1. INTRODUCTION 2. PROBLEM FORMULATION ROMAI J., 6, 2(2010), 1 13

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1 Contents 1 A product formula approach to an inverse problem governed by nonlinear phase-field transition system. Case 1D Tommaso Benincasa, Costică Moroşanu 1 v

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3 ROMAI J., 6, 2(21), 1 13 A PRODUCT FORMULA APPROACH TO AN INVERSE PROBLEM GOVERNED BY NONLINEAR PHASE-FIELD TRANSITION SYSTEM. CASE 1D Tommaso Benincasa, Costică Moroşanu University of Bologna, Bologna, Italy University Alexandru Ioan Cuza of Iaşi, Romania Abstract In this paper we study an inverse problem, in one space dimension case, connected with the industrial solidification process called casting wire, as an optimal control problem governed by nonlinear phase-field system with nonhomogeneous Cauchy-Neumann boundary conditions. We prove the convergence of an iterative scheme of fractional steps type for the optimal control problem. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to the idea of a numerical algorithm in order to approximate the original optimal control problem. Keywords: nonlinear parabolic systems, phase-field models, optimality conditions, applications, timedependent initial-boundary value problems, fractional steps method, inverse problems. 2 MSC: 35K55, 49N15, 62P3, 65M12, 65M INTRODUCTION Phase-field models, strongly studied in recent years, describe the phase transitions between two different phases in a pure material by a system of nonlinear parabolic equations. These models can be viewed as extensions of the classical Stefan problem in two phases. Consequently, the interface boundary between the phases can be constructed from so-called phase function and, phenomena associated with surface tension and supercooling are incorporated into the model. The mathematical literature concerning the optimal control problems associated with such models is in a deep process of development as the models are suitable for many modern applications. One of them is the subject of this paper. 2. PROBLEM FORMULATION Denote by Ω = (, b 1 ) R, < b 1 < +. Let T > and we set: Q = (, T) Ω, Σ = {(t, x) Q, t = f (x)}, Σ = (, T) {b 1 }, 1

4 2 Tommaso Benincasa, Costică Moroşanu where t = f (x) is considered to be the equation of the moving boundary separating the liquid and solid phases, = f (b ), < b < b 1 (see Figure 1.1). Consider the following nonlinear parabolic system in one space dimension: ρc u t + l 2 ϕ t = ku xx, in Q, τϕ t = ξ 2 ϕ xx + 1 2a (ϕ ϕ3 ) + 2u, in Q, subject to the non-homogeneous Cauchy-Neumann boundary conditions and to the initial conditions (1.1) u x + hu = w(t), ϕ x = on Σ, (1.2) u x =, ϕ x = on Σ, (1.3) u(, x) = u (x), ϕ(, x) = ϕ (x) on Ω = [b, b 1 ], (1.4) where u is the reduced temperature distribution, ϕ is the phase function used to distinguish between the phase of Ω, u, ϕ : Ω R are given functions, w : [, T] R is the boundary control (the temperature surrounding at x = b 1 ), w U where U = {v L ([, T]), R v(t) a.e. t [, T]}; the positive parameters ρ, c, τ, ξ, l, k, h, a, have the following physical meaning: ρ - is the density, c - is the heat capacity, τ - is the relaxation time, ξ - is the length scale of the interface, l - denotes the latent heat, k - the heat conductivity, h - the heat transfer coefficient and a is an probabilistic measure on the individual atoms (a depends on ξ). Figure 1.1.Geometrical image of the elements in inverse problem (P inv ). The mathematical model (1.1), introduced by Caginalp [3], has been established in the literature as an extension of the classical two phase Stefan problem to capture the effects of surface tension, supercooling, and superheating.

