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This chapter first introduces the constitutive relations which are commonly used in electromagnetic theory for the mathematical modelling of complex electromagnetic media. These constitutive relations are to be understood as operators connecting the electric flux density and the magnetic flux density with the electric and the magnetic fields. When they are introduced into the Maxwell equations, this chapter obtains differential equations (PDEs) that govern the evolution of the electromagnetic fields. This chapter also seeks to formulate and discuss the scope of the various problems related to the Maxwell equations that will be treated in this volume. It introduces and formulates in terms of differential equations various problems of interest related to the Maxwell equations: time-harmonic problems, scattering problems, time-domain evolution problems, random and stochastic problems, controllability problems, homogenisation problems, and others.

This chapter first presents a model for the Maxwell equations for complex random media in terms of stochastic integrodifferential equations. It then provides the appropriate functional framework that can allow for the treatment of the problem as an abstract stochastic integrodifferential equation in Hilbert space. In exploiting this framework, the chapter provides some solvability and well-posedness results for this model, using a semigroup-based approach. Hereafter, this chapter proposes alternative approaches to solvability and well posedness for the stochastic integrodifferential equations that arise in the modelling of random complex electromagnetic media, using either the finite-dimensional approximation (Faedo-Galerkin method) or the Wiener chaos approach.

This chapter presents a theory of homogenisation for random bianisotropic media exhibiting an ergodic structure. It shows that for such a medium there exists a homogenised system of the Maxwell type. To begin, the chapter presents an introduction to the necessary notions from the theory of ergodicity that will be used in this treatment of homogenisation. It then showcases a model for a random complex medium, on which this chapter's analysis will be based. The chapter also details a formal two-scale approach to set ideas and understand the basic mechanisms that will lead to a homogenised system, as well as to identify the coefficients of the homogenised system. Finally, this chapter presents some rigorous results on the homogenisation of random complex electromagnetic media.

This chapter addresses the problem of controllability for stochastic complex electromagnetic media. The starting point is the stochastic integrodifferential equations that model the evolution of the fields, with a control procedure to be selected so that the system is driven to a desired final state. The controllability problem for stochastic media is more complicated than that for deterministic media and includes subtleties that must be addressed to reach a satisfactory answer. Hence, this chapter sets the model and then discusses the subtle issues introduced by the stochasticity in the controllability problem. It then proposes two different approaches towards controllability: an approach using PDEs with random coefficients and an approach using backward stochastic evolution equations (BSEEs). Afterward, the chapter lists several comments concerning boundary controllability and optimal control.