Motivated by problems arising in magnetic drug targeting, we propose to generate an almost constant Kelvin (magnetic) force in a target subdomain, moving along a prescribed trajectory. This is carried out by solving a minimization problem with a tracking type cost functional. The magnetic sources are assumed to be dipoles and the control variables are the magnetic field intensity, the source location and the magnetic field direction. The resulting magnetic field is shown to effectively steer the drug concentration, governed by a drift-diffusion PDE, from an initial to a desired location with limited spreading.From step 3 there exists {(v τ , w τ )} τ >0 such that v τ v and w τ w in H 1 (0, T ) andwhere we have used thatᾱ τ is a minimizer for J τ together with (4.5).5.-Convergence: Since (ᾱ τ ,θ τ ) is a minimizer we deduce the inequality J τ (ᾱ τ ,θ τ ) ≤ J τ (Π τᾱ , Π τθ ), whence applying first step 3 and next step 2 we see that

No abstract

No abstract

In this paper we are concerned with two topics: the formulation and analysis of the eigenvalue problem for the curl operator in a multiply-connected domain, and its numerical approximation by means of finite elements. We prove that the curl operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space for which curl v • n = 0 on the boundary. Additional conditions must be imposed when the physical domain is not topologically trivial: we show that a viable choice is the vanishing of the line integrals of v on suitable homological cycles lying on the boundary. A saddle-point variational formulation is devised and analyzed, and a finite element numerical scheme is proposed. It is proved that eigenvalues and eigenfunctions are efficiently approximated, and some numerical results are presented in order to test the performance of the method.

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.