5 A product formula approach to an inverse problem... 3 As regards the existence, it is known that under appropriate conditions on u, ϕ and w, the state system (1.1)-(1.4) has a unique solution u, ϕ W = Wp 2,1 (Q) L (Q), p > 3/2 (see Proposition 2.1 in [6]). Given the positive numbers d 1, d 2, we define: the pure liquid region : {(t, x) Q, u(t, x) > d 2 and ϕ(t, x) 1 + d 1 }, the pure solid region : {(t, x) Q, u(t, x) < d 2 and ϕ(t, x) 1 d 1 }, the separating region : {(t, x) Q, u(t, x) < d 2, ϕ(t, x) 1 + d 1 }. and we set Q = { (t, x) Q, f (x) t T }. Consider the following inverse problem: (P inv ) Given Σ f ind the boundary control w L ([, T]) such that Q is in the liquid region, Q 1 = Q \ Q is in the solid region and a neighbourhood of Σ is the separating region between the liquid and the solid region. (P inv ) is in general ill posed and a common way to treat this inverse problem is to reformulate it as an optimal control problem with an appropriate cost functional. Consequently, we will concern in the present paper with an optimal control problem associated to the inverse problem (P inv ), namely: (P) Minimize L (w) = β [ (u(t, x) δ2 ) +]2 χ 2 Q dtdx Q Q (ϕ(t, x) 1 δ 1 ) 2 χ Q dtdx T w 2 (t) dt, on all (u, ϕ) solution of the system (1.1)-(1.4) and for all w U β > is a given constant. In the above statement we denoted by u + the positive part of u, i.e. { u + u, if u >, =, if u. We point out that problem (P) is an optimal problem with boundary control w(t) depending on time variable t [, T], being dictated by the industrial solidification process like casting wire. 3. APPROXIMATING PROCESS We associate to the nonlinear system (1.1)-(1.4) the following approximanting scheme (ε > ): ρcu ε t + l 2 ϕε t = ku ε xx τϕ ε t = ξ 2 ϕ ε xx + 1 2a ϕε + 2u ε, in = { (t, x) Q, ε t T }, (1.5)

6 4 Tommaso Benincasa, Costică Moroşanu u ε x + hu ε = w(t), ϕ ε x = on Σ ε = [ε, T] {b 1 }, (1.6) u ε x =, ϕ ε x = on Σ ε = {(t, x) Q, ε t T}, (1.7) u ε (ε, x) = u (x) ϕ ε +(ε, x) = z(ε, ϕ ε (ε, x)) on Ω ε. (1.8) where z(ε, ϕ ε (ε, x)) is the solution of the Cauchy problem: z (s) + 1 2a z3 (s) =, s (, ε), z() = ϕ ε (ε, x), ϕ ε (, x) = ϕ (x), (1.9) and ϕ ε +(ε, x) = lim t ε ϕ ε (t, x), ϕ ε (ε, x) = lim t ε ϕ ε (t, x). The convergence and weak stability of the approximating scheme (1.5)-(1.9), in a more general case (w(t, x) in place of w(t)), was studied in the paper [2]. Corresponding to the approximating scheme (1.5)-(1.9), we will consider the approximating optimal control problem: (P ε ) Minimize L ε(w) = β [ (u ε (t, x) δ 2 ) +]2 χ 2 Q dtdx Q Q (ϕ ε (t, x) 1 δ 1 ) 2 χ Q dtdx on all (u ε, ϕ ε ) solution of (1.5)-(1.9) corresponding to w U. T w 2 (t) dt, The main result of the present paper (Theorem 4.1) says that problem (P) can be approximated for ε by the sequence of optimal control problems (P ε ) and so the computation of the approximate boundary control w(t) can be substituted by computation of an approximate control of (P ε ). The plan of the paper is the following: In Section 2 we shall prove the convergence results regarding the sequence of optimal control problem (P ε ). Such a convergence scheme was studied (for an optimal control problem governed by nonlinear parabolic variational inequalities) by Barbu [1]. For other works in this context see [6] and references therein. Necessary optimality conditions for the approximating process (P ε ) (Theorem 5.1) and, a conceptual algorithm of gradient type are established in the last Section. 4. THE CONVERGENCE OF PROBLEM (P ε ) The main result of this paper is Theorem 4.1. Let {w ε} be a sequence of optimal controllers for problem (P ε ). Then lim inf ε Lε (w) = inf {L (w); w U} (2.1)

