Optimization and Discretization in 2D Problems of Electromagnetic Invisible Cloaking

Abstract

We study control problems for the 2D electromagnetic eld model describing scattering TM-polarized electromagnetic waves in unbounded homogeneous medium containing a penetrable inhomogeneous dielectric obstacle with the boundary partially coated for masking. These problems arise when developing the design technologies of electromagnetic cloaking devices using optimization method. Two constitutive parameters: variable refraction index of the obstacle and surface conductivity of the coated part of the boundary play the role of controls. Solvability of control problems is proved, the optimality system which describes the necessary conditions of extremum is derived, uniqueness and stability of optimal solutions are established. Two numerical algorithms are proposed and discussed. The rst of them is based on strategy \optimize-then-discretize" and the second one is based on opposite strategy \discretize-then-optimize".

Keywords

• TM-polarization discretization
• control problem

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di culty of their technical realization. For example, the design of the TO-based cloaks involves extreme values of constitutive parameters and spatially varying distributions of the permittivity and permeability tensors which are very di cult to implement [18].

That is why the another cloak design strategy began develop recently. It obtained the name of inverse design as it is related with solving inverse electromagnetic (or acoustic) problems. The optimization method forms the core of this inverse design methodology. This enables us to solve some substantial limitations of previous cloaking solutions. A growing number of papers is devoted to applying the inverse design methodology in various cloaking problems. Among them we mention [14, 16, 17] where numerical optimization algorithms are applied for nding the unknown material parameters of TO-based cloak. It was shown there that the optimized multi-layer cloak essentially outperforms the similarly sized metamaterial cloak designed by using the TO approach. In [18] the authors review the invisibility cloak design methodologies and discuss the recent transition from forward design to inverse design. We also mention papers [1{4] where the mathematical apparatus for solving impedance cloaking problem on the basis of optimization approach is developed.

This paper is devoted to theoretical analysis of control problems for 2D electromagnetic wave scattering model. These problems arise when optimization method is applied for solving cloaking problems for respective 2D electromagnetic scattering model.

We begin with formulation of the direct scattering problem. Let be a bounded domain in R2 with a connected complement c := R2n and Lipschitz boundary consisting of two parts D and I . We consider the scattering problem for TM-polarized electromagnetic waves in homogeneous medium containing penetrable inhomogeneous dielectric obstacle with coated partially (for masking) boundary. Mathematically this problem is reduced to nding functions w in and u = uinc + us in c satisfying equations w + k2n(x)w = 0 in ; 4u + k2u = 0 in c; mixed transmission conditions on the boundary ( 1 ) ( 2 ) ( 3 ) w uj = 0; and the Sommerfeld radiation condition in R2 lim r!1 pr( Here uinc is the incident wave, us is the scattered wave, k is the wave number, k2 = "0 0!2 where ! is an angular frequency, "0 and 0 are constant electric permittivity and magnetic permeability, n(x) > 0 is a variable index of refraction of dielectric obstacle , is the surface conductivity of the coated part I of , i is an imaginary unit, is the outward (relative to ) unit normal on .

One can nd the formulation and brief analysis of problem ( 1 ){( 3 ) in [6]. Besides, in [6] the inverse scattering problem of recovering the shape and surface conductivity of a partially coated dielectric in nite cylinder from the far eld data was also studied. The control problems considered in our paper consist of minimization of certain cost functionals dependent on the state (electromagnetic eld) and unknown functions (controls or design parameters) satisfying equations ( 1 ){( 3 ). As the cost functional we choose one of the following:

Z Z I1(U ) =

Q jU udj2dx; I2(U ) = r jU

udj2d : ( 4 ) ( 5 ) Here U is the function equalled to w in and to u in c, function ud 2 L2(Q) (or ud 2 L2( r)) describes the eld measured in some subdomain Q c or on the boundary r of the disc Br of the radius r containing inside. In the case when ud = uinc the functional I1(U ) (or I2(U )) has the sense of squared mean-square integral norm of the scattered eld us over Q (or over r). As controls we choose index of refraction n and surface conductivity . We assume that n and are elements of Sobolev spaces Hr( ) and Hs( I ) and de ne the following regularized functional: Jj (U; n; ) = 0 Ij (U ) + 2 21 knk2Hr( ) + 22 k k2Hs( I ): Here j = 1 or 2, 0, 1 and 2 are nonnegative parameters specifying the relative importance of each term in ( 5 ). We want to nd controls n; and the associated state { electromagnetic eld U = (w; u), such that the functional Jj (U; n; ) de ned in ( 5 ) is minimized subject to state equations ( 1 ){( 3 ).

