Time domain decomposition in final value optimal control of the Maxwell system

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002042

TIME DOMAIN DECOMPOSITION IN FINAL VALUE OPTIMAL CONTROL OF THE MAXWELL SYSTEM

John E. Lagnese

and G. Leugering

Abstract. We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time control applications.

Mathematics Subject Classification. 65N55, 49M27, 35Q60.

Received December 17, 2001. Revised February 16, 2002.

1. Introduction

Problems of optimal control of electromagnetic waves arise in a variety of applications, e.g. in stealth technology, design and control of antennas, diffraction optics, magnetotellurics and related fields. In many important cases the objective to be met relates to the final values of both the electric and the magnetic fields involved. Hence, the problem of exact controllability has attracted considerable interest. See Lagnese [9] as an early paper and Phung [15] or Belishev and Glasman [2] for more recent ones on this topic. The list of references is far from being exhaustive. As a more realistic requirement, one typically asks for controls that bring the final states close to a given target configuration, that is, one wants to achieve optimal controls. While the papers mentioned focus on the case of constant permeabilities and permittivities, modern applications require dealing with heterogeneous materials. In addition, in most cases real-time requirements are to be met.

Therefore, in order to obtain adjoint-based gradients and sensitivities in real-time, the large scale (or global) heterogeneous problem has to be reduced to smaller, more standard ones that can be processed in parallel.

Domain decomposition of the Maxwell system for the purpose of simulation has been considered ine.g. Alonso and Valli [1] and Santos [16]. Domain decompositions with respect to optimality systems have been investigated by Lagnese [8]. In this paper we approach the problem of time domain decompositions of the optimality systems.

Time decomposition methods (TDDM) appear to be a promising tool in real-time applications, as the sometimes long time horizon, which would be prohibitive in terms of numerical calculations, can be split into smaller time

Keywords and phrases:Maxwell system, optimal control, domain decomposition.

∗Research of J.E.L. supported by the National Science Foundation through grant DMS-9972034.

∗∗Research of G.L. supported by DFG grants Le595/12-2 and Le595/13-1

1Department of Mathematics, Georgetown University, Washington, DC 20057, USA; e-mail: lagnese@math.georgetown.edu

2Fachbereich Mathematik, Technische Universit¨at Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany;

e-mail: leugering@mathematik.tu-darmstadt.de

c EDP Sciences, SMAI 2002

intervals, even down to the underlying time grid. Thus the problem can be recast into the framework of receding horizon control problems, or into what has come to be known as instantaneous control problems. An early reference to the possible utility of TDDM for the wave equation, but with no analysis provided, may be found in Benamou [3]. The transmission conditions suggested by Benamou contain a Robin-type condition in time which can be viewed as an approximation to transparent transmission conditions studied by Ganderet al. [6].

Nevertheless, TDD-methods have been analyzed in the literature only very recently; see Heinkenschloss [7] and J.-L. Lions [13]. The common approach is to perform some sort of shooting method. That is to say, at the break points one restarts the dynamics with initial values that have to be optimized in order to achieve the correct continuity with respect to the time variable at the break points. While Heinkenschloss formulates the problem as an equality constrained optimal control problem with a Lagrangian relaxation, which is solved by a Gauss-Seidel type preconditioning of a GMRES-solve for a Shur-complement-type equation, Lions penalizes the defect of continuity across the break points. We, instead, mimic an augmented Lagrangian relaxation from spatial decompositions of elliptic problems, a strategy we have successfully used for wave equations in [10] (see also the references therein). This procedure leads to a sequence of local-in-time problems on the subintervals, which, in fact, turn out to be optimality systems for local-in-time optimal control problems. The procedure is completely parallel. We show convergence of the iteration and provide some usefula posterioriestimates of the error in the approximation. We emphasize that this novel time domain decomposition method can be combined with the corresponding spatial domain decomposition method such that the resulting iteration scheme provides a decomposition into space-time subdomains or, after discretization, even to space-time atoms on the finite element level.

2. Setting the problem

Let Ω be a bounded, open, connected set in IR³with piecewise smooth, Lipschitz boundary Γ, and letT >0.

We consider the Maxwell system

(εE⁰−rotH+σE =F

µH⁰+ rotE=G inQ:= Ω×(0, T) H_τ−α(ν∧E) =J on Σ := Γ×(0, T)

E(0) =φ, H(0) =ψ in Ω.