7 and A product formula approach to an inverse problem... 5 lim L (w ε) = inf {L (w); w U}. (2.2) ε Moreover, every weak limit point of {w ε} is an optimal controller for problem (P). Remark 4.1. Theorem 4.1 amounts to saying that (P ε ) approximates problem (P) and, an optimal controller {w ε} of (P ε ) is a suboptimal controller for problem (P). The main ingredient in the proof of the Theorem 4.1 is the following Lemma. Lemma 4.1. If {w ε} is a sequence of optimal controllers for problems (P ε ) then there exists {ε n } such that where (u ε n, ϕ ε n, w ε n ) = (u w εn ε n w ε n w weakly star in L (Σ), (2.3) u ε n u strongly in L 2 ((, T); H 1 (Ω)), (2.4) ϕ ε n ϕ strongly in L 2 ((, T); H 1 (Ω)), (2.5), ϕ w εn ε n, w ε n ) is the solution to (1.5)-(1.8) corresponding to w = w ε n and (u, ϕ, w ) = (u w, ϕ w, w ) is the solution to (1.1)-(1.4) corresponding to w = w. Proof. Details on the demonstration of this Lemma can be found in the work [6, Lemma 3.1, pp. 11]. We omit them. We can now give the proof of Theorem 4.1. Proof. Let {w ε} be an optimal controller for problem (P ε ) and let (u ε, ϕ ε, w ε) be the corresponding solution of (1.5)-(1.8) with w = w ε. Lemma 4.1 above allows us to conclude that there exist w L ([, T]) and {ε n } such that relations (2.3)-(2.5) are valid. Since: u β ( (u(t, x) δ2 ) +)2 χ 2 Q dtdx, Q ϕ 1 (ϕ(t, x) 1 δ 1 ) 2 χ 2 Q dtdx, w 1 2 Q T w 2 (t) dt are convex continuous functions, it follows that these are weakly lower semicontinuous functions. Hence L (w ) lim inf n Lε n (w ε n ). (2.6)

8 6 Tommaso Benincasa, Costică Moroşanu Let w be an optimal controller for problem (P). Since w ε n is an optimal controller for problem (P ε n ) it follows that L ε n (w ε n ) L ( w ε ). But (see (2.4) and (2.5)) uε w n u w ϕε w n ϕ w strongly in L 2 ((, T); H 1 (Ω)) and so, the latter inequalities implies From (2.6)-(2.7) we get lim n Lε n ( wε ) L ( w ε ). (2.7) Hence L (w ) lim inf n Lε n (w ε n ) lim sup n L ε n (w ε n ) L ( w ). lim inf ε n Lε n (w ε n ) = L ( w ) = inf{l (w), w U} and then (2.1) holds. To prove (2.2) we set: ū ε = u w ε, ϕε = ϕ w ε (we recall that w ε is chosen to be optimal in (P ε )). On a subsequence {ε n } we have w ε w weakly star in L ([, T]), ū εn u strongly in L 2 ((, T), H 1 (Ω)), ϕ εn ϕ strongly in L 2 ((, T), H 1 (Ω)), where (u, ϕ, w ) satisfy (1.1)-(1.4), i.e., (u, ϕ) = (u w, ϕ w ). Therefore, we derive L (w ) inf P and, because {ε n } was chosen arbitrarily, (2.2) follows. Now, taking into account that w ε is an optimal controller for problem (P ε ), it follows that L ε (w ε) L ε (w) w U. On the other part, on the basis of relation (2.6), we can put L (w ) lim inf ε L ε (w ε) and thus, along with previous inequality, we may conclude that Consequently L (w ) lim L ε ε (w) w U. L (w ) L (w) w U i.e., the weak limit point w is a suboptimal controller for problem (P). This completes the proof of Theorem 4.1.

