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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.