Control problem for a non-linear thermoelasticity system

SUMMARY The control problem for a three-dimensional non-linear thermoelasticity system is considered. The sys-tem may represent, among others, the dynamical model of shape memory materials. As controls we take distributed heat sources and body forces. The goal functional refers to the desired evolution of displacement, strain and temperature. The continuity and di(cid:2)erentiability of solutions with respect to controls is studied. The existence of optimal controls is proved and the necessary optimality conditions are formulated. The existence of adjoint state variables is proved under additional regularity of data. Copyright ? 2004 John Wiley & Sons, Ltd.


INTRODUCTION
The main objective of the paper consists in proving the existence and characterizing the control laws for optimization problems concerning fairly general non-linear three-dimensional (3-D) thermoelastic systems. The main representative of such systems describes the behaviour of shape memory materials (SMM) and its study was the primary motivation of this work.
The shape memory materials have a peculiar property that their free energy functions posses, depending on temperature, variable number of stable minima in terms of strain. Above certain temperature there is only one minimum corresponding to the strain-free state, and below it the minima occur also for several non-zero strains.
Thus, at a temperature below critical, an external force may cause shift of the state from the strain-free conÿguration to another stable shape, and the subsequent heating causes the appearance of elastic forces striving to restore the initial conÿguration. This property, known as shape memory e ect, is a consequence of structural phase transitions between low-temperature 2186 I. PAW LOW AND A.ŻOCHOWSKI martensitic phases and high-temperature austenitic phase. It is used in many applications, see e.g. References [1,2].
As we see, the choice of control variables is natural, namely the intensity and location of external heat sources and forces. The goal functional should refer to a desired evolution of a structure made of SMM. Therefore it can depend in particular on the variable conÿguration (displacement) and strain, which in turn is related to the material phases, as well as on temperature distribution.
The generality of the problem statement is due to the fact that the system under consideration expresses balance laws of linear momentum and energy with constitutive relations characteristic for a broad class of materials. In particular, we admit governing elastic energy function corresponding to several types of SMM models, like 3-D Falk-Konopka model for metallic alloys [3] and 3-D 'averaged' model for a polymer material [4].
In 1-D case the problem is identical to the well-known Falk's model for martensitic phase transitions of the shear type [5].
Questions related to thermodynamical background of thermoelastic systems under consideration, the existence and uniqueness of global in time solutions have been addressed in the previous papers [6][7][8]. Here we study the stability and di erentiability properties of these solutions with respect to control variables. Furthermore, we prove the existence result for an optimal control problem and formulate the necessary optimality conditions. We point out that our analysis of the stability and di erentiability properties is based on the technique developed in Reference [8] for the global in time existence.
Similar control problems but for special kinds of 2-D systems, namely plates activated by shape memory reinforcements, have been treated in References [4,9,10].

State equations
Let ⊂ R n , n = 2 or 3, be a bounded domain with a smooth boundary @ , occupied by an elastic body in a reference conÿguration. Let also I = (0; T ); Q t = (0; t) × , t = {t} × , S t = (0; t) × @ ; and n stands for the unit outward normal to @ . Let u : Q T → R n be the displacement vector, and Â : Q T → R + the absolute temperature. We denote by U = ( ij ) with ij (u) = 1 2 (u i=j +u j=i ) the linearized strain tensor, and by U t = U(u t ) the strain rate tensor.
Throughout the paper we use the notation f =i = @f=@x i , f t = @f=@t. The state equations to be considered express balances of linear momentum and energy which, under simplifying assumption of constant mass density ≡ 1, are given by 2187 and boundary conditions u = 0; Qu = 0 (5) We shall refer to (1)-(7) as problem (P). The quantities in (P) have the following meaning: F(U; Â)-elastic energy, c(U; Â)-speciÿc heat coe cient, c v ; k; and Ä-positive constants corresponding to thermal speciÿc heat, heat conductivity, viscosity and interface energy.
The vector b is a distributed external force and g a distributed heat source which represent possible mechanical and thermal controls.
The linear map where ; are the LamÃ e constants and I = ( ij ) is the unit matrix, represents Hooke's law for the homogeneous isotropic material. Here A = (A ijkl ) with is the fourth-order elasticity tensor. The second-order di erential operator Q deÿned by is known as operator of linearized elasticity. In the divergence operator ∇ · we use the convention of the contraction over the last index, ∇ · (AU(u)) = @ j (A ijkl kl (u)) = A ijkl @ j kl (u) = A∇U(u) Moreover, the summation convention over repeated indices is used, and the following notation: Problem (P) is associated with the free energy functional of the Ginzburg-Landau form, with the three terms representing thermal, elastic and interfacial energy. The main characteristic feature of (9) as a model of shape memory materials is the nonlinearity of the elastic energy. Namely, F(U; Â) is a multiple-well in U with the shape changing qualitatively with Â. The second characteristic feature is the presence of strain-gradient term which accounts for interaction e ects on phase interfaces.
A typical example of the elastic energy is the Falk-Konopka model [3] in the form of sixth-order polynomial in terms of ij : where J k i (U); i = 1; : : : ; i k , are kth order crystallographical invariants, that is appropriate combinations of the strain tensor components ij , and with constant parameters k i ; Â c . Form (10) represents a generalization of the well known 1-D Landau-Devonshire energy proposed for shape memory alloys by Falk [5], where i ¿0 are constant parameters, and Â c ¿0 is a critical temperature. Our formulation (1)-(7) constitutes an analog of 1-D dynamical Falk's model [5].