(2.1)

Here ⁰ =∂/∂t,∧is the standard vector product operation,ν denotes the exterior pointing unit normal vector to Γ,H_τ is the tangential component of H, that is,

H_τ=H−(H·ν)ν =ν∧(H∧ν),

and α∈ L^∞(Γ), α(x) ≥ α₀ > 0. Further, ε = (ε^jk(x)), µ = (µ^jk(x)) and σ = (σ^jk(x)) are 3×3 Hermitian matrices withL^∞(Ω) entries such thatεandµare uniformly positive definite andσ≥0 in Ω. The functionsF, G∈L¹(0, T;L²(Ω)) are given whileJ is a control input and is taken from the class

U =L²_τ(Σ) :={J|J ∈L²(0, T;L²_τ(Γ))}

where L²_τ(Γ) denotes vector valuedL²(Γ) functions having a zero normal component.

WhenJ = 0 andα= 1, the boundary condition (2.1)₃is known as the Silver–M¨uller boundary condition. It is the first approximation to the so-called transparent boundary condition, which corresponds to the transmission of electromagnetic waves through the boundary without reflections. In general, whenα >0 on a set on positive measure in Γ, the boundary condition (2.1)₃isdissipative. That is, if one defines the electromagnetic energy by

E(t) = Z

Ω

(εE·E+µH·H) dx,

then, in the absence of external inputsF, G, J, the functionalE(t) is nonincreasing. Moreover, the boundary condition (2.1)₃ is alsoregularizing. That is to say, ifE(0)<∞, ifF, Gand J have regularity assumed above, and if the support of J is contained in the support ofα, the solution of (2.1) satisfiesE(t)<∞for all t≥0.

On the other hand, if α≡0 andJ6≡0 then, in general, solutions of (2.1) will have less regularity.

In what follows, function spaces ofC-valued functions are denoted by capital Roman letters, while function spaces of C³-valued functions are denoted by capital script letters. We use α·β to denote the natural scalar product inC³, i.e.,α·β =P₃

j=1α_jβ_j, and writeh·,·ifor the natural scalar product in various function spaces such asL²(Ω) andL²(Ω). A subscript may sometimes be added to avoid confusion. The spacesL²(Ω) andL²(Ω) denote the usual spaces of Lebesque square integrableC-valued functions andC³-valued functions, respectively, and, similarly, for the Sobolev spaces H^s(Ω), H^s(Ω). We also denote byL²_ε(Ω) the space L²(Ω) with weight matrix ε and hφ, ψiε :=hεφ, ψi the scalar product of φ and ψ in that space. With this notation, the energy spaceH:=L²_ε(Ω)× L²_µ(Ω).

When (φ, ψ)∈ H, (F, G)∈L¹(0, T;H) andJ∈ L²_τ(Σ), it may be proved that the system (2.1) has a unique solution with regularity (E, H)∈C([0, T];H),ν ∧E|_Σ∈ L²_τ(Σ) and, moreover, the linear map from the data to ((E, H), ν∧E|Σ) is continuous in the indicated spaces (see,e.g. [11]). Therefore, given (E_T, H_T)∈ H, we may consider the final value optimal control problem

J∈Uinf J(J), U :=L²_τ(Σ) (2.2)

subject to (2.1), where

J(J) = 1 2

Z

Σ|J|²dΣ +z

2k(E(T), H(T))−(E_T, H_T)k²_H (2.3) withz a positive penalty parameter. Since the cost functionalJ is convex and in view of the properties of the map ((φ, ψ),(F, G), J) 7→(E, H), it is standard theory that there exists a unique optimal control J_opt. It is shown in the next section thatJ_opt is given by

J_opt=ν∧P|Σ, (2.4)

where (P, Q) is the solution of the backwards runningadjoint system (εP⁰−rotQ−σP = 0

µQ⁰+ rotP= 0 in Q Q_τ+α(ν∧P) = 0 on Σ (P(T) =z(E(T)−E_T)

Q(T) =z(H(T)−H_T) in Ω.

(2.5)

The purpose of this paper is to develop a convergent time domain decomposition method (TDDM) to approxi- mate the solution of theoptimality system(2.1, 2.4, 2.5), and to derive certaina posterioriestimate of the error in the approximation.