The rest of the paper is organised as follows. In Section 2 we reduce unbounded problem ( 1 ){( 3 ) to an equivalent problem considered in bounded domain and prove the correct solvability of the latter problem. In Section 3 we prove the existence of a solution of control problem and derive an optimality system. Based on analysis of the optimality system we prove further in Section 4 uniqueness and stability of optimal solutions. Then in Section 5 we propose and discuss two numerical algorithms for solving our control problem. 2

Functional Spaces. Solvability of Direct Problem Let us introduce the function spaces to be used in the subsequent analysis. Let BR be the disc of radius R containing , e := c \ BR. We will use the spaces Hs( ),

H1( e), H1(BR), L2(Q), L2( I ), H1=2( R), H 1=2( R), L1( I ), Hs( I ) with norms k ks; , k k1; e , k k1;BR , k kQ, k k I , k k1=2; R , k k 1=2; R , k kL1( I ) and k ks; I , respectively. The scalar products and norms in Hr( ), L2( ), Hs( I ) and L2( I ) will be denoted by ( ; )r; , k kr; , ( ; ) , k k , ( ; )s; I , k ks; I and ( ; ) I , k k I , respectively.

Along with the space H1( ) we will consider it's subspace H1( ; ) := fw : w 2 H1( ); w 2 L2( )g equipped with the norm kwk2H1( ; ) = kwk21; + k wk2 . It is well known (see [9, p. 28]) that any function w 2 H1( ; ) has the trace 1w @w=@ j 2 H 1=2( ) and the following Green formula holds: ( w; w) = (rw; rw) + k2n w dx = Here and below denotes the complex conjugate of . In a similar manner, we multiply the second equation in ( 1 ) by j e , integrate over e, apply the Green formula for the Here and below integral over (or over R) denotes the duality pairing h ; i between H1=2( ) and H 1=2( ) (or between H1=2( R) and H 1=2( R)). Similar formula holds and for the domain e.

We also need the space X = H1(BR) with the norm k kX := k k1;BR and the space Hinc Hinc( e) = fu 2 H1( e) : u + k2u = 0 in eg with the norm kuk1; e . These spaces will be used for describing properties of the weak solutions of problem ( 1 ){( 3 ) and for describing incident waves uinc, respectively. By X we denote dual of space X. Let Ln10 ( ) = fn 2 L1( ) : n(x)0g annd0g,HHs0nr(0 (I ) )== ffn 2 Hr( ) : n(x) n0g, L10 ( I ) = f 2 L1( I ) : (x) 2 Hs( I ) : (x) 0g where n0 = const > 0, 0 = const > 0; r > 0; s > 0. These sets will serve for describing properties of index of refraction n and conductivity . We note that continuous compact embeddings Hr( ) L1( ) at r > 1 and Hs( I ) L1( I ) at s > 1=2 take place (if I 2 C1;1) and the following estimates hold: knkL1( )

Cr0 knkr; 8n 2 Hr( ); k kL1( I )

Csk ks; I 8 2 Hs( I ): ( 7 ) Here Cr0 (or Cs) is the constant dependent on r and (or on s and I ).

Now we are in position to study problem ( 1 ){( 3 ). We begin with reducing problem ( 1 ){( 3 ) to an equivalent problem considered in the disc BR. For this purpose we de ne the Dirichlet-to-Neumann (DtN) operator T : H1=2( R) ! H 1=2( R) that maps every function g 2 H1=2( R) to a function @u~=@ 2 H 1=2( R) where u~ is a solution of the exterior Dirichlet problem for the Helmholtz equation u~ + k2u~ = 0 in cnBR with condition u~j R = g. It is well known that problem ( 1 ){( 3 ) is equivalent to problem ( 1 ), ( 2 ) considered in the disc BR under the following boundary condition for scattered eld us on R: We will refer to ( 1 ) in [ e, ( 2 ) and ( 8 ) as problem 1.

