Chapter 12 Networks of Antennas: Power Optimization Stéphane Labbé 12.1 Introduction of the Problem In this text, we illustrate the process leading from a physical problem to an effective simulation. This process will display to three types of models. The first one, the most simple, will provide the opportunity to familiarise with the model and the existence of solutions to the problem, a second, more complex, will illustrate the necessity of numeric computations and at last we will give a complete formulation. The example we chose is the optimisation of a network of antennas. For the sake of simplification, the antennas taken into consideration will be assimilated to discrete dipolar systems. The goal of this study is to give an algorithm for antennas placement and power regulation. In a first part, we will focus on the modelling of the problem. We will start by setting the problem and choose the notations, and will, then, focus on the modelling of an antenna, explaining the link between the electromagnetic equations and a dipolar antenna. In the second, third and fourth parts, we will treat the three optimisation problems. 12.2 A Network of Antennas: Modelling The first work to perform in order to model a situation, is to set the problem in mathematical terms. The built model will enable the study and optimisation of the situation parameters. This first milestone of the modelling and simulation process is very important and must be carefully treated. The key of the modelling process S. Labbé () Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP (Institute of Engineering Univ. Grenoble Alpes), Grenoble, France e-mail: stephane.labbe@univ-grenoble-alpes.fr © Springer Nature Switzerland AG 2020 E. Lindner et al. (eds.), Mathematical Modelling in Real Life Problems, Mathematics in Industry 33, https://doi.org/10.1007/978-3-030-50388-8_12 155