an unconditional energy-stable fully discrete scheme

This work is devoted to the study of a fully discrete scheme for a repulsive chemotaxis with quadratic production model. By following the ideas presented in [Guilen-Gonzalez et al], we introduce an auxiliary variable (the gradient of the chemical concentration), and prove that the corresponding Finite Element (FE) backward Euler scheme is conservative and unconditionally energy-stable. Additionally, we also study some properties like solvability, a priori estimates, convergence towards weak solutions and error estimates. On the other hand, we propose two linear iterative methods to approach the nonlinear scheme: an energy-stable Picard method and Newton's method. We prove solvability and convergence of both methods towards the nonlinear scheme. Finally, we provide some numerical results in agreement with our theoretical analysis with respect to the error estimates.

math.NA↗