Control of traveling localized spots

The ability to control a desired dynamics or pattern in reaction-diffusion systems has attracted considerable attention over the last decades and it is still a fundamental problem in applied nonlinear science. Besides traveling waves, moving localized spots -- also called dissipative solitons -- represent yet another important class of self-organized spatio-temporal structures in non-equilibrium dissipative systems. In this work, we focus attention to control aspects and present an efficient method to control both the position and orientation of these moving localized structures according to a prescribed protocol of movement. In detail, we present two different approaches to guide a localized spot along a pre-given trajectory. First, an analytical solution for the control -- being an open-loop control -- is proposed which attempts to shift the spot without disturbing its profile. The control signal is expressed in terms of the uncontrolled spot profile and its propagation velocity; rendering detailed informations about the reaction kinetics unnecessary. Secondly, the standard formulation of optimal control with an objective functional involving the Tikhonov regularization is used. We confirm, within numerical accuracy, that the analytical position control is indeed the solution to an unregularized optimal control problem and even close to the regularized one. Consequently, the analytic expressions are excellent initial guesses for the numerical calculation of regularized optimal control problems, thereby achieving a substantial computational speedup.