Abstract
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.
Highlights
Localized spots, sometimes referred to as auto-solitons [22], dissipative solitons [39], or bumps [24], are a subclass of traveling waves that spontaneously evolve in two-dimensional (2D) dissipative nonlinear systems driven far from thermodynamic equilibrium
Chemo-mechanical, electrical or neural activity is ubiquitous in spatially extended nonlinear systems driven far from thermodynamic equilibrium
We have demonstrated that the control signals, which one has to apply to solve these tasks, can be analytically expressed by the Goldstone modes of the uncontrolled spot
Summary
Sometimes referred to as auto-solitons [22], dissipative solitons [39], or bumps [24], are a subclass of traveling waves that spontaneously evolve in two-dimensional (2D) dissipative nonlinear systems driven far from thermodynamic equilibrium. Sustained moving hot-spot activity spontaneously formed near the wall of catalytic packed-bead and flow reversal reactors, respectively, may pose severe safety hazard problems Another example is the control of localized neural activity including so-called bump solutions to neural field equations describing ensembles of synaptically coupled neurons [11, 24, 56]. In some technical applications like catalytic reactors, it is necessary to avoid the collision of high-temperature spots with the reactor walls or their pinning at heterogeneities of the catalyst’s support to maintain operational safety [52] Another example of open-loop position control is the enhancement of the CO2 production rate during the low-pressure catalytic oxidation of CO on Pt(110) single crystal surfaces by dragging reaction pulses and fronts using a focused laser beam with a speed differing from their natural propagation velocity in the absence of control [41, 54]. In the spectrum of the linear stability operator of the uncontrolled stable solution a sufficiently large gap should exist between the symmetry-induced neutral eigenvalues on the imaginary axis and the remaining eigenvalues with negative real part
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