Assumptions
The problem (P) is studied under several conditions concerning data and constitutive functions. We assume that (A1) Domain ⊂ R n ; n = 2; 3, with the boundary @ of the class C 3 .
(A3) The function F(U; Â) is of class C 3 on S 2 × [0; ∞), where S 2 denotes the set of symmetric tensors of second order in R n . We assume the splitting where F 1 (U; Â) is a concave function with respect to Â, such that F 1 (U; Â) is linear in Â over a certain interval [0; Â 1 ), Â 1 = const, and has the polynomial growth Â r for Â¿Â 1 .
(A4) Growth conditions: There exists a positive constant such that for Â¿Â 1 and large values of ij the following conditions are satisÿed: where p n = n + 2 and q n is the Sobolev exponent for which the imbedding of W 1 2 ( ) into L qn ( ) is continuous, that is q n = 2n=(n − 2) for n¿3 and q n is any ÿnite number for n = 2. We note that 0¡q6 q n p n 2n The above conditions imply the following growth of F(U; Â): We add some comments on the above conditions. The restrictions concern Â-growth exponent of F 1 , -growth exponent of F 2 and the condition relating -growth of F 1 with its Â-growth and -growth of F 2 . The most restrictive are the conditions r¡1=2 and q65=2 in 3-D. In 2-D, since q n is any ÿnite number, arbitrary polynomial growth is admissible.
In particular, in 3-D the above conditions are satisÿed for, q = 5 2 ; q= 1; r= 3 14 In addition we assume the structural lower bound for the part F 2 (U) of the free energy.
(A5) There exist positive constants c; such that Let us stress that growth assumptions (A4) allow us to use model (10) only in the ÿnite range of strains and temperatures. The next assumption concerns the structural simpliÿcation of the energy equation by neglecting the non-linear elastic contribution −ÂF 1=ÂÂ (U; Â) in the speciÿc heat coe cient. This allows application of the classical parabolic theory in the existence proof.
We point out that because of the applied technique we were unable either to allow F 1 ( ; Â) linear in Â or, assuming Â-growth condition, to incorporate the arising non-linearity in the speciÿc heat coe cient.

I. PAW LOW AND A.ŻOCHOWSKI
We are looking for the solution in the anisotropic Sobolev space } with a parameter p related to L p -integrability. The assumptions on the initial data and the source terms correspond to this space.
(A7) The initial conditions satisfy for 1¡p¡∞ the inclusions and the compatibility relations. The source terms satisfy b ∈ L p (Q T ); g∈ L p (Q T ); g¿0 a:e: in Q T

Existence results
We recall here the existence and uniqueness results for problem (P) proved in Reference [8]. there exists for p n 6p¡∞ a solution (u; Â) ∈ V (p) to problem (P) for any T ¿0. Moreover, Â¿0 in Q T and the following a priori estimates hold: p (QT ) 6 with a constant depending on the data of the problem, and time T .
We note some properties of the solution which follow directly from the classical imbeddings.

Corollary 2.1
For a solution to problem (P) the following holds: u, ∇u, ∇ 2 u, u t , Â are H older continuous in Q T , ∇ 3 u; ∇u t ; ∇Â ∈ L p (Q T ), p n 6p¡∞, and The proof of uniqueness requires the continuity of ∇u t in Q T , which holds provided p¿p n .

Theorem 2.2
Let the assumptions of Theorem 2.1 be satisÿed for Then the solution to the problem (P) is unique for any T ¿0. The solution to problem (P) has in case p n ¡p¡∞ the following properties: ∇ 3 u, ∇u t , ∇Â are H older continuous in Q T and satisfy the bounds, The existence proof in Reference [8] is based on the parabolic decomposition (see Reference [4]) of the problem (P). The same decomposition is used here for the proof of the stability and di erentiability results. Choosing numbers ; ÿ so that system (1) with initial conditions (3) and boundary conditions (5) is equivalent to the following two sets of BVPs for a vector ÿeld w: w = 0 on S T and the displacement u: The condition for parameters Ä and required by Theorem 2.1, assures that ; ÿ¿0.