Our TDDM is introduced in Section 4, and it is shown that each of the local problems entering into the algorithm is itself an optimality system. Convergence of the algorithm is established in Section 5. A posteriori estimates of the error in the approximation in terms of the mismatch of the iterates at the break points are derived in Section 6.

Remark 2.1. Instead ofL²_τ(Σ) one may choose as the control spaceUthoseL²_τ(Σ) functions that are supported in eΓ×(0, T), where eΓ is a subset of Γ of positive 2-dimensional measure. The analysis presented below may easily be adapted to this more general setting with only minor modifications.

3. The optimality system

The necessary and sufficient condition for optimality is that the directional derivative of J at J_opt in the direction of ˆJ is equal to zero. ThereforeJ_opt is the solution of the variational equation

Z

Σ

J_opt·JˆdΣ +zh(E(T)−E_T, H(T)−H_T),( ˆE(T),Hˆ(T))iH= 0, ∀Jˆ∈ L²_τ(Σ), (3.1)

where ( ˆE,Hˆ) is the solution of

(εEˆ⁰−rot ˆH+σEˆ = 0

µHˆ⁰+ rot ˆE= 0 inQ Hˆ_τ−α(ν∧E) = ˆˆ J on Σ E(0) = ˆˆ H(0) = 0 in Ω.

Let (P, Q) be the solution of (2.5). Then (P, Q)∈C([0, T];H) andν∧P|_Σ∈ U. We have

0 = Z _T

0 {hεP⁰−rotQ−σP,Eˆi+hµQ⁰+ rotP,Hˆi}dt. (3.2) By utilizing Green’s formula

hrotφ, ψi=hφ,rotψi+ Z

Γ

(ν∧φ)·ψ_τdΓ

=hφ,rotψi − Z

Γ

φ_τ·(ν∧ψ)dΓ

(3.3)

we may rewrite (3.2) as

0 =zh(E(T)−E_T, H(T)−H_T),( ˆE(T),Hˆ(T))i_H+ Z

Σ

(ν∧P)·JˆdΣ, ∀Jˆ∈ L²_τ(Σ). (3.4)

Thus (2.4) follows from (3.1) and (3.4).

4. Time domain decomposition

We introduce a partition of the time interval [0, T] by setting

0 =T₀< T₁<· · ·< T_K < T_K+1=T (4.1) and thereby decompose [0, T] into K+ 1 subintervals I_k := [T_k, T_k+1], k = 0, . . . , K. We further introduce locally defined functions E_k = E|Ik, H_k = H|Ik, and so forth. We proceed to decompose the optimality system (2.1, 2.4, 2.5) into the following local systems defined onI_k,k= 0, . . . , K:

(εE_k⁰ −rotH_k+σE_k =F_k

µH_k⁰ + rotE_k=G_k inQ_k:= Ω×I_k H_kτ−α(ν∧E_k) =ν∧P_k on Σ_k := Γ×I_k,

(4.2)

(εP_k⁰ −rotQ_k−σP_k = 0

µQ⁰_k+ rotP_k = 0 in Q_k Q_kτ+α(ν∧P_k) = 0 on Σ_k.

(4.3)

The “initial” values (E_k(T_k), H_k(T_k)) and (P_k(T_k+1), Q_k(T_k+1)) for (4.2 4.3) are given through the coupling conditions

E_k(T_k) =E_k−1(T_k), H_k(T_k) =H_k−1(T_k), k= 1, . . . , K,

P_k(T_k+1) =P_k+1(T_k+1), Q_k(T_k+1) =Q_k+1(T_k+1), k=K−1, . . . ,0, (4.4) together with

E₀(0) =φ, H₀(0) =ψ,

P_K(T) =z(E_K(T)−E_T), Q_K(T) =z(H_K(T)−H_T). (4.5) We now uncouple the local problems by uncoupling (4.4) through an iteration as follows:

(βE_kⁿ⁺¹(T_k+1)−P_kⁿ⁺¹(T_k+1) =µⁿ_k,k+1

βH_kⁿ⁺¹(T_k+1)−Qⁿ⁺¹_k (T_k+1) =η_k,k+1ⁿ , k= 0, . . . , K−1 (βE_kⁿ⁺¹(T_k) +P_kⁿ⁺¹(T_k) =µⁿ_k,k−1

−βH_kⁿ⁺¹(T_k)−Qⁿ⁺¹_k (T_k) =η_k,k−1ⁿ , k= 1, . . . , K,

(4.6)

where β >0 and

µⁿ_k,k+1=βE_k+1ⁿ (T_k+1)−P_k+1ⁿ (T_k+1) η_k,k+1ⁿ =βH_k+1ⁿ (T_k+1)−Qⁿ_k+1(T_k+1)

µⁿ_k,k−1=βE_k−1ⁿ (T_k) +P_k−1ⁿ (T_k) η_k,k−1ⁿ =−βH_k−1ⁿ (T_k)−Qⁿ_k−1(T_k).

