PARTIAL EXACT CONTROLLABILITY FOR INHOMOGENEOUS MULTIDIMENSIONAL THERMOELASTIC DIFFUSION PROBLEM

. The problem of stabilization and controllability for inhomogeneous multidimensional thermoelastic diﬀusion problem is considered for anisotropic material. By introducing a nonlinear feedback function on part of the boundary of the material, which is clamped along the rest of its boundary, we prove that the energy of the system decays to zero exponentially or polynomially. Both rates of decay are determined explicitly by the physical parameters. Via Russell’s “Controllability via Stabilizability” principle, we prove that the considered system is partially controllable by a boundary function determined explicitly.


1.
Introduction. The research conducted in the development of high technologies after the second world war, confirmed that the field of diffusion in solids can not be ignored. Diffusion can be defined as the random walk of a set of particles from regions of high concentration to regions of lower concentration. Thermodiffusion in an elastic solid is due to coupling of the fields of strain, temperature and mass diffusion. The processes of heat and mass diffusion play a important role in many engineering applications.
The anisotropic thermoelastic diffusion materials with variables coefficients are used in several engineering applications such that metallurgical industry where these materials are processed under strong temperature gradients. An important example is the thermo-mechanical modeling of aluminium electrolytic cell [10]. Another important application of this kind of materials is the thermo-mechanical modeling of slabs during direct chill castings of alloys [9]. In all these processes and others, it is very important to take into account the mechanical heat dissipation and the non-homogeneity of the materials. The stabilization and the controllability analysis of these problems becomes very difficult due to all these aspects. For this reason the qualitative properties of solutions to thermoelastic diffusion systems becomes that the isotropic thermoelastic system is partially controllable with boundary controls without smallness restrictions on the coupling parameters. Lasiecka in [18] provided different techniques to control several coupled systems.
The main novelties of the present paper are the following.
(i) The question of analyzing the stabilization or the controllability under the weaker feedback forces of the form (8) is an interesting open problem. This analysis has been performed in [3,15] in elasticity context and in [25,28] in isotropic thermoelasticity but, to our knowledge, this issue has not been addressed for inhomogeneous and anisotropic system of thermoelasticity with diffusion. The coupled system of equations considered in this study consists of a hyperbolic equation coupled with two parabolic equations. This poses new mathematical difficulties due to the nonlinear boundary conditions in stabilization or controllability issues. Under the classical polynomial growth assumption on the nonlinear boundary feedback near the origin, by using multiplier techniques and Lyapunov methods, we show that the considered system decays to zero at an exponential or polynomial rate. For the polynomial decay, some necessary modifications are imposed by the inhomogeneous and anisotropic nature of the material and the complicated coupling of our equations since they do not fall directly in the abstract frame of classical systems. Moreover, the proof of the polynomial stability has never been done previously as clearly with the weaker feedback forces given by (8) for any material. We show that the polynomial stability recover the exponential stability as a limit special case.
(ii) However, our approach, even if it uses ingredients of literature, relies essentially on the generalized Young's inequality and it is simpler. This allows us to get more explicit expressions for the nonlinearity entering in the differential inequality governing the decay of the energy and the function of control. We show how the explicit expressions of the decay rates of the energy play an important role in the choice of material leading to faster decay and in the selection of the correct conditions imposed for stabilization or controllability.
(iii) To the best of our knowledge, there is no papers in literature discussing the controllability of the thermoelastic diffusion problem. In this paper, via Russell's "Controllability via Stabilizability" principle we show the partial exact controllability of the considered problem with smallness restriction (see (91)) on the coupling tensors, stress-temperature and stress-diffusion. This condition shows the influence of the rate of exponential decay on the controllability. The boundary feedback function of the control will be determined explicitly. This represents a pleasant feature from the physical viewpoint.
The rest of the paper is organized as follows. In Section 2, we present the basic equations of the considered system and we give sufficient conditions needed in the sequel. In Section 3, we describe briefly the well-posedness of the corresponding system. In Section 4, we show that the considered system decays to zero at an exponential or polynomial rate depending on the assumptions on the feedback and on the system's parameters. Finally, in Section 5, we show the partial exact controllability of the considered system.
