English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Dynamic phenomena in social and biological sciences can often be modeled byemploying reaction-diffusion equations. When addressing the control of thesemodes, from a mathematical viewpoint one of the main challenges is that,because of the intrinsic nature of the models under consideration, thesolution, typically a proportion or a density function, needs to preserve givenlower and upper bounds. Controlling the system to the desired finalconfiguration then becomes complex, and sometimes even impossible. In thepresent work, we analyze the controllability to constant steady-states ofspatially homogeneous semilinear heat equations, with constraints in the state,and using boundary controls, which is indeed a natural way of acting on thesystem in the present context. The nonlinearities considered are among the mostfrequent: monostable and bistable ones. We prove that controlling the system toa 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. Whensuch an obstruction does not arise, we build sophisticated control strategiescombining the dissipativity of the system, the existence of traveling waves andsome connectivity of the set of steady-states. This connectivity allowsbuilding paths that the controlled trajectories can follow, in a long time,with small oscillations, preserving the natural constraints of the system. Thiskind of strategy was successfully implemented in one-space dimension, wherephase plane analysis techniques allowed to decode the nature of the set ofsteady-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 restrictedto the original domain.

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