On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I.Well-posedness and Convergence of the Method of Lines

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.


Introduction and Motivation
Consider the general one-phase Stefan problem ( [14,25]): find the temperature function u(x, t) and the free boundary x = s(t) from the following conditions a(s(t), t)u x (s(t), t) + γ(s(t), t)s ′ (t) = χ(s(t), t), where a, b, c, f , φ, g, γ, χ, µ are known functions and a(x, t) ≥ a 0 > 0, s 0 > 0 (1. 6) In the physical context, f characterizes the density of the sources, φ is the initial temperature, g is the heat flux on the fixed boundary and µ is the phase transition temperature. Assume now that some of the data is not available, or involves some measurement error. For example, assume that the heat flux g(t) on the fixed boundary x = 0 is not known and must be found along with the temperature u(x, t) and the free boundary s(t). In order to do that, some additional information is needed. Assume that this additional information is given in the form of the temperature measurement along the boundary x = 0: u(0, t) = ν(t), for 0 ≤ t ≤ T (1.7) Inverse Stefan Problem (ISP): Find the functions u(x, t) and s(t) and the boundary heat flux g(t) satisfying conditions (1.1)-(1.7). ISP is not well posed in the sense of Hadamard. If there is no coordination between the input data, the exact solution may not exist. Even if it exists, it might be not unique, and most importantly, there is no continuous dependence of the solution on the data. Inverse Stefan problem was first mentioned in [9], in the form of finding a heat flux on the fixed boundary which provides a desired free boundary. This problem is similar to noncharacteristic Cauchy problem for the heat equation. The variational approach for solving this ill-posed inverse Stefan problem was performed in [6,7]. First result on the optimal control of the Stefan problem appeared in [35]. It consists of finding optimal value of the external temperature along the fixed boundary, in order to ensure that the solutions of the Stefan problem are close to the measurements taken at the final moment. In [35] existence result was proved. In [37] the Frechet differentiability and the convergence of the difference schemes was proved for the same problem and Tikhonov regularization was suggested. Later development of the inverse Stefan problem was along these two lines: Inverse Stefan problems with given phase boundaries were considered in [1,3,5,8,10,11,12,17,31,15]; optimal control of Stefan problems, or equivalently inverse problems with unknown phase boundaries were investigated in [2,13,18,19,20,21,22,24,28,26,29,30,34,15]. We refer to monography [15] for a complete list of references of both types of inverse Stefan problems, both for linear and quasilinear parabolic equations. The main methods used to solve inverse Stefan problem are based on variational formulation, method of quasi-solutions or Tikhonov regularization which takes into account ill-posedness in terms of the dependence of the solution on the inaccuracy involved in the measurement (1.7), Frechet differentiability and iterative conjugate gradient methods for numerical solution. Despite its effectiveness, this approach has some deficiencies in many practical applications: • Solution of the inverse Stefan problem is not continuously dependent on the phase transition temperature µ(t): small perturbation of the phase transition temperature may imply significant change of the solution to the inverse Stefan problem. Accordingly, any regularization which equally takes into account instability with respect to both ν(t) from measurement (1.7), and the phase transition temperature µ(t) from (1.5) will be preferred. It should be also mentioned that in many applications the phase transition temperature is not known explicitly. In many processes the melting temperature of pure material at a given external action depends on the process evolution. For example, gallium (Ga, atomic number 31) may remain in the liquid phase at temperatures well below its mean melting temperature ( [25]).
• Numerical implementation of the iterative gradient type methods within the existing approach requires to solve full free boundary problem at every step of the iteration, and accordingly requires quite high computational cost. Iterative gradient method which requires at every step solution of the boundary value problem in a fixed region would definitely be much more effective in terms of the computational cost.
The main goal of this project is to develop a new variational approach based on the optimal control theory which is capable of addressing both of the mentioned issues and allows the inverse Stefan problem to be solved numerically with least computational cost by using conjugate gradient methods in Hilbert spaces. In this paper we prove the existence of the optimal control and convergence of the family of time-discretized optimal control problems to the continuous problem both with respect to cost functional and control. We employ Sobolev spaces framework which allows to reduce the reguarity and structural requirements on the data. We address the problems of convergence of the fully discretized family of optimal control problems, Frechet differentiability and iterative conjugate gradient methods in Hilbert spaces in an upcoming paper.
In the next section we formulate a new variational formulation of the inverse problem which takes into account the described deficiencies.

Optimal Control Problem
Consider a minimization of the cost functional on the control set δ, l, R, β 0 , β 1 are given positive numbers, and u = u(x, t; v) be a solution of the Neumann problem (1.1)-(1.4).

Discrete Optimal Control Problem
Let ω τ = {t j = j · τ, j = 0, 1, . . . , n} be a grid on [0, T ] and τ = T n . Consider a discretized control set under the standard notation for the finite differences: Introduce two mappings Q n and P n between continuous and discrete control sets: where s k = s(t k ), g k = g(t k ), k = 0, 1, ..., n.
Introduce Steklov averages where d stands for any of the functions a, b, c, f , and h stands for any of the functions ν, µ.
Given v = (s, g) ∈ V R we define Steklov averages of traces ∈ V n R we define Steklov averages χ k s n and (γ s n (s n ) ′ ) k through (1.12) with s replaced by s n from (1.11).
Next we define a discrete state vector through time-discretization of the integral identity (1.9) is called a discrete state vector if (c) For arbitrary k = 0, 1, ..., n, u(x; k) ∈ W 1 2 [0, s k ] iteratively continued to [0, l] as u(x; k) = u(2 n s k − x; k), 2 n−1 s k ≤ x ≤ 2 n s k , n = 1, n k , n k ≤ N = 1 + log 2 l δ (1.14) where [r] means integer part of the real number r.
Consider a discrete optimal control problem of minimization of the cost functional on a set V n R subject to the state vector defined in Definition 1.3. Furthermore, formulated discrete optimal control problem will be called Problem I n .

