INTERNAL STABILIZATION OF MAXWELL’S EQUATIONS IN HETEROGENEOUS MEDIA

We consider the internal stabilization of Maxwell's equations 
with Ohm's law with space variable coefficients in a bounded 
region with a smooth boundary. Our result is mainly based on an 
observability estimate, obtained in some particular cases by the 
multiplier method, a duality argument and a weakening of norm 
argument, and arguments used in internal stabilization of scalar 
wave equations.


Introduction
Let Ω be an open bounded domain in R 3 with a boundary Γ of class C 2 .For the sake of simplicity we further assume that Ω is simply connected and that Γ is connected.
But to our knowledge the internal stabilization of Maxwell's equations with Ohm's law is only considered for constant coefficients λ and µ [17].Therefore our goal is to consider the internal stabilization of Maxwell's equations with Ohm's law for space variable coefficients λ and µ.We then give sufficient conditions guaranteeing the exponential decay of the energy.Our method actually combines arguments used in the stabilization of scalar wave equation with locally distributed (internal) damping [24] with the use of an internal observability estimate for the standard Maxwell equations obtained for constant coefficients by Phung [17] using microlocal analysis and extended here to some subsets ω of Ω and space variable coefficients.This observability estimate is obtained using a vectorial multiplier method (see [11] in the scalar case and [22] for constant coefficients), a duality argument from [1,12] and a weakening of norm argument (as in [11] in the scalar case).
The schedule of the paper is the following one: Well-posedness of the problem is analysed in Section 2 under appropriate conditions on Ω, λ, µ and σ using semigroup theory.Section 3 is devoted to the proof of the observability estimate when ω is a (small) neighbourhood of the boundary.Finally we conclude in Section 4 by the exponential stability of our system.

Well-posedness of the problem
Introduce the Hilbert spaces equipped with the inner product Now define the operator A as follows: S. Nicaise and C. Pignotti 793 where, as usual, For any D B in D(A) we take We then see that formally problem (1.1) to (1.5) is equivalent to when Φ = D B and Φ 0 = D0 B0 .We will prove that this problem (2.6) has a unique solution using Lumer-Phillips' theorem [16] by showing the following lemma.
Lemma 2.1.A is a maximal dissipative operator.
Proof.We start with the dissipativeness of A, in other words we need to show that (AΦ,Φ) H ≤ 0, ∀Φ ∈ D(A). (2.7) With the above notation we have By Green's formula and the boundary condition Let us now pass to the maximality.For that purpose it suffices to show that for all f g in H, there exists a unique D B in D(A) such that Equivalently, we have ) This last problem has a unique solution D in H 0 (curl,Ω) because its variational formulation is This problem has a unique solution by the Lax-Milgram lemma because the bilinear form defined as the left-hand side is coercive on H 0 (curl,Ω) because λ(1 + σ) ≥ λ 0 .
Since it is well-known that D(A) is dense in H (see [9,Section 7] or [10]), by Lumer-Phillips' theorem (see, e.g., [16,Theorem I.4.3]),we conclude that A generates a C 0semigroup of contraction T(t).Therefore we have the following existence result.Theorem 2.2.For all Φ 0 ∈ H, the problem (2.6) has a weak solution For our further use we also need the next result.
Proof.Denoting by A 0 the above operator A corresponding to σ = 0, the above problem (2.16) to (2.20) is equivalent to when Φ = D B and F = f 0 .As A 0 generates a C 0 -semigroup of contraction T 0 (t), problem (2.22) has a unique mild solution Φ ∈ C([0,∞),H) given by (see [16,Section 4.4.2]) (2.23) This identity implies that ds. ( We conclude by integrating the square of this estimate in t ∈ (0,T), using Cauchy-Schwarz's inequality and taking into account the assumption (1.6).
We end this section by showing that the energy of our system is decreasing.
Lemma 2.4.Let (D 0 ,B 0 ) be an initial pair in H and let (D,B) be the solution of the system (1.1), (1.2), (1.3), (1.4), and (1.5).Then the derivative of the energy (defined by (1.8)) is Proof.Deriving (1.8) we obtain then, by (1.1) and (1.2), We conclude by integrating by parts in the first term of this right-hand side and using the boundary condition (1.5).
From this lemma we directly conclude that the energy is non-increasing.

