Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Ruiz-Balet D, Zuazua Iriondo E (2019)

Publication Language: English

Publication Type: Other publication type

Publication year: 2019

DOI: 10.1016/j.matpur.2020.08.006

Abstract

Dynamic phenomena in social and biological sciences can often be modeled by employing reactiondiffusion equations. Frequently in applications, their control plays an important role when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc. When addressing these issues from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0, 1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady-states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steady-state may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves and some connectivity of the set of steady-states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady-states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. The results are illustrated by numerical simulations of these models that find several applications, such as the extinction of minority languages or the survival of rare species in sufficiently large reserved areas.

Authors with CRIS profile

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

Additional Organisation(s)

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

Involved external institutions

University of Deusto / Universidad de Deusto ES Spain (ES)

How to cite

APA:

Ruiz-Balet, D., & Zuazua Iriondo, E. (2019). Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations.

MLA:

Ruiz-Balet, Domènec, and Enrique Zuazua Iriondo. Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations. 2019.

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