Remark 2.1
The decomposition of system (1) is applied for technical reasons. It allows sequential improvement of a priori estimates in small steps. It should be underlined, however, that system (1) is parabolic in the sense [17], what can be checked in the same way as in Lemma 7.1 [8].
It seems possible to generalize the results for any Ä; , but it would require the proof of regularity properties for fourth-order parabolic systems with the right-hand side in the divergence form, similar to Lemma 7.2 [8].

STABILITY
In this section, we prove the stability of solutions (u; Â) of problem (P) with respect to control parameters (b; g). Let (u 1 ; Â 1 ) and (u 2 ; Â 2 ) be the solutions corresponding to (b 1 ; g 1 ) and (b 2 ; g 2 ), respectively. We have the following where is a constant depending on the data of the problem, and time T .

Proof
To simplify notation we set Energy inequalities: In the ÿrst step we obtain estimates for v. To this purpose we multiply (14) by v t and integrate over Q t to get Integrating by parts the second integral gives Similarly, for the third integral, after applying twice integration by parts and using symmetry property for A, we get, Finally, after integrating by parts, the fourth integral in (18) is Combining (18)-(21) and using initial conditions (16) yields, Moreover, in view of (16), we have, Adding (22) and (23) and using estimate which follows from the regularity assumption for F and the uniform bounds on U i ; Â i in Q T , by Young's inequality we arrive at Choosing = a ? =2, the use of Gronwall's inequality implies v t L∞(0;T ;L2( )) + U(v) L∞(0;T ;L2( )) + Qv L∞(0;T ;L2( )) + U(v t ) L2(QT ) Hence, recalling the ellipticity property of the operator Q, v L∞(0;T ;W 2 2 ( )) 6 ( The estimates for Á follow from multiplying Equation (15) by Á and integrating over Q t : In view of the estimate which follows from uniform estimates on U i ; Â i ; U i t and using (25), we get, Hence, by Gronwall's inequality, where for simplicity we use the following abbreviation: Now, combining (28) and (25), yields, We note the following consequences of (29). By the imbedding W 1 2 ( ) ⊂ L qn ( ), and by the imbedding (see e.g. Reference [18]) L ∞ (0; T ; L 2 ( )) ∩ L 2 (0; T ; W 1 2 ( )) ⊂ L 2pn=n (Q T ); 2¡ 2p n n = 2(n + 2) n ¡q n we have, First estimates: Here we make use of the parabolic decomposition of (14). Let (w 1 ; u 1 ) and (w 2 ; u 2 ) denote the corresponding solutions of the decomposed problems (11), (12), and let The functions (y; v) satisfy the following BVPs: where is a constant depending on ; + 2 ; ; T and p. Hence, in view of (24), Consequently, using estimates (30) and (31) it follows, Now, with the help of classical parabolic theory [19], we can obtain additional bounds on Á. To this purpose applying (31), (34) and (27) we estimate the right-hand side of (15): Iterative improvement of estimates: Now, returning to system (32), (33), in view of (36) and (39) we can obtain further improvement of estimates. Namely, since U i ; Â i ; ∇U i and ∇Â i are continuous in Q T , Because q n p n =n¿p n , from Solonnikov theory of parabolic systems [17] (see also Therefore, by imbedding, Repeating estimation (37), in view of (35), (38) and (41), it follows that, Consequently, the classical parabolic theory implies, Á W 2; 1 p (QT ) 6 (D(p n ) + D(p)) 6 D(p) for p n 6p¡∞ Hence, by imbedding, This completes the proof.

DIFFERENTIABILITY
Let us consider two control pairs (b i ; g i ) ∈ L p (Q T ) × L p (Q T ), g i ¿0 a.e. in Q T , i = 1; 2; such that where 06 6 0 . Let (u i ; Â i ) ∈ V (p), p¿p n , be the unique solutions of problem (P) corresponding to (b i ; g i ). According to Theorem 3.1 we have the following stability estimate: for p¿p n . Consequently, by imbeddings, similar bounds hold pointwise in Q T for the di erences u 2 − u 1 , Â 2 − Â 1 , ∇(u 2 t − u 1 t ), ∇ k (u 2 − u 1 ), k = 1; 2; 3, and ∇(Â 2 − Â 1 ). Our goal is to ÿnd a pair (v; Á) ∈ V (p) such that in the sense of the space V (p). For simplicity we introduce the notation: Using formal approximation by Taylor series we obtain the following system of equations for the pair (v; Á): where is a constant depending on the data of the problem, and time T . Hence which means that the pair (v; Á) is a Gateaux derivative of the solution with respect to the parameters (b; g).