9 A product formula approach to an inverse problem NECESSARY OPTIMALITY CONDITIONS IN (P ε ) Let (u ε, ϕ ε, w) be the solution of (1.5)-(1.8) and let w L [, T]) be arbitrary but fixed and >. Set w = w + w and let (u,ε, ϕ,ε ) be the solution of (1.5)-(1.8) corresponding to w, that is: ρcu,ε t τϕ,ε t + l 2 ϕ,ε t = ku,ε xx, = ξ 2 ϕ,ε xx + 1 2a ϕ,ε + 2u,ε, in, subject to non-homogeneous Cauchy-Neumann boundary conditions: and initial conditions: (3.1) u,ε x + hu,ε = w, ϕ,ε x = on Σ ε, (3.2) u,ε x =, ϕ,ε x = on Σ ε, (3.3) u,ε (ε, x) = u (x), ϕ,ε + (ε, x) = z (ε, ϕ (x)) on Ω ε, (3.4) where z (ε, ϕ (x)) is the solution of the Cauchy problem: ( z (s) ) + 1 ( z (s) )3 =, s (, ε), 2a z () = ϕ,ε (ε, x), ϕ,ε (, x) = ϕ (x). Subtracting (1.5)-(1.8) from (3.1)-(3.4) and dividing by >, we get (3.5) ρc ( u,ε u ε ) + l ( ϕ,ε ϕ ε t 2 )t = k( u,ε u ε ), xx in, τ ( (3.6) ϕ,ε ϕ ε )t = ξ2( ϕ,ε ϕ ε )xx + 1 ( ϕ,ε ϕ ε ) ( u,ε u ε ) + 2, 2a ( u,ε u ε )x + h( u,ε u ε ) w w =, ( ϕ,ε ϕ ε ) = on x Σε, (3.7) ( u,ε u ε )x =, ( ϕ,ε ϕ ε ) = on x Σε, (3.8) u,ε (ε, x) u ε (ε, x) =, ϕ,ε + (ε, x) ϕ ε +(ε, x) = z (ε, ϕ (x)) z(ε, ϕ (x)) on Ω ε. (3.9)

10 8 Tommaso Benincasa, Costică Moroşanu Letting tend to zero in (3.6)-(3.9) we get the system in variation (3.1)-(3.13) below ρcũ ε t + l 2 ϕε t = kũ ε xx, in, (3.1) τ ϕ ε t = ξ 2 ϕ ε xx + 1 2a ϕε + 2ũ ε, where ũ ε = lim ũ ε x + hũ ε = w, ϕ ε x = on Σ ε, (3.11) ũ ε x =, ϕ ε x = on Σ ε, (3.12) ũ ε (ε, x) =, ϕ ε +(ε, x) = η(ε, x) on Ω ε. (3.13) u,ε u ε, etc., and z (ε, ϕ η(ε, x) = lim (x)) z(ε, ϕ (x)) = = z (ε, ϕ (x)) ϕ ε (ε, x) + z(ε, ϕ (x)) = z(ε, ϕ (x)) with η(, x) the solution of the Cauchy problem η (s, x) + 3 2a z2 (s, x)η(s, x) =, s (, ε), η(, x) = ϕ ε (ε, x), (3.14) that is η(s, x) = exp ( ε 3 2a z(t, )2 dt ) ϕ ε (ε, x). (3.15) We now introduce the adjoint state system. For this, the system (3.1) can be written in the abstract form: (ũε ) (ũε ) ϕ ε = A ϕ ε in where (here ϕ = ϕ xx ) D(A) = A = 1 ρc t ( l ) k τ 2 τ l ( ξ ) 2τρc 2a 1( ξ ) τ 2a { (ψ, γ) H 2 (Ω) H 2 (Ω); ψ ν + hψ L2 ( Ω),, } γ ν =.