(4.7)

It is readily seen that in the limit of (4.6, 4.7) one recovers (4.4), so that (4.6, 4.7) are consistent with (4.4).

For the convenience of the reader we write down the complete set of uncoupled systems:

(ε(E_kⁿ⁺¹)⁰−rotH_kⁿ⁺¹+σE_kⁿ⁺¹=F_k

µ(H_kⁿ⁺¹)⁰+ rotE_kⁿ⁺¹=G_k in Q_k (ε(P_kⁿ⁺¹)⁰−rotQⁿ⁺¹_k −σP_kⁿ⁺¹= 0

µ(Qⁿ⁺¹_k )⁰+ rotP_kⁿ⁺¹= 0 inQ_k

(4.8)

(H_kτⁿ⁺¹−α(ν∧E_kⁿ⁺¹) =ν∧P_kⁿ⁺¹

Qⁿ⁺¹_kτ +α(ν∧P_kⁿ⁺¹) = 0 on Σ_k (4.9)

E₀ⁿ⁺¹(0) =φ, H₀ⁿ⁺¹(0) =ψ (P_Kⁿ⁺¹(T) =z(E_Kⁿ⁺¹(T)−E_T)

Qⁿ⁺¹_K (T) =z(H_Kⁿ⁺¹(T)−H_T) in Ω

(4.10)

subject to (4.6, 4.7).

We wish to show that for (µⁿ_k,k+1, η_k,k+1ⁿ ) and (µⁿ_k,k−1, η_k,k−1ⁿ ) given inH, the local problems are well posed for k= 0, . . . , K. To do so we shall show that the local problem with index k is in fact an optimality system concentrated on the intervalI_k,k= 0, . . . , K.

Proposition 4.1. Assume that(µⁿ_k,k+1, ηⁿ_k,k+1)∈ H,(µⁿ_k,k−1, ηⁿ_k,k−1)∈ H. Set Jk(J_k, h_k,k−1, g_k,k−1) = 1

2 Z

Σk

|J_k|²dΣ + 1 2β

kβE_k(T_k+1)−µⁿ_k,k+1k²_ε

+kβH_k(T_k+1)−ηⁿ_k,k+1k²_µ+k(h_k,k−1, g_k,k−1)k²_H , k= 1, . . . , K−1, (4.11) JK(J_K, h_K,K−1, g_K,K−1) = 1

2 Z

ΣK

|J_K|²dΣ +z

2k(E_K(T), H_K(T))−(E_T, H_T)k²_H+ 1

2βk(h_K,K−1, g_K,K−1)k²_H, (4.12) J0(J₀) = 1

2 Z

Σ0

|J₀|²dΣ + 1 2β

kβE₀(T₁)−µⁿ_0,1k²_ε+kβH₀(T₁)−η_0,1ⁿ k²_µ · (4.13)

For k = 0. . . , K, the system (4.8–4.10, 4.6) is the optimality system for the optimal control problem inf

UkJ_k, where Uk =L²_τ(Σ_k)× Hfor k= 1, . . . , K, U0=L²_τ(Σ₀), subject to

(εE_k⁰ −rotH_k+σE_k =F_k

µH_k⁰ + rotE_k=G_k inQ_k H_kτ−α(ν∧E_k) =J_k onΣ_k

(4.14)

and







E_k(T_k) = 1

β(h_k,k−1+µⁿ_k,k−1) H_k(T_k) =−1

β(g_k,k−1+ηⁿ_k,k−1)inΩ, k= 1, . . . , K, (4.15)

E₀(0) =φ, H₀(0) =ψ inΩ. (4.16)

Remark 4.1. In the local optimal control problem inf_UkJk(J_k, h_k,k−1, g_k,k−1), the control J_k is, in the ter- minology of J.-L. Lions and O. Pironneau, the effective control and the controlsh_k,k−1, g_k,k−1 are virtual, or artificial controls; see [12–14].

