Optimizing the Kelvin force in a moving target subdomain
This work addresses the control of magnetic forces for moving targets, which is relevant for applications like magnetic drug delivery, but the results are incremental as they extend existing optimization frameworks to a specific moving subdomain scenario.
The authors propose a minimization problem to generate a desired Kelvin force in a moving target subdomain, using dipole approximation with fixed source locations and bounded magnetic field intensity. They prove existence, local uniqueness, and convergence of a semi-discrete numerical scheme, with computational results demonstrating applicability to magnetic drug delivery.
In order to generate a desired Kelvin (magnetic) force in a target subdomain moving along a prescribed trajectory, we propose a minimization problem with a tracking type cost functional. We use the so-called dipole approximation to realize the magnetic field, where the location and the direction of the magnetic sources are assumed to be fixed. The magnetic field intensity acts as the control and exhibits limiting pointwise constraints. We address two specific problems: the first one corresponds to a fixed final time whereas the second one deals with an unknown force to minimize the final time. We prove existence of solutions and deduce local uniqueness provided that a second order sufficient condition is valid. We use the classical backward Euler scheme for time discretization. For both problems we prove the $H^1$-weak convergence of this semi-discrete numerical scheme. This result is motivated by $Γ$-convergence and does not require second order sufficient condition. If the latter holds then we prove $H^1$-strong local convergence. We report computational results to assess the performance of the numerical methods. As an application, we study the control of magnetic nanoparticles as those used in magnetic drug delivery, where the optimized Kelvin force is used to transport the drug to a desired location.