Let us multiply the rst equation in ( 1 ) by j where integrate over and apply the Green formula ( 6 ). We obtain 2 X is a test function, R: Z ( 6 ) ( 8 ) ( 9 ) 8 Z

R

Z

I domain e and add with ( 9 ). Using the boundary conditions in ( 2 ) and condition ( 8 ) for us we arrive at the following identity with respect to function U := (w; u) 2 X: a (U; ) := a0(U; ) an(U; ) a (U; ) = hf; i denotes the pair (n; ), a0, an, a and f are sesquilinear and Here and below index linear forms de ned by

Z a0(U; ) := r rU dx +

Z e (r rU k2 U )dx

T U d ; ( 11 ) an(U; ) = k2(nU; ) := k2 Z n U dx; a (U; ) := i( U; ) I := i

U d ; ( 12 ) hf; i :=

Z

R

T uincd +

Z

R d ; = (n; ): ( 13 ) The solution U 2 X of problem ( 10 ) is called a weak solution of problem 1.

Using the embedding theorems, trace theorem and the properties of DtN operator T it is easy to derive the following estimates for forms a0, an, a , f : ka0k

C1; kank

C1knkL1( ); ka k

C1k kL1( I ); kf kX

C1kuinck1; e : ( 14 ) Here C1 is a constant dependent on , k and R. We note that the sesquilinear form a introduced in ( 10 ) de nes operator A : X!X by hA U; i = a (U; ) for all U 2 X, 2 X and problem ( 10 ) for U 2 X is equivalent to equation

A U = f:

Using properties of forms a0, an, a and operator T one can prove arguing as in [2] that the operator A is an isomorphism. Denote by A 1 : X ! X the inverse of the operator A . Let C~ = kA 1k. It follows from the results above that for any element f 2 X equation ( 15 ) has a unique solution U 2 X which satis es the estimate kU kX C~ kf kX . Besides, in the case when index of refraction n and conductivity belong to nonempty bounded sets K1 Hnr0 ( ), r > 1 and K2 Hs0 ( I ), s > 1=2 one can show proceeding as in [2] that the solution U of ( 15 ) satis es the estimate kU kX C~0kf kX where constant C~0 is independent of . Using estimate kf kX C1kuinck1; e from ( 14 ) and setting C0 = C~0C1 we rewrite this estimate as kU kX

C0kuinck1; e 8 = (n; ) 2 K1

K2: ( 15 ) ( 16 ) Let us formulate the result obtained as the next theorem.

Theorem 1. Let 2 C0;1, I 2 C1;1 and let K1 Hnr0 ( ) and K2 Hs0 ( I ) be nonempty bounded sets where r > 1, s > 1=2. Let uinc 2 Hinc be an incident eld. Then for any pair (n; ) 2 K1 K2 problem ( 10 ) has a unique solution U 2 X which satis es estimate ( 16 ) with constant C0 independent of .

Solvability of Control Problem. Optimality System In this Section we formulate and study our control problem. We assume that controls n and can change in certain sets K1 and K2. More precisely, the following conditions are assumed to hold:

(j) 2 C0;1 I 2 C1;1; 0 > 0; K1 Hnr0 ( ) and K2 Hs0 ( I ) are nonempty convex closed sets where n0=const > 0, 0=const > 0, r > 1, s > 1=2.

Let K = K1 K2, = (n; ). De ning the operator G : X K Hinc ! X by hG(U; ; uinc); i = a0(U; ) k2(nU; ) i( U; ) I hf; i for all 2 X consider the constrained minimization problem

J (U; ) := 0 I(U ) + 2 Here c1; c2; c3 are some constants which are independent of m 2 N = f1; 2; :::g. By de nition of Zad the pair (Um; m) satis es the identity a0(Um; ) k2(nmUm; ) i( mUm; ) I = hf; i 8 2 X; m 2 N: ( 18 ) It follows from the estimates above that there exist weak limits n 2 K1 Hns0 ( ), 2 K2 Hs0 ( I ) and U 2 X of some subsequences of the sequences fnmg, f mg and fUmg. Using this fact and compactness of embeddings Hr( ) L1( ) at r > 1, Hs( I ) L1( I ) at s>1=2 we conclude (passing if necessary to subsequencies) that Um ! U 2 X weakly in X and Umj ! U j weakly in L2( ); Umj I ! U j I weakly in L2( I ); nm ! n 2 K1 strongly in L1( ); m ! 2 K2 strongly in L1( I ):

Let us pass to the limit in ( 18 ) when m ! 0. Using ( 14 ) we obtain that the pair (U ; ) where := (n ; ) satis es a0(U ; ) k2(nU ; ) i(

U ; ) I = hf; i 8 This means that G(U ; ; uinc) = 0. Since J is weakly lower semicontinuous on X K, we have J (U ; ) = J which proves the theorem.