Proof. We give the proof only for k = 1, . . . , K −1 since the proofs in the two remaining cases are similar.

The necessary and sufficient condition that J_k, h_k,k−1, g_k,k−1 be optimal is that the directional derivative of J_k atJ_k,h_k,k−1, g_k,k−1 in the direction of ˆJ_k,ˆh_k,k−1,gˆ_k,k−1be zero. This leads to the variational equation

0 = Z

Σk

J_k·Jˆ_kdΣ +hβE_k(T_k+1)−µⁿ_k,k+1,Eˆ_k(T_k+1)iε

+hβH_k(T_k+1)−ηⁿ_k,k+1,Hˆ_k(T_k+1)iµ+ 1

βh(hk,k−1, g_k,k−1),(ˆh_k,k−1,gˆ_k,k−1)iH,

∀( ˆJ_k,ˆh_k,k−1,gˆ_k,k−1)∈ Uk, (4.17)

where

(εEˆ_k⁰ −rot ˆH_k+σEˆ_k = 0

µHˆ_k⁰ + rot ˆE_k = 0 inQ_k Hˆ_kτ−α(ν∧Eˆ_k) = ˆJ_k on Σ_k Eˆ_k(T_k) = 1

βˆh_k,k−1, Hˆ_k(T_k) =−1

βˆg_k,k−1 in Ω.

(4.18)

Introduce (P_k, Q_k) as the solution of

(εP_k⁰−rotQ_k−σP_k = 0

µQ⁰_k+ rotP_k= 0 in Q_k Q_kτ +α(ν∧P_k) = 0 on Σ_k (P_k(T_k+1) =βE_k(T_k+1)−µⁿ_k,k+1

Q_k(T_k+1) =βH_k(T_k+1)−η_k,k+1ⁿ in Ω.

(4.19)

We have

0 = Z _Tk+1

Tk

{hεP_k⁰−rotQ_k−σP_k,Eˆ_ki+hµQ⁰_k+ rotP_k,Hˆ_ki}dt (4.20)

=h(Pk(t), Q_k(t)),( ˆE_k(t),Hˆ_k(t))iH ^Tk+1

t=Tk

+ Z

Σk

[Q_kτ·(ν∧Eˆ_k) + (ν∧P_k)·Hˆ_kτ]dΣ

where we have used (3.3). By utilizing (4.18) and (4.19) it is seen that (4.20) may be written 0 =h(βE_k(T_k+1)−µⁿ_k,k+1, βH_k(T_k+1)−ηⁿ_k,k+1),( ˆE_k(T_k+1),Hˆ_k(T_k+1))i_H

−

(P_k(T_k), Q_k(T_k)), 1

βˆh_k,k−1,−1 βˆg_k,k−1

H

+ Z

Σk

(ν∧P_k)·Jˆ_kdΣ. (4.21)

It now follows from (4.21) and (4.17) that

J_k=ν∧P_k|Σk, h_k,k−1=−P_k(T_k), g_k,k−1=Q_k(T_k), from which the conclusion of Proposition 4.1 follows immediately.

Corollary 4.1. Let(µ⁰_k,k+1, η⁰_k,k+1)|^K−1_k=0 and(µ⁰_k,k−1, η_k,k−1⁰ )|^K_k=1 be given arbitrarily inH. Then the iterative procedure described by (4.6–4.10) is well defined for n= 0,1, . . .

5. Convergence of the iterates

We consider the iterative procedure with the basic step given by (4.6–4.10). In fact, we shall consider a relaxation of the iteration step (4.6, 4.7). Thus we introduce a relaxation parameter ∈[0,1) and consider the

iterative step (4.6, 4.7) with under relaxation:

βE_kⁿ⁺¹(T_k+1)−P_kⁿ⁺¹(T_k+1) = (1−)µⁿ_k,k+1+(βE_kⁿ(T_k+1)−P_kⁿ(T_k+1)) βH_kⁿ⁺¹(T_k+1)−Qⁿ⁺¹_k (T_k+1) = (1−)η_k,k+1ⁿ +(βH_kⁿ(T_k+1)−Qⁿ_k(T_k+1))

βE_kⁿ⁺¹(T_k) +P_kⁿ⁺¹(T_k) = (1−)µⁿ_k,k−1+(βE_kⁿ(T_k) +P_kⁿ(T_k))

−βH_kⁿ⁺¹(T_k)−Qⁿ⁺¹_k (T_k) = (1−)η_k,k−1ⁿ +(−βH_kⁿ(T_k)−Qⁿ_k(T_k)).

