| Issue |
Math. Model. Nat. Phenom.
Volume 16, 2021
Control of instabilities and patterns in extended systems
|
|
|---|---|---|
| Article Number | 46 | |
| Number of page(s) | 27 | |
| DOI | https://doi.org/10.1051/mmnp/2021036 | |
| Published online | 26 July 2021 | |
Control of traveling localized spots*
1
Institut für Theoretische Physik, Hardenbergstraße 36, EW 7-1, Technische Universität Berlin,
10623
Berlin, Germany.
2
Technische Universität Berlin, Institut für Mathematik,
Str. d. 17. Juni 136, MA 4-5,
10623
Berlin, Germany.
3
Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Straße 38,
01187
Dresden, Germany.
** Corresponding author: h.engel@physik.tu-berlin.de
Received:
7
October
2020
Accepted:
10
June
2021
Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction–diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot’s concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.
Mathematics Subject Classification: 35B06 / 35C07 / 93C10 / 49M41 / 35Q93
Key words: Pattern formation / Goldstone mode control / reaction–diffusion system / instabilities / traveling spot solutions
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© The authors. Published by EDP Sciences, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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