We note that the assertion of Theorem 2 is valid for both functionals I1(U ) and I2(U ) since they are nonnegative and are weakly lower semicontinuous.

The next stage in the study of control problem ( 17 ) is to establish su cient conditions on the input data under which its solution is unique and stable for particular cost functionals. For this purpose we make use the approach developed in [2, 3]. It is based on the derivation and analysis of an optimality system describing the rst-order necessary conditions for an extremum in problem ( 17 ). Arguing as in [3] one can prove the following result.

Theorem 3. Let under conditions (j) the triple (U^ ; n^; ^) 2 X K be a solution of problem ( 17 ) where the functional I(U ) is continuously di erentiable with respect to U in the point U^ . Then there exists a unique nonzero Lagrange multiplier P 2 X which satis es the Euler-Lagrange equation a0( ; P ) k2(n^ ; P ) i(^ ; P ) I = ( 0=2)hIU0 (U^ ); i 8 and the minimum principle holds which is equivalent to inequalities

Direct problem ( 10 ), the Euler-Lagrange equation (20) which has the sense of adjoint problem for the adjoint state P 2 X and variational inequalities (21), (22) constitute the optimality system for control problem ( 17 ). The optimality system plays an important role in studying the properties of solutions of the control problem. On its basis, e cient numerical algorithms of solving problem ( 17 ) can be developed. In addition, using analysis of the optimality system one can establish the su cient conditions on the initial data providing the uniqueness and stability of solutions of particular extremal problems. 4

Uniqueness and Stability of Optimal Solutions We assume that the incident eld uinc can change in a bounded set Kinc Hinc. Denote by (U1; n1; 1) 2 X K a solution of ( 17 ) corresponding to given eld uinc = ui1nc 2 Kinc. By (U2; n2; 2) 2 X K we denote a solution of problem

J~(U; ) = 0 I~(U ) + 2

+ 22 k ks2; I ! inf; G(U; ; u~inc) = 0; (U; ) 2 X

K; It is obtained from ( 17 ) by replacing functional I(U ) by another functional I~(U ) and replacing incident eld uinc by another incident eld u~inc = ui2nc 2 Kinc. We assume (20) (21) (22) (23) that the set K is bounded and derive one important inequality for the di erence of solutions of problems ( 17 ) and (23). We note rstly that by Theorem 1 the following estimates hold for Ul, l = 1; 2: kUlkX

MU = C0 sup kuinck1; e ; uinc 2 Kinc:

Denote by Pl 2 X , l = 1; 2 Lagrange multipliers corresponding to solutions (Ul; nl; l). By Theorem 3 Pl; l = 1; 2 satisfy identity In a similar manner we obtain the following inequality for ; U and P : Let us subtract (25) at l = 2 from (25) at l = 1. Setting = U we obtain k2Re[(n U; P1)

+ (n U2; P ) ]

Re[i( U; P1) I + i( U2; P ) I ] a0(U; P ) k2(n2U; P )

k2(nU; P1) i( U; P1) I = ( 0=2)hIU0 (U1) 1knkr2; :

2 1k ks; I : i( 2U; P ) I I~U0 (U2); U i: We set = P in (27), subtract from (32) and add the real part of obtained result with (30) and (31). Using relation (nU; P1) + (nU2; P ) (nU1; P ) = (nU; P1) (nU; P ) = (nU; P2) and analogous one for terms in (32) containing we arrive at the inequality ( 0=2)Re[hIU0 (U1)

I~U0 (U2); U i]

1knkr2;

+Re[k2(nU; P1 + P2) + i( U; P1 + P2) I

2k ks2; I + hf; P i]: Let us formulate obtained results as the Lemma. Lemma 1. Let in addition to conditions (j) K and Kinc Hinc be bounded sets and let the triples (U1; n1; 1) and (U2; n2; 2) be solutions of problems ( 17 ) at uinc = ui1nc 2 Kinc and (23) at u~inc = ui2nc 2 Kinc, respectively. Let functionals I(U ) and I~(U ) be continuously di erentiable and let Pl be Lagrange multipliers corresponding to (Ul; nl; l), l = 1; 2. Then the estimate (29) for the di erence U := U1 U2 and inequality (33) for di erences de ned in (26) hold.

