Control of traveling localized spots

S. Martens¹ ; C. Ryll² ; J. Löber^1, 3 ; F. Tröltzsch² ; H. Engel¹

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.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 46

Résumé

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.

Détail

DOI : 10.1051/mmnp/2021036

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