We write partial derivatives with respect to the spatial variable as and with respect to the time derivative in the following waẏ Here we will adopt double index convention (that is double index means summation over the repeated i.e., Let us denote by Ω ⊂ R n the connected bounded domain in which our material will be configurable with boundary Γ = ∂Ω of class C 2 . For any x of the body at a time t, we will denote by u, θ and P the displacement, the thermal and the chemical potential, respectively. We denote by ν(x), x ∈ Γ = ∂Ω the exterior normal vector to Γ. We assume also that there exists a point x 0 ∈ R n for which we denote by m the vector field m( The notation m(x) · ν(x) represents the inner product in R n of the vectors m(x) and ν(x). As example of the existence of such point x 0 we can consider the domain given by Figure 1. By our choice of x 0 we easily get that hypotheses (1) holds. Note that both Γ 0 and Γ 1 are non-empty and satisfy Since Γ 0 is a compact subset, there exists two positive constants a 0 , a 1 for which Let us define by the radius of the smallest ball, with center in x 0 , containing Ω.

PARTIAL EXACT CONTROLLABILITY FOR INHOMOGENEOUS 205
In the absence of exterior forces, heat and chemical sources, the thermoelastic diffusion equations for inhomogeneous and anisotropic material are given by [4] We will consider the following boundary conditions with i, j, k, l ≥ 1, Q 1 > 0 and Q 2 > 0. In the first part of this paper, the function φ is given by where g i is at our disposal. In the second part, the function φ i will be determined to get the partial exact controllability of the system. We assume that the initial conditions are (9) In the above equations at a point x ∈ Ω, ρ(x) denotes the mass density and C ijkl (x), α ij (x), β ij (x) are tensor fields which represent the elasticity of the material, the stress-temperature, the stress -diffusion. By c(x), d(x) and r(x) we denote respectively the specific heat, the measure of thermal-diffusion effect and the measure of diffusive effect. k ij (x) is the conductivity tensor of thermal field and h ij (x) is the conductivity tensor of diffusion field.
We assume also that C ijkl (x), α ij (x), β ij (x), k ij (x) and h ij (x) are smooth over Ω and satisfy the following conditions There exists positive constants γ 1 , α, ς and κ such that We set The density ρ(x) is continuous over Ω and satisfies the following conditions where γ 0 is a positive constant. The functions c(x), r(x) and d(x) are continuous over Ω and satisfy

MONCEF AOUADI AND KAOUTHER BOULEHMI
Note that (15) implies that This condition is needed to stabilize the thermoelastic diffusion system (see [4] for more details). Finally, we assume that the function g i ∈ C(R n ), nondecreasing on Γ 0 and satisfies for all u i ∈ R the following hypothesis [28] for some positive constants k 2 , k 3 , p and q with 0 < q ≤ 1.

The Riesz representation theorem ensures that
. Let ., . denote the duality pairing between (H 1 Γ1 (Ω)) n and [(H 1 Γ1 (Ω)) n ] or H 1 0 (Ω) and H −1 (Ω). Also we define a nonlinear operator B, by Note that there exist constants c > 0 and b ≥ 0 such that for u i ∈ R, g i satisfies [28] for more details). Using the above notations we conclude that the field equation (5)-(9) is equivalent to an abstract Cauchy problem where the nonlinear operator A is considered on (17), (18) and (24) and Based on the above properties of the operators A, B, C, D and suppose that g i satisfies (24), it is standard matter to show that the nonlinear operator A is mdissipative on H (for more details see [11,28,29] and references therein).
Lemma 3.1. Suppose that (2) holds and g i satisfies (16) and (24), then the domain (19) and (20) we have for some s > 3 2 . The proof is essentially the same as that of Lemma 9.2 of [2] and Lemma 3.3 of [28]. Hence it is omitted here for the sake of brevity.
