Control of certain parabolic models from biology and social sciences

Ruiz-Balet D, Zuazua E (2022)

Publication Language: English

Publication Type: Journal article, Online publication

Publication year: 2022

Journal

Mathematical Control and Related Fields American Institute of Mathematical Sciences (AIMS)

Book Volume: 12

Pages Range: 955-1038

Journal Issue: 4

DOI: 10.3934/mcrf.2022032

Open Access Link: https://doi.org/10.3934/mcrf.2022032

Abstract

These lecture notes address the controllability under relevant state constraints of reaction-diffusion equations. Typically the quantities modeled by reaction-diffusion equations in socio-biological contexts (e.g. population, concentrations of chemicals, temperature, proportions etc.) are positive by nature. The uncontrolled models intrinsically preserve this nature thanks to the maximum principle. For this reason, any control strategy for such systems has to preserve these state constraints. We restrict our study in the case of scalar equations with monostable and bistable nonlinearities. The presence of constraints produces new phenomena such as a possible lack of controllability, or existence of a minimal controllability time. Furthermore, we explain general ways for proving controllability under state constraints. Among different strategies, we discuss how to use traveling waves and connected paths of steady states to ensure controllability. We devote particular attention to the construction of such connected paths of steady-states. Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to provide intuition to the reader.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Imperial College London / The Imperial College of Science, Technology and Medicine GB United Kingdom (GB)

How to cite

APA:

Ruiz-Balet, D., & Zuazua, E. (2022). Control of certain parabolic models from biology and social sciences. Mathematical Control and Related Fields, 12(4), 955-1038. https://dx.doi.org/10.3934/mcrf.2022032

MLA:

Ruiz-Balet, Domènec, and Enrique Zuazua. "Control of certain parabolic models from biology and social sciences." Mathematical Control and Related Fields 12.4 (2022): 955-1038.

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