(5.1)

We introduce the errors

Ee_kⁿ =E_kⁿ−E_k, He_kⁿ =H_kⁿ−H_k, Pe_kⁿ=P_kⁿ−P_k, Qeⁿ_k =Qⁿ_k−Q_k. These satisfy

(ε(Ee_kⁿ⁺¹)⁰−rotHe_kⁿ⁺¹+σEe_kⁿ⁺¹= 0

µ(He_kⁿ⁺¹)⁰+ rotEe_kⁿ⁺¹= 0 inQ_k (ε(Pe_kⁿ⁺¹)⁰−rotQeⁿ⁺¹_k −σPe_kⁿ⁺¹= 0

µ(Qeⁿ⁺¹_k )⁰+ rotPe_kⁿ⁺¹= 0 inQ_k

(5.2)

(He_kτⁿ⁺¹−α(ν∧Ee_kⁿ⁺¹) =ν∧Pe_kⁿ⁺¹

Qⁿ⁺¹_kτ +α(ν∧P_kⁿ⁺¹) = 0 on Σ_k (5.3)

Eeⁿ⁺¹₀ (0) =He₀ⁿ⁺¹(0) = 0

Pe_Kⁿ⁺¹(T) =zEe_Kⁿ⁺¹(T), Qeⁿ⁺¹_K (T) =zH_Kⁿ⁺¹(T) in Ω (5.4) subject to

(βEe_kⁿ⁺¹(T_k+1)−Pe_kⁿ⁺¹(T_k+1) = ˜µⁿ_k,k+1

βHe_kⁿ⁺¹(T_k+1)−Qeⁿ⁺¹_k (T_k+1) = ˜η_k,k+1ⁿ , k= 0, . . . , K−1 (βEe_kⁿ⁺¹(T_k) +Pe_kⁿ⁺¹(T_k) = ˜µⁿ_k,k−1

−βHe_kⁿ⁺¹(T_k)−Qeⁿ⁺¹_k (T_k) = ˜η_k,k−1ⁿ , k= 1, . . . , K

(5.5)

where, for n≥1,

µ˜ⁿ_k,k+1= (1−)(βEe_k+1ⁿ (T_k+1)−Pe_k+1ⁿ (T_k+1)) +(βEe_kⁿ(T_k+1)−Pe_kⁿ(T_k+1)) η˜ⁿ_k,k+1= (1−)(βHe_k+1ⁿ (T_k+1)−Qeⁿ_k+1(T_k+1)) +(βHe_kⁿ(T_k+1)−Qeⁿ_k(T_k+1))

µ˜ⁿ_k,k−1= (1−)(βEe_k−1ⁿ (T_k) +Pe_k−1ⁿ (T_k)) +(βEe_kⁿ(T_k) +Pe_kⁿ(T_k)) η˜ⁿ_k,k−1= (1−)(−βHe_k−1ⁿ (T_k)−Qeⁿ_k−1(T_k)) +(−βHe_kⁿ(T_k)−Qeⁿ_k(T_k)),

(5.6)

µ˜⁰_k,k+1=µ⁰_k,k+1−(βE(T_k+1)−P(T_k+1)) η˜_k,k+1⁰ =η_k,k+1⁰ −(βH(T_k+1)−Q(T_k+1))

µ˜⁰_k,k−1=µ⁰_k,k−1−(βE(T_k) +P(T_k)) η˜_k,k−1⁰ =η_k,k−1⁰ + (βH(T_k) +Q(T_k)).

(5.7)

We shall prove the following convergence result:

Theorem 5.1. Let β >0. Then (a) for any ∈[0,1)

ν∧Pe_kⁿ|Σk →0 strongly inL²_τ(Σ_k),k= 0, . . . , K (Pe_Kⁿ,Qeⁿ_K)→0 inC(I_K;H)

(Ee₀ⁿ,He₀ⁿ)→0 inC(I₀;H) ν∧Ee₀ⁿ|Σ0 →0 strongly inL²_τ(Σ₀).