11 A product formula approach to an inverse problem... 9 Then A = 1 ρc l 2τρc ( l ) 2 k τ τ ( ξ a ) 1( ξ ) τ 2a, { D(A ) = (ψ, γ) H 2 (Ω) H 2 (Ω); ψ ν + hψ =, γ Thus, the adjoint state system is ( ) ( ) p ε p = A ε q ε t q ε ( + u ε L ε (w) ϕ ε L ε (w) ν = 2ρc l ) } ψ. ν i.e. q ε t lξ2 p ε t + k ρc pε xx l τρc pε + 2 τ qε = β(u ε d 2 ) + χ Q,, p ε x + hp ε =, on Σ ε, p ε (ε, x) =, p ε (T, x) = x Ω T, 2τρc pε xx l 4aτρc pε + ξ2 τ qε xx+ 1 2aτ qε =(ϕ ε 1 d 1 ) χ Q, in, (3.16) q ε x = l 2ρc pε x, on Σ ε, q ε x =, on Σ ε, q ε (ε, x) = exp ( ε 3 2a z2 (t, )dt ) q ε +(ε, x), q ε (T, x) =, x Ω T. Let us introduce the cost functional (3.17) L ε 1 (w) = Lε (w) I U(w) where, as usually, I U (w) is the indicator function of the set U. If w is an optimal controller of problem (P ε ), then L ε 1 (w + w) L ε 1 (w ) that leads to (letting tend to zero) >.

12 1 Tommaso Benincasa, Costică Moroşanu β Q ũ ε (u ε d 2 ) + χ Q dtdx + T + Q w w dt + I U (w, w) w T U (w ). ϕ ε (ϕ ε 1 d 1 ) χ Q dtdx+ (3.18) Multiplying (3.16) 1 by ũ ε and (3.17) 1 by ϕ ε, using integration by parts and Green s formula, we get p ε t ũ ε dt dx + k p ε ũ ε ρc xxdt dx l p ε ũ ε dt dx+ (3.19) τρc + 2 τ + ξ2 τ Σ ε q ε ũ ε dt dx + k ( pεũ ε x p ε ρc xũ ε) dt dγ = β (u ε d 2 ) + χ Q ũ ε dtdx, q ε t ϕ ε dt dx lξ2 2τρc Σ ε p ε ϕ ε l xxdt dx 4aτρc + lξ2 ( p ε 2τρc x ϕ ε p ε ϕ x) ε ξ 2 ( ) dt dγ + qε ϕ ε x ϕ ε q ε x dtdγ+ τ q ε ϕ ε xxdt dx + 1 2aτ Σ ε p ε ϕ ε dt dx+ (3.2) q ε ϕ ε dt dx = (ϕ ε 1 d 1 ) χ Q ϕ ε dtdx. Now we multiply (3.11) by p ε, (3.16) 2 by ũ ε, by subtraction we get p ε xũ ε ũ ε xp ε = p ε w. (3.21) Adding (3.19)-(3.2) and taking into account (3.11) 2, (3.17) 2, (3.21), we obtain p ε t ũ ε dt dx + q ε t ϕ ε dt dx + k p ε wdt dγ + ρc Σ ε + p ε[ k ρcũε xx lξ2 2τρc ϕε xx + l 4aτρc ϕε q ε[ ξ 2 τ ϕε xx + 1 2aτ ϕε + 2 τũε] dt dx = l τρcũε] dt dx +