Based on Lemma 1 we are able now to study uniqueness and stability of solutions of control problem

Jj (U; n; )! inf; G(U; ; uinc) = 0; (U; )2X K; = (n; ); j = 1; 2 (34) for particular cost functionals de ned in ( 4 ), ( 14 ). We begin with the case j = 1 corresponding to functional I1(U ) := kU udk2Q. Denote by (U1; n1; 1) the solution of (34) at j = 1 corresponding to given functions ud = u1d 2 L2(Q) and uinc = ui1nc 2 Kinc Hinc. By (U2; n2; 2) we denote the solution of (34) at j = 1 corresponding to perturbed functions u~d = u2d 2 L2(Q) and u~inc = ui2nc 2 Kinc. Setting ud = u1d u2d in addition to relation (26) we have hI10(Ul); i = 2(Ul uld; )Q; hI10(U1)

I~10(U2); U )i = 2(kU k2Q (U; ud)Q): Then the identity (25) for Lagrange multiplier Pl 2 X and inequality (33) for di erences (26) take the form

Firstly we estimate multipliers P1, P2 and the term hfi; P i entering into the righthand side of (36). To this end we consider the problem (35) for Lagrange multiplier Pl which is equivalent to the following equation: A l Pl = 0fl 2 X , hfl; i = ( ; Ul uld)Q; l = 1; 2. Here A l is an adjoint operator of A l . It is de ned by hA l P; i = a l ( ; P ) = hA l ; P i for all P 2 X, 2 X. Since j( ; Ul uld)Qj [kUlkX + d d max(ku1kQ, ku2kQ)]k kX then using the properties of adjoint operators and Theorem 1 we derive the estimate kPlkX ~

C0 0MU0 ; l = 1; 2; MU0 = MU + max(ku1dkQ; ku2dkQ): Taking into account ( 14 ), (26) and (37) we deduce that jhf; P ij

C1kuinckX kP1 + P2kX

0akuinckX ; a := 2C0MU0 : all c

Using estimates ( 7 ), ( 14 ), (29), (37) and Young's inequality 2cd 0, d 0, > 0 at = 1 we have c2 + (1= )d2 for jk2(nU; P1 + P2) j

C1knkL1( )kU kX kP1 + P2kX 2C1C0C~0 0MU0 Cr0 knkr; (Cr0 MU knkr;

+kuinck1; e ) 0b(4Cr02MU2 knkr2; + Cs2MU2 k ks; I + kuinc 2 2 k1; e ); ji( U; P1 + P2) I j

C1k kL1( I )kU kX kP1 + P2kX 0b(Cr02MU2 knkr2;

+ 4Cs2MU2 k ks2; I + kuinck21; e ); b := C02MU0 MU 1:

We assume that the following conditions take place: where " 2 (0; 1) is an arbitrary constant. Using (41) we derive from (39), (40) 1(1 ") > 5 0bCr02MU2 ; 2(1

") > 5 0bCs2MU2 Rejk2(nU; P1 + P2)

+ i( U; P1 + P2) I j 1(1 ")knkr2; + 2(1 ")k ks; I + 2 0bkuinc 2 2 k1; e : Taking into account (38) and (42) from (36) we obtain Here function '( ) is de ned by 0kU k2Q 0Re(U; ud)Q " 1knkr2;

2 " 2k ks; I +

0'(kuinck1; e ): '(kuinck1; e ) = akuinck1; e + 2bkuinck21; e 1=2 where constants a and b are de ned in (38) and (39). Omitting nonpositive term " 1knkr2; " 2k ks2; I in the right-hand side of (43) we have kU k2Q kU kQkudkQ + akuinck1; e + 2bkuinck12; e . This is a quadratic inequality for kU kQ. Solving it for kU kQ = kU1 U2kQ we obtain the estimate kU1

U2kQ

d ku1 u2dkQ + '(kui1nc uinc 2 k1; e ): If ui1nc = uinc (45) transforms to the estimate kU1

2

Using estimate (45) and inequality kU kQkudkQ from Young's inequality we obtain from (43) that U2kQ ku1d u2dkQ. kU k2Q +(1=4)kudk2Q which follows From (46) and (29) we deduce the estimates: " 1knkr2;