(b) For any∈(0,1)and for k= 0, . . . , K,

(Eeⁿ_k,He_kⁿ)→0 and (Pe_kⁿ,Qeⁿ_k)→0 inC(I_k;H) ν∧Ee_kⁿ|Σk →0 strongly inL²_τ(Σ_k).

Remark 5.1. Part (a)_1,2 states that for any ∈ [0,1) the effective local optimal controls {ν ∧P_kⁿ|_Σk}^K_k=0 converge strongly to the global optimal controlν ∧P|_Σ, and the deviation k(E_Kⁿ(T), H_Kⁿ(T))−(E_T, H_T)k_H of the local optimal trajectory (E_Kⁿ, H_Kⁿ) at time T from the target state (E_T, H_T) converges to the deviation k(E(T), H(T))−(E_T, H_T)k_H of the global optimal trajectory at timeT from the target state. Ana posteriori estimate of the error in the approximation is derived in the next section.

Remark 5.2. When= 0 it is possible to show that (

(Ee_kⁿ,He_kⁿ)→0 weakly* inL^∞(0, T;H)

ν∧Eeⁿ_k|_Σk →0 weakly inL²_τ(Σ_k), k= 1, . . . , K (Pe_kⁿ,Qeⁿ_k)→0 weakly* inL^∞(0, T;H),k= 0, . . . , K−1

(5.8)

provided the following backwards uniqueness property is valid: if (E, H) satisfies the Maxwell system (2.1)_1,2 withF =G= 0 and the boundary condition (2.1)₃withJ = 0, and ifE(T) =H(T) = 0, thenE(0) =H(0) = 0.

Whether or not this uniqueness property holds seems to be an open question.

Proof of Theorem 5.1. Although the technical details differ, the proof given below is structurally similar to proofs given in earlier works such as [3–5], and in papers by the present authors (see,e.g.[8, 10] and references therein), all of which dealt with space domain decomposition of either direct or optimal control problems of one type or another. A key role in the convergence proofs in all of the papers is played by a fundamental recursion formula such as (5.22) below.

SetX =H^2K with the standard product norm. Let

X ={(µ_k,k+1, η_k,k+1)^K−1_k=0 ,(µ_k,k−1, η_k,k−1)^K_k=1} ∈ X

and let (E_k, H_k), (P_k, Q_k) be the solution of (4.8–4.10, 4.6) with the superscriptsn+ 1 andnremoved and with zero data

F_k=G_k= 0, φ=ψ=E_T =H_T = 0. (5.9)

Define the linear mapping T : X 7→ X by TX=

n(βE_k+1(T_k+1)−P_k+1(T_k+1), βH_k+1(T_k+1)−Q_k+1(T_k+1))^K−1_k=0,

(βE_k−1(T_k) +P_k−1(T_k),−βH_k−1(T_k)−Q_k−1(T_k))^K_k=1 o·

Note thatX is a fixed point ofT if and only if the transmission conditions (4.4) are satisfied, that is, if and only if {(E_k, H_k),(P_k, Q_k)}^K_k=0is the solution of the global optimality system with vanishing data (5.9). Since, in this case, the optimal control is obviouslyJ_opt= 0, it follows that the only fixed point ofT isX= 0.

The significance of the mapT is that, if we set

Xⁿ={(˜µⁿ⁻¹_k,k+1,η˜_k,k+1ⁿ⁻¹ )^K−1_k=0 ,(˜µⁿ⁻¹_k,k−1,η˜ⁿ⁻¹_k,k−1)^K_k=1} (5.10) and let{(Eeⁿ_k,He_kⁿ),(Pe_kⁿ,Qeⁿ_k)}^K_k=0 be the solution of (5.2–5.5) withnreplaced byn−1, then

Xⁿ ={(βEe_kⁿ(T_k+1)−Pe_kⁿ(T_k+1), βHe_kⁿ(T_k+1)−Qeⁿ_k(T_k+1))^K−1_k=0 , (βEeⁿ_k(T_k) +Pe_kⁿ(T_k),−βHe_kⁿ(T_k)−Qeⁿ_k(T_k))^K_k=1}

TXⁿ={(βEe_k+1ⁿ (T_k+1)−Pe_k+1ⁿ (T_k+1), βHe_k+1ⁿ (T_k+1)−Qeⁿ_k+1(T_k+1))^K−1_k=0 ,

(βEe_k−1ⁿ (T_k) +Pe_k−1ⁿ (T_k),−βHe_k−1ⁿ (T_k)−Qeⁿ_k−1(T_k))^K_k=1} and the relaxed iteration step (5.5, 5.6) may be expressed as the relaxed fixed point iteration

Xⁿ⁺¹= (1−)TXⁿ+Xⁿ:=TXⁿ. (5.11)

The following result shows thatT is nonexpansive.