13 = β A product formula approach to an inverse problem (u ε d 2 ) + χ Q ũ ε dtdx + (ϕ ε 1 d 1 ) χ Q ϕ ε dtdx, i.e., making use of equations in (3.1), the last relation leads to ( p ε t ũ ε + p ε ũ ε t + q ε t ϕ ε + q ε ϕ ε ) k t dt dx + p ε wdt dγ = ρc Σ ε = β (u ε d 2 ) + χ Q ũ ε dtdx + (ϕ ε 1 d 1 ) χ Q ϕ ε dtdx, By Fubini s theorem and definition of distributional derivative, the latter relation give us k p ε wdt dγ = β (u ε d 2 ) + χ ρc Q ũ ε dtdx + (ϕ ε 1 d 1 ) χ Q ϕ ε dtdx, Σ ε and then (3.18) becomes or T k p ε w dtdγ + ρc Σ ε T w w dt + I U (w, w) w T U (w ) [ k ρc pε (s, b 1 ) + w (s) ] w(s) ds + I U (w, w) w T U (w ). The last inequality is equivalent with r(t) I U (w ) a.p.t. (t, x) [, T], where r(t) = k ρc pε (t, b 1 ) + w(t), and thus we can conclude that w (t) = {, if r(t) >, R, i f r(t) <. (3.22) Summing up, we have proved the following maximum principle for problem (P ε ) Theorem 5.1. Let (u,ε, ϕ,ε, w ) be optimal in problem (P ε ). Then the optimal control is given by (3.22) where (p ε, q ε ) satisfy along with u,ε, ϕ,ε the dual system (3.16) (3.17).

14 12 Tommaso Benincasa, Costică Moroşanu Now we will present a numerical algorithm of gradient type in order to compute the approximating optimal control stated by Theorem 5.1. Algorithm InvPHT1D (Inverse PHase Transition case 1D) P. Choose w () U and set iter= ; Choose ε > ; P1. Compute z(ε, ) from (1.9); P2. Compute (u ε,iter, ϕ ε,iter ) from (1.5)-(1.8); P3. Compute (p ε,iter, q ε,iter ) from (3.16)-(3.17); P4. For t [, T], compute r iter (t) = k ρc pε,iter (t, b 1 ) + w iter ; P5. Set w iter (t) = {, if r iter (t) >, R, if r iter (t) <. P6. Compute iter [, 1] solution of the minimization process: min {L ε (witer + (1 ) w iter, [, 1]; Set w iter+1 = iter w iter + (1 iter ) w iter ; P7. If w iter+1 w iter η /* the stopping criterion */ then STOP else iter:= iter+1; Go to P1. In the above, the variable iter represents the number of iterations after which the algorithm InvPHT1D found the optimal value of the cost functional L ε (w) in (Pε ). Remark 5.1. The stopping criterion in P7 could be where η is a prescribed precision. 6. CONCLUSIONS L ε (witer+1 ) L ε (witer ) η, The main novelty brought by this work is that the computation of the approximate solution corresponding to the nonlinear system (1.1) is replaced with calculation of the approximate solution for an ordinary equation and a linear system (compare step P1 in [5] with the steps P1-P2 in present paper). Numerical implementation of the conceptual algorithm InvPHT1D remain an open problem. We wish only to draw attention to the type of boundary condition considered here (see (1.2)) namely that they fully cover industrial application proposed by us for numerical simulations - a matter for further investigation.

15 A product formula approach to an inverse problem References [1] V. Barbu, A product formula approach to nonlinear optimal control problems, SIAM J. Control Optim., 26(1988), [2] T. Benincasa, C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with nonhomogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 3, 3-4(29), [3] G. Caginalp, An analysis of a phase field model of a free boundary, in Arch. Rat. Mech. Anal., 92(1986), [4] 4. M. Heinkenschloss, F. Tröltzsch, Analysis of the Lagrange-SQP- Newton Method for the Control of a Phase Field Equation, Control & Cybernetics, 28, 2(1999), [5] C. Moroşanu, Numerical approach of an inverse problem in the phase field equations, An. Şt. Univ. Al.I. Cuza Iaşi, T XXXIX, s. I-a, f.4(1993), [6] C. Moroşanu, Boundary optimal control problem for the phase-field transition system using fractional steps method, Control & Cybernetics, 32, 1(23), 5-32.

16 14 Tommaso Benincasa, Costică Moroşanu