Control of reaction-diffusion models in biology and social sciences
Ruiz-Balet D, Zuazua Iriondo E (2024)
Publication Language: English
Publication Status: Accepted
Publication Type: Unpublished / Preprint
Future Publication Type: Other publication type
Publication year: 2024
Publisher: Math. Control Relat. Fields
URI: https://dcn.nat.fau.eu/wp-content/uploads/NotesSichuan-DRuizBalet_EZuazua.pdf
DOI: 10.3934/mcrf.2022032
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/NotesSichuan-DRuizBalet_EZuazua.pdf
Abstract
These lecture notes address the controllability under state constraints of reaction-diffusion
equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.
The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.
We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.
The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.
We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems’ dynamics and, in particular, the use of traveling waves and steady-states’ paths, can be employed to build controls driving the system to the desired target.
We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.
Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
United Kingdom (GB)How to cite
APA:
Ruiz-Balet, D., & Zuazua Iriondo, E. (2024). Control of reaction-diffusion models in biology and social sciences. (Unpublished, Accepted).
MLA:
Ruiz-Balet, Domènec, and Enrique Zuazua Iriondo. Control of reaction-diffusion models in biology and social sciences. Unpublished, Accepted. 2024.
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