Lemma 5.1. For anyX ∈ X,

kTXk²_X =kXk²_X−2F_β (5.12)

where

F_β = 2βX^K

k=0

Z

Σk

|ν∧P_k|²dΣ + 2βzk(E_K(T_K+1), H_K(T_K+1))k²_H. (5.13)

Proof. We have

kXk²_X =

K−1X

k=0

k(βEk(T_k+1)−P_k(T_k+1), βH_k(T_k+1)−Q_k(T_k+1))k²_H

+ XK k=1

k(βE_k(T_k) +P_k(T_k),−βH_k(T_k)−Q_k(T_k))k²_H. (5.14)

Let us define

Ek,β(t) =β²(kE_k(t)k²_ε+kH_k(t)k²_µ) +kP_k(t)k²_ε+kQ_k(t)k²_µ. One finds after a little calculation that (5.14) may be written

kXk²_X =

K−1X

k=1

Ek,β(T_k+1) +Ek,β(T_k)

+E0,β(T₁) +EK,β(T_K)

−2β^K−1X

k=0

Re

hE_k(t), P_k(t)i_ε+hH_k(t), Q_k(t)i_µTk+1

t=Tk

+2βRe

hE_K(T_K), P_K(T_K)iε+hH_K(T_K), Q_K(T_K)iµ

. (5.15)

Form

0 = Z _Tk+1

Tk

hεE_k⁰ −rotH_k+σE_k, P_ki+hµH_k⁰ + rotE_k, Q_ki dt

=h(E_k(t), H_k(t)),(P_k(t), Q_k(t))i_H|^T_t=T^k+1_k+ Z

Σk

|ν∧P_k|²dΣ,

(5.16)

where we have used (3.3). In particular we have h(E_K(T_K), H_K(T_K)),(P_K(T_K), Q_K(T_K))iH

=h(E_K(T_K+1), H_K(T_K+1)),(P_K(T_K+1), Q_K(T_K+1))i_H+R

ΣK|ν∧P_K|²dΣ

=zk(E_K(T_K+1), H_K(T_K+1))k²_H+R

ΣK|ν∧P_K|²dΣ. (5.17)

It follows from (5.15–5.17) that

kXk²_X=Eβ+Fβ (5.18)

where

Eβ=

K−1X

k=1

Ek,β(T_k+1) +Ek,β(T_k)

+E0,β(T₁) +EK,β(T_K). (5.19) A similar calculation yields

kTXk²_X =

K−1X

k=0

k(βEk+1(T_k+1)−P_k+1(T_k+1), βH_k+1(T_k+1)−Q_k+1(T_k+1))k²_H

+ XK k=1

k(βEk−1(T_k) +P_k−1(T_k),−βH_k−1(T_k)−Q_k−1(T_k))k²_H

=Eβ+ 2β

K−1X

k=0

Re

hE_k(t), P_k(t)iε+hH_k(t), Q_k(t)iµ_Tk+1

t=Tk

−2βRe

hE_K(T_K), P_K(T_K)iε+hH_K(T_K), Q_K(T_K)iµ

=E_β− F_β.

(5.20)

Thus (5.12) follows from (5.18) and (5.20).

To continue with the proof of Theorem 5.1, we set

E_k,βⁿ (t) =β²(kEe_kⁿ(t)k²_ε+kHe_kⁿ(t)k²_µ) +kPe_kⁿ(t)k²_ε+kQeⁿ_k(t)k²_µ E_βⁿ=

K−1X

k=1

E_k,βⁿ (T_k+1) +E_k,βⁿ (T_k)

+E_0,βⁿ (T₁) +E_K,βⁿ (T_K)

=

K−1X

k=0

E_k,βⁿ (T_k+1) +E_k+1,βⁿ (T_k+1)

F_βⁿ= 2βX^K

k=0

Z

Σk

|ν∧Pe_kⁿ|²dΣ + 2βzk(Ee_Kⁿ(T_K+1),He_Kⁿ(T_K+1))k²_H.