2 + " 2k ks; I

0[(1=2)kudkQ + '(kuinck1; e )]2: kn1 n2kr; p 0=" 1 ; k 1 2ks; I p 0=" 2 ; U2kX

C0(Cr0 MU p 0=" 1 + CsMU p 0=" 2 + kui1nc uinc 2 k1; e ) (47) kU1 where

d = (1=2)ku1 u2dkQ + '(kui1nc uinc 2 k1; e ): The estimates (47) have the sense of stability estimates of the solution (U^ ; n^; ^) of problem (34) at j = 1 with respect to small perturbations of functions ud 2 L2(Q) and uinc 2 Hinc. We formulate the obtained result as (39) (40) (41) (42) (43) (44) (45) (46) (48) Theorem 4. Let in addition to conditions (j) K := K1 K2 and Kinc Hinc be bounded sets and let the triple (Ul; nl; l) 2 X K be a solution of problem (34) at j = 1 corresponding to given functions uld 2 L2(Q) and ulinc 2 Kinc, l = 1; 2, where Q e is a nonempty open subset. Suppose that conditions (41) take place. Then the stability estimates (45) and (47) hold where is given by (48).

Similar result holds and for problem (34) at j = 2 corresponding to I2(U ). 5

Numerical Algorithms

Optimality system ( 10 ), (20), (21), (22) derived above can be used to design e cient numerical algorithms for solving control problem (34). The simplest one (Algorithm 1) for I1(U ) can be obtained by applying simple iteration method for solving the optimality system. The m-th iteration of this algorithm consists of nding unknown values U m, P m, nm+1 and m+1 for given nm and m by sequentially solving following problems: a0(U m; ) k2(nmU m; ) i( mU m; ) I = hf inc; i 8 2 X; a0( ; P m) k2(nm ; P m) i( m ; P m) I =

0( ; U m 1(nm+1; n 2( m+1; nm)r; m)s; I For discretization and solving problems (49), (50) one can use open source software free FEM++ (www.freefem.org) based on using nite element method. For discretization of (51), (52) it is convenient to look for solutions n and as n(x) =

N X nj 'j (x); x 2 j=1

M ; (x) = X k k(x); x 2 1: k=1 (49) (50) (51) (52) (53) Here 'j 2 Hr( ) and k 2 Hs( I ) are nonnegative basis functions in H+r( ) and H+s( I ) and nj 0 and k 0 are unknown coe cients. Similar algorithm which is based on the strategy \optimize-then-discretize" can be used and for functional I2(U ).

Now we discuss another algorithm (Algorithm 2) which is based on the opposite strategy: \discretize-then-optimize". The idea of this algorithm consists of seeking unknown controls { refraction index n and surface conductivity in the form (53). Here nj and k are unknown coe cients which one should de ne from the condition of minimum of the discrete analogue of functional I1(U ) (or I2(U )) in ( 4 ) which has the form

I1(n1; : : : ; nN ; 1; : : : ; M ) = jU (n1; : : : ; nN ; 1; : : : ; M ) udj2dx: (54) Z

Q Here U (n1; : : : ; nN ; 1; : : : ; M ) is a solution of the direct problem ( 1 ){( 3 ) for the case when parameters n(x) and (x) have the form (53). In such a case the discrete analogue of problem (34) takes the form 0 I1(n1; : : : ; nN ; 1; : : : ; M ) + 2

N 1 X nj2 + 2 j=1 where nj 0; k 0; j = 1; : : : ; N; k = 1; : : : ; M .

Problem (55) represents the nite-dimensional problem of conditional minimization which can be solved numerically using known methods of solution of discrete extremum problems. The formal comparison of both algorithms shows that Algorithm 1 is more complicated and more expensive in terms of CPU time and memory space. This is due to the fact that Algorithm 1 is based on solving the optimality system ( 10 ), (20), (21), (22) which involves a coupled system of state and adjoint equation together with two variational inequalities for sought for controls. 6

Concluding Remarks

We studied control problems for the 2D electromagnetic eld model describing scattering TM-polarized electromagnetic waves by a penetrable inhomogeneous dielectric obstacle. These problems arise when optimization method is applied for solving cloaking problems for respective scattering model. The refraction index n(x) of the inhomogeneous medium lling the obstacle and the boundary conductivity (x) of the coated part of the boundary play the role of controls. We studied some new properties of solutions of the direct problem, proved the solvability of control problems and derived the optimality systems describing the necessary conditions of extremum. Based on analysis of the optimality system we established the uniqueness and stability estimates of optimal solutions. Besides, we proposed two numerical algorithms for solving our cloaking problems. Separate paper by the authors will be devoted to comparative study of properties of these algorithms and to detailed analysis of results of numerical experiments.

Acknowledgments. The rst and the third authors were supported by Russian Foundation for Basic Research (project no. 16-01-00365-a), the second author was supported by the Russian Science Foundation (project no. 14-11-00079).

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