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An analysis of the mapping properties of three commonly used domain integro-differential operators for electromagnetic scattering by an inhomogeneous dielectric object embedded in a homogeneous background is presented in the Laplace domain. The corresponding three integro-differential equations are shown to be equivalent and well-posed under finite-energy conditions. The analysis allows for non-smooth changes, including edges and corners, in the dielectric properties. The results are obtained via the Riesz-Fredholm theory, in combination with the Helmholtz decomposition and the Sobolev embedding theorem.

The z-coil of an MRI-scanner is modeled as a set of circular loops of strips, or rings, placed on one cylinder. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time harmonic. The frequency is low enough to allow for a quasi-static approximation. The rings have a thin rectangular cross-section; the thickness is so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur, resulting in a so-called edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and the self-inductance decreases. From the Maxwell equations, an integral equation for the current distribution in the rings is derived. The Galerkin method is applied, using Legendre polynomials as global basis functions, to solve this integral equation. This method shows a fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N ≥ 1) rings, and explicit results are presented for N = 1, N = 2 and N = 24.

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In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sucient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagin's maximumprinciple and dynamic programming. Sucient and sometimes necessary conditions for the existence of positive stabilizing controls are incorporated. Convergence properties between the nite and innite horizon case are presented. Besides these analytical methods, we describe briey a method for the approximation of the optimal controls for the nite and innite horizon problem.

The current distribution in a parallel set of thin conducting sheets due to an external applied source is investigated. All sheets are placed in one plane. The source, and all excited fields, are time harmonic. The frequency is low enough to allow for an electro quasi-static approximation (neglecting the displacement current). The conducting sheets are infinitely long and the current is uniform in the longitudinal direction of the sheets. The sheets have a thin rectangular cross-section, so thin that the current can be assumed uniform in the thickness-direction. Hence, the current distribution only depends on the transverse coordinate. Due to the mutual induction between the sheets, the current distribution over the width of the cross-section becomes non-uniform: it accumulates at the edges of the sheets. It is especially this so-called edge-effect, and its dependence on the applied frequency and the distances between the sheets, that is the aim of this investigation. From the Maxwell equations, a set of integral equations for the current distribution in the sheets is derived. These integral equations are solved, as far as possible by analytical means, by writing the current distribution in each sheet as a series of Legendre polynomials. The general method is worked out for N (N ≥ 1) sheets, but explicit results are presented for N = 1 and N = 3. It turns out that the edge-effect becomes stronger for increasing frequencies. For this solution, only a very restricted number of Legendre polynomials is needed.

[1] To describe the behavior of a finite array, the current distribution on each of its elements is represented by a finite number of eigencurrents or modes. Subsequently, the eigencurrents of the array are expanded in terms of these element eigencurrents. For uniform linear arrays of loops and dipoles, the array-eigencurrent expansions and their associated eigenvalues are investigated. We focus on their parameter independent and diagonalizing features, and on their interpretation in terms of far-field characteristics and (standing) wave behavior.

A transverse gradient coil (x-or y-coil) of an MRI-scanner is modeled as a network of curved circular strips placed at the surface of a cylinder. The current in this network is driven by a time-harmonic source current. The low frequency applied allows for an electro-quasi-static approach. The strips are thin and the current is assumed to be uniformly distributed in the thickness direction. For the current distribution in the width direction of the strips, an integral equation is derived. Its logarithmically singular kernel represents inductive effects related to the occurrence of eddy currents. For curved circular strips of width much smaller than the radius of the cylinder one may locally replace the curved circular strip by a tangent plane circular strip. This plane geometry preserves the main characteristics of the transverse current distribution through the strips. The current distribution depends strongly on the in-plane curvature of the strips. The Petrov-Galerkin method, using Legendre polynomials, is applied to solve the integral equation and shows fast convergence. Explicit results are presented for two examples: a set of 1 strip and one of 10 strips. The results show that the current distributions are concentrated near the inner edges and that resulting edge-effects, both local and global, are non-symmetric. This behavior is more apparent for higher frequencies and larger in-plane curvatures. Results have been verified by comparison with finite-element results.