An observability estimate
Let us consider the solution (D h ,B h ) of the standard Maxwell system: For our next purposes, we need that the following internal observability estimate holds: The subset ω of Ω is such that there exist a time T > 0 and a constant C > 0 such that where This estimate was proved by Phung [17,Theorem 3.4] using microlocal analysis, when µ and λ are constant and ω = ω ∩ Ω such that ω controls geometrically Ω.We will extend such an estimate to variable coefficients and some open subsets ω using the multiplier method.For that purpose, we further require that there exist x 0 ∈ Ω and a positive constant c 0 such that for all x ∈ Ω.
We first reduce the estimate to the estimate of the electric field.
Lemma 3.1.Fix T > 0. Let (D h ,B h ) be the solution of (3.1), (3.2), (3.3), (3.4), and (3.5) with initial datum (D 0 ,B 0 ) ∈ H 1 .Then there exists C > 0 such that (3.9) S. Nicaise and C. Pignotti 797 Proof.We adapt step 1 of the proof of [17,Theorem 3.4] to our setting.Recall that the Hilbert space H T (curl,div, Ω), defined in (2.15), equipped with its natural norm is compactly embedded into (L 2 (Ω)) 3 [20].Therefore there exists a unique Indeed the above compactness property and the hypotheses on Ω and Γ guarantee that the left-hand side of (3.11) is coercive on H T (curl,div, Ω).On the other hand since div B h = 0 in Ω we easily see that the solution ψ of (3.11) satisfies (3.10) (see [2, Theorem 1.1]).Setting A = curlψ, we deduce that div Moreover taking w = ψ in (3.11) we see that this last estimate following from the compact embedding of H T (curl,div, Ω) into (L 2 (Ω)) 3 .In other words we have Using (3.2), (3.3), (3.5) and (3.12) to (3.14), we see that ) The first identity and the fact that Ω is simply connected imply that with ϕ ∈ H 1 (Ω).The properties (3.18), (3.19) and the fact that Γ is connected imply that ϕ is constant and therefore we conclude that Now by integration by parts in t, we get The identity (3.21) then yields  (3.31) The conclusion follows from (3.29).
Since it remains to estimate T 0 Ω |D h (x,t)| 2 dx dt we are looking at D h as solution of the following second order system: Consider the set continuously embedded into H 1 (Ω) 3 (see, e.g., [5,Section I.3.4]) and compactly embedded into L 2 (Ω) 3 [20].Let us set (3.37) The bilinear form a is symmetric and strongly coercive on ᐂ, moreover ᐂ is compactly embedded into Ᏼ (see [10]).By spectral analysis, the above problem has a unique solution The energy of the solution of that system is given by

.38)
A simple application of Green's formula shows that and therefore the energy E D is constant.Using a vectorial multiplier method we first prove the following lemma.An analogous lemma was proved in [22] in the case of constant coefficients.Lemma 3.2.Let D h be the solution of the system (3.32),(3.33), (3.34), and (3.35) with (D 0 ,D 1 ) ∈ ᐂ × Ᏼ, and let q : Ω → R 3 a C 1 vector field.Then for any time T > 0 the following identity holds: curl λD h i curl λD h j ∂ i q j dx dt Proof.By (3.32) Integrating by parts we obtain S. Nicaise and C. Pignotti 801 and then Analogously, we can rewrite curl λD h j q i ∂ i curl λD h j dx dt curl λD h i curl λD h j ∂ i q j − curl λD h 2 div q 802 Internal stabilization of Maxwell's equations and then curl λD h i curl λD h j ∂ i q j − µ curl λD h 2 div q + curl λD h 2 q • ∇µ dx dt.
(3.46) Therefore (3.40) follows observing that the boundary term can be rewritten using For any ε > 0 let us denote by ᏺ ε (Γ) the neighborhood of Γ of radius ε, that is, , for some > 0 and λ, µ satisfy (1.6), (3.8), then there exist T 0 > 0 and C > 0 such that for T > T 0 we have Proof.From (3.40), using the standard multiplier q(x) = m(x) = x − x 0 , we obtain for any T > 0 (3.50) Using the assumption (3.8), the above identity implies (3.51)Note that by (1.6) Now, set ω 0 = ᏺ ε/4 (Γ) and apply (3.40) using as multiplier q 804 Internal stabilization of Maxwell's equations We obtain for a suitable constant C > 0.Then, from (3.54) and (3.56), (3.57) Now, let g : Ω → R be a C 1 function with 0 ≤ g(x) ≤ 1, and , for any positive time T, by integration by parts, we have (3.60)By Young's inequality we can estimate Moreover, using the inequality consequence of the compact embedding of H N (curl,div, Ω) into L 2 (Ω) 3 , we have Therefore, using (3.61) and (3.63) in (3.60), we obtain for suitable positive constants C,C .Finally, by (3.57) and (3.64) we have for some constant C > 0. So, we can deduce the existence of a time T 0 such that for T > T 0 In a second step using a duality argument as in [1] (see also [12,Lemma 10]) we prove the following estimate.Lemma 3.4.Let D h be the solution of the system (3.32),(3.33), (3.34), and (3.35) with (D 0 ,D 1 ) ∈ ᐂ × Ᏼ.If ω = ᏺ (Γ) and ω = ᏺ /2 (Γ), for some > 0, then there exists C > 0 such that for any η > 0 we have for all w ∈ H N (curl,div, Ω).This solution z satisfies (due to the compact embedding of H N (curl,div, Ω) in L 2 (Ω) 3 and to the properties of Ω and Γ) , for some > 0 and λ, µ satisfy (1.6), (3.8), then there exist T 1 > 0 and C > 0 such that for T > T 1 we have for any η > 0. By the conservation of energy, this yields

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As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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