Proof
Let By deÿnition, (z; ') satisfy the system: with initial and boundary conditions, In view of regularity of solutions (u i ; Â i ) there exists the unique solution (z; ') ∈ V (p) to problem (49)-(52) for any p¿p n . We shall show that By assumptions on F(U; Â) and the regularity of (u i ; Â i ) the following bounds are valid: From now on we will follow closely the proof of Theorem 3.1.
In view of (54) the right-hand side of equation (50) is bounded by

Iterative improvement of estimates:
We return now to the decomposed system (61), (62). In view of the bounds which follow from the regularity of (u i ; Â i ), where (u; Â) is the solution of (P) corresponding to f = (b; g). Theorem 3.1 ensures that the map S is Lipschitz continuous. We have also the following weak continuity property.

Lemma 5.1
Under assumptions of Theorem 2.2 the map S is continuous from U ad (weak) into V (p) (weak).

Proof
Consider a sequence (b n ; g n ) ∈ U such that, (b n ; g n ) → (b; g) weakly in U Let (u n ; Â n ) be a sequence of solutions of (P n ) corresponding to (b n ; g n ). Since (b n ; g n ) is uniformly bounded in U, the a priori bounds of Theorem 2.1 imply that (u n ; Â n ) is also uniformly bounded in the space V (p). Therefore, after selecting the subsequence, (u n ; Â n ) → (u; Â) weakly in V (p) Since p¿p n , by the compact imbeddings, u n ; ∇u n ; ∇ 2 u n ; ∇ 3 u n ; u n t ; ∇u n t ; Â n ; ∇Â n are convergent in spaces of H older continuous functions. Therefore we have pointwise convergence for all terms entering the right-hand sides of Equations (1)-(2). Then we can pass to the weak limit in (P n ) to conclude that (u; Â) satisÿes (P).
By virtue of the stability estimate (44) the result of Theorem 4.1 can be easily reformulated in terms of S in the following way: Let and S(f); S(f + f) be the corresponding solutions of (P). Then, where the coe cients F =UU ; F =UÂ ; H 1 ; H 2 ; H 3 are evaluated at (u; Â) = S(f).

Remark 5.1
The constant on the right-hand side of estimate (74) does not depend on the norms of v; Á. Therefore the operator S (f) : U ad → V (p) is the FrÃ echet derivative of the operator S.
We consider the following cost functional: where the function (s 1 ; s 2 ; s 3 ) is assumed to be of a class C 1 ( R 3 + ), Lipschitz continuous, and the weight coe cient is positive. Moreover, s ∈ N and 2s¿p n . The functions u; Â are given reference solutions of problem (P).

Theorem 5.1
There exists an optimal controlf ∈ U ad minimizing the cost functional (79) of the problem (P), 2208 I. PAW LOW AND A.ŻOCHOWSKI For example this continuity holds if the following conditions are satisÿed: • D s t D r x u are continuous for 2s + r66; • D s t D r x Â are continuous for 2s + r64.

Proof
Due to our regularity assumptions we may di erentiate (92) and obtain for q t the linear parabolic equation with continuous coe cients. Moreover, from (92) it follows, by substituting t = T , that what supplies the necessary end condition. For simplicity of reasoning let us now change the time direction by substitution t := T − t, so that the end conditions become initial ones. System (91)-(92) transforms to c v q t − k q − H 3 q = −F =UÂ : U(r) + 2 in Q T with unchanged boundary conditions. By standard parabolic theory we have from (99) the estimate, q W 2; 1 2 (QT ) 6 + U(r) L2(QT ) Similarly, after di erentiating (99) with respect to time, and using our regularity assumptions as well as initial condition (97), we get similar estimates for q t : q t W 2; 1 2 (QT ) 6 + U(r t ) L2(QT ) + U(r) L2(QT ) These bounds are crucial for the proof. We shall concentrate on obtaining an a priori estimate for the solutions in the spaces given by the formulation of the theorem. The rest of the steps required by the Leray-Schauder theorem is easy due to the linearity of the problem. First we multiply (98) by r t and integrate over Q t , using initial and boundary conditions. As a result for the left-hand side we obtain the identity, The right-hand side consists of ÿve terms, R := R 1 + R 2 + R 3 + R 4 + R 5 which will be considered one by one. For the ÿrst we have, and again the ÿrst part may be absorbed by L 2 . For the third term we have, after application of (100), The last term is simple, Qt r t · 1 dx dt 6 ( r t L2(Qt ) + 1) Because of the strong ellipticity of the operator Q we have also, taking into account homogeneous boundary conditions, Qt |U(r)| 2 dx dt 6 Qt |Qr| 2 dx dt Therefore, by suitable choice of 1 ; 3 ; 4 we get, Qt |r t | 2 dx dt + Qt |Qr| 2 dx dt + 1