By the classical theory of nonlinear semigroups (see [8], Chap. 3), we have Theorem 3.2. Let Γ 0 and Γ 1 are non-empty satisfying (1) and (2). Suppose that the function g i satisfies (16), (21) and (24). Then we have (i) For every initial condition (u 0 , u 1 , θ 0 , P 0 ) ∈ H, the problem (5)-(9) has a unique mild solution satisfying (ii) Further, if g i satisfies (19) and (20), then, for every initial condition (u 0 , u 1 , θ 0 , P 0 ) ∈ D(A), the problem (5)-(9) has a unique classical solution satisfying 4. Decay of solutions. Firstly, we need to perform some technical lemmas. To do this end, we introduce the following energy functional Recall that Γ 0 is a compact subset satisfying (3). The conductivity terms k ij and h ij are definite positive and satisfy (12). The function g i satisfies (17) and (18). Finally we assume that (15) holds, Q 1 > 0 and Q 2 > 0. Based on the above properties of the operators A, B, C, D, we multiply (5) 1 , (5) 2 and (5) 3 respectively by v i , θ, P ∈ H 1 Γ1 (Ω) and we integrate over Ω we conclude that d dt First, let us introduce the following functionals where µ = 1, · · · , n and m µ is the µ−component of m. By a direct calculation, we get Proof. Differentiating the function F(t) with respect to time, we get By integration by parts, the first term of the right hand side of (35) can be written in the form while the second term is given by Since over Γ 1 we have u i = 0, then we obtain so the second term of r. h. s of (37) is given by Using the symmetry on the coefficient which implies (C ijkl u k,l u i,j ) ,µ = C ijkl,µ u k,l u i,j + 2C ijkl u k,l u i,jµ , 210 MONCEF AOUADI AND KAOUTHER BOULEHMI the last term of (37) becomes Then using (39) and (40), (37) becomes Substituting Eqs. (36) and (41) into (35) we obtain (34).
Proof. After differentiating the function K(t) with respect to time and using (33), (34) and (1) we get Using (3) and (14) 2 the first term of the r. h. s of (44) can be estimated as Using Young's inequality and (38), the second term of the r. h. s of (44) can be written as where R 0 is given by (4). The third term of the r. h. s of (44) can be estimated as follows where is a positive constant. Using (13), the fourth term of the r. h. s of (44) becomes where M 2 is given by (43). From (14) 3 and (11) 1 the fifth and sixth terms of the r. h. s of (44) are given by The two last terms of the r. h. s of (44) can be estimated as follows and 2 where M 3 is given by (43). By substituting Eqs. (45)-(51) into (44), we get

MONCEF AOUADI AND KAOUTHER BOULEHMI
where 2 and M 1 are given by (43). It is easy to see that there exist a constant where 1 is given by (43). Then we infer from (53) that From (22) and by taking = 2 1 in (52), the second term of the r. h. s of (52) can be estimated as follows From (11) 2 , the third term of the r. h. s of (52) can be estimated as follows Using (7) 1 and Young's inequality, we get Substituting (54)-(56) into (52), we get (42).
We are now ready to prove the stability results. To this end, let us introduce for any positive number δ, the Lyapunov functional where σ = p+1−2q 2q is a nonnegative constant that will be determined later. The method of the prove is analogous to the one developed in [28] with some necessary modifications imposed by the inhomogeneous and anisotropic nature of the material and the complicated coupling in our equations. Evidently, the function L(t) is actually a generalized energy functional which is closely related to the energy functional E(t). We first show that L and E are equivalent. Then, by solving the differential inequality d dt for different values of σ, our decay results follows. In fact, the conditions (16), (17) and (19) imply that q ≤ p. Therefore, under the assumption 0 < q ≤ 1, we have 2q ≤ p + 1. If 2q = p + 1, then q − 1 = p − q ≥ 0 implies that p = q = 1. Therefore, we have only two cases: p = q = 1 (σ = 0) and 2q < p + 1 (σ > 0). Since we have where 0 = max 2R0 ρ1 + (n−1) α , Differentiating L(t) with respect to time, using (58) and (31), we get From (31) and (42) of Lemma 4.2, we infer that where 3 = Λ 2 max{ 1 κ , 1 ς }, 4 = 0 max{Q 1 , Q 2 } and 5 = min{Q 1 , Q 2 }. We now distinguish the cases p = q = 1 and p + 1 > 2q.
