Controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model. In the present paper, under the main assumptions that the discretized semigroup is uniformly analytic, and that the control operator is mildly unbounded, we prove that the semidiscrete approximation models are uniformly controllable. Moreover, we provide a computationally efficient way to compute the approximation controls. An example of application is implemented for the one-and two-dimensional heat equation with Neumann boundary control.

In this paper, we present a model describing the dynamics of a population of ice floes with arbitrary shapes and sizes, which are exposed to atmospheric and oceanic skin drag. The granular model presented is based on simplified momentum equations for ice floe motion between collisions and on the resolution of linear complementarity problems to deal with ice floe collisions. Between collisions, the motion of an individual ice floe satisfies the linear and angular momentum conservation equations, with classical formula applied to account for atmospheric and oceanic skin drag. To deal with collisions, before they lead to interpenetration, we included a linear complementarity problem based on the Signorini condition and Coulombs law. The nature of the contact is described through a constant coefficient of friction l, as well as a coefficient of restitution 0 e 1 ð Þdescribing the loss of kinetic energy during the collision. In the present version of our model, this coefficient is fixed. The model was validated using data obtained from the motion of interacting artificial wood floes in a test basin. The results of simulations comprising few hundreds of ice floes of various shapes and sizes, exposed to different forcing scenarios, and under different configurations, are also discussed. They show that the progressive clustering of ice floes as the result of kinetic energy dissipation during collisions is well captured, and suggest a collisional regimes of floe dispersion at small scales, different from a large-scale regime essentially driven by wind forcing.

The microwave permeability of composites made of ferrite particles in a non-magnetic matrix is a!ected by particle shape and size. To account for the second e!ect, we have developed a micromagnetic model for calculating the microwave polarizability of a particle with non-uniform magnetization. This model is described in detail: a generalized demagnetizing operator associated with the equilibrium magnetic con"guration inside the particle is introduced; the polarizability is de"ned as the perturbation of this con"guration by the external magnetic excitation; the permeability of a macroscopic sample is calculated using a space averaging procedure. The numerical methods are introduced and results for two systems are discussed: domain modes in thin "lms and ferrite composites with planar anisotropy.1999 Elsevier Science B.V. All rights reserved.