Remark 4.5. We have proved that the decay rate is governed by a dissipative ordinary differential equation, even if the nonlinearity does degenerate at the origin faster than any polynomial. This allows to show, in particular, that if the nonlinearity degenerates at the origin exponentially then a logarithmic decay holds. To do that, one can proceed as in [28,19].
As stated in [24], this is equivalent to steering every initial state (u 0 , u 1 ) of the displacement in the function space to the state u(T ),u(T ) = (0, 0), disregarding the values of the temperature and the chemical potential. Let σ be the smallest positive constant such that [25,26] div u −1 ≤ σ u L 2 (Ω) , ∀u ∈ L 2 (Ω).
We now state the main result of this section.
where ω and Λ are the constants in Theorem 4.3, η and ϑ are given by (82) and σ is defined by (76). Let T 0 be large enough such that and T ≥ T 0 , Then for any (u 0 , u 1 ) ∈ W, there exists a boundary control function φ(x, t) defined by such that the solution to the problem (5)- (7) with the initial conditions (74), satisfies (75). Moreover, g satisfies where C is a positive constant independent of (u 0 , u 1 ).
where (w, ϕ, ψ) solution to (85), z solution to (77) and (ξ, χ) solution to (80). Then (u, θ, P ) satisfies where φ is given by (93) and g satisfies (16)- (20). Now, we define the linear operator N from W into W by where (z 0 , z 1 ) ∈ W are defined by (77) 4 . Moreover, we have Applying Hölder's inequality to the last integral in (101), we get Using the same argument, the last term of (100) can be estimated as Substituting the two last inequalities in (100), we get Since N − I is an isomorphism from W onto itself and from (92) and (98) we obtain that 0 < N − I ≤ 2. Thus it follows that Σ0 (m j · ν j )|g i (ẇ i ) + g i (ż i )| 2 dΣ ≤ C (N − I) −1 (u 0 i , u 1 i ) 2 (use (99)) ≤ C (u 0 i , u 1 i ) 2 . for a positive constant C. Using (3), our conclusion follows. 6. Conclusion. (i) In this paper the partial exact controllability is proved for the multidimensional thermoelastic diffusion problem for inhomogeneous and anisotropic material with smallness restriction (91) on the coupling tensors, stress-temperature and stress-diffusion. This condition states that more the magnitudes of the coupling tensors are small or more the energy of the system decreases exponentially faster to zero (ω large enough), more the system is controllable. From the numerical values of explicit exponential decay rate, Aouadi and Moulahi found in [5] that the decay rate is best for copper, than for aluminum alloy, silicon, diamond, steel, indium arsenide, for germanium and finally for gallium arsenide, sequentially. The choice of materials in applications should be based on this. Also the boundary feedback function of the control has been determined explicitly. These results represent a pleasant feature from the physical viewpoint.
(ii) The explicit expressions of the decay rates ω and λ play an important role in stabilization and controllability issues. They describe the dependence of the decay rate on the assumptions on the feedback and the physical parameters leading to the best choice of materials realizing the fastest decay of the energy. This is useful to select the correct setting of the conditions imposed for stabilization or controllability. For example, if we take a linear boundary feedback g(s) = k 3 s where k 3 is the parameter appearing in conditions (17) and (18) and let k 3 → 0, then it follows from (62) 1,3 that ω → 0 and from (64) 3,6 that λ → 0. Thus we lose the stability and consequently the controllability of the problem (see (91)).
(iii) In addition, by these expressions, we can analyze the limit of the polynomial decay rate 1 + σλt − 1 σ as p, q tends to 1 and recover the exponential decay of the case p = q = 1. Indeed, it is easy to see that lim p,q→1 δ 2 (p, q) = δ 1 and lim p,q→1 λ(p, q) = ω 2 .
Consequently we have lim p,q→1 Despite the exponential stability is a result more important than the polynomial stability in our study, it turns out that it is a special case of the latter.