Formulation of the Main Result
Let Throughout the whole paper, with the exeption of Section 3.1, we assume the following conditions are satisfied by the data: Sequence of discrete optimal control problems I n approximates the optimal control problem I with respect to functional, i.e. where

Preliminary Results
In a Lemma 2.1 below we prove existence and uniqueness of the discrete state vector [u([v] n )] n (see Definition 1.3) for arbitrary discrete control vector [v] n ∈ V n R . In a Lemma 2.2 we remind a general approximation criteria for the optimal control problems from ( [36]). In a Lemma 2.3 we prove some properties of the mappings Q n and P n between continuous and discrete control sets.
To prove an existence we apply Galerkin method. Consider an approximate solution where {ψ i } is a fundamental system in W 1 2 [0, s k ] and the coefficients {d i } solve the following system Homogeneous system corresponding to (2.5 Let us multiply each equation in (2.6) by d i and add with respect to i: As before, from (2.7) it follows that u N ≡ 0, and therefore the homogeneous system (2.6) has only the trivial solution. This proves the uniqueness of the approximate solution u N (x). Let us now prove uniform estimation of the sequence {u N (x)}. Multiply (2.4) by d i and add with respect to i = 1, . . . , N: We estimate the four integrals on the left-hand side of (2.8) as we did before to prove (2.3) and derive for all τ ≤ τ 0 2 . By Morrey's inequality we have where the constant C is independent of N and τ . By using Cauchy inequalities with appropriately chosen ǫ > 0, from (2.9) and (2.10) it easily follows that is a solution of (1.13) and in view of uniqueness the whole sequence u N converges weakly in W 1 2 [0, s k ] to u(x; k). Lemma is proved. The following known criteria will be used in the proof of Theorem 1.2.
Lemma 2.2 [36] Sequence of discrete optimal control problems I n approximates the continuous optimal control problem I if and only if the following conditions are satisfied: (1) for arbitrary sufficiently small ǫ > 0 there exists number and for any fixed ǫ > 0 and for all v ∈ V R−ǫ the following inequality is satisfied: (2.12) (2) for arbitrary sufficiently small ǫ > 0 there exists number (2.13) (3) the following inequalities are satisfied: In the next lemma we prove that the mappings Q n and P n introduced in Section 1.3 satisfy the conditions of Lemma 2.2.

Lemma 2.3
For arbitrary sufficiently small ǫ > 0 there exists n ǫ such that . By applying Cauchy-Bunyakovski-Schwarz (CBS) inequality and Fubini's theorem we have R . We simplify the notation and assume v = (s, g) = P n ([v] n ). Through direct calculations we derive where C is independent of τ . By using CBS inequality we have In a similar way we calculate From (2.24) it follows that given ǫ > 0 we can choose n ǫ such that for any n > n ǫ where C is independent of n.
and hence for the first component from (2.29), (2.28) easily follows.

First Energy Estimate and its Consequences
Throughout this section we assume that , a satisfies (1.6) and b, c, f satisfy the conditions imposed in Section 1.4. The main goal of this section to prove the following energy estimation for the discrete state vector.

2)
where C is independent of τ and 1 + be an indicator function of the positive semiaxis.
We split the proof into two Lemmas.
where C is independent of τ .
Proof. By choosing η(x) = 2τ u(x; k) in (1.13) and by using the equality Using (1.6), Cauchy inequalities with appropriately chosen ǫ > 0, and Morrey inequality (2.10) from (3.4) we derive that where C 1 is independent of τ . Assuming thatτ < C 1 , from (3.5) it follows that By induction we have For arbitrary 1 ≤ j ≤ k ≤ n we have as τ → 0. Accordingly for sufficiently small τ we have By applying CBS inequality from (3.7)-(3.9) it follows that where C 2 is independent of τ . Having (3.10), we perform summation of (3.5) with respect to k from 1 to n and derive

12)
where C is independent of τ .
Proof. By induction it follows that the first two terms on the left hand side are estimated by the first two terms on the right hand side with the constant C = 2 N , where N is defined in (1.14).
We have where (3.14) Hence, from (3.13)-(3.17) it follows that where C is independent of τ . From ( Proof. In addition to quadratic interpolation of [s] n from (1.11), consider two linear interpolations: s n 1 (t) =s n (t + τ ), 0 ≤ t ≤ T. It can be easily proved that both sequencess n ands n 1 are equivalent to the sequence s n in W 1 2 [0, T ] and converge to s strongly in W 1 2 [0, T ]. In particular, sup n s n where C * is independent of n. We estimate the last term on the right-hand side of (3.1) as follows: By applying CBS inequality we have From the results on traces of the elements of space V 2 (D) ( [23,4,27]) it follows that for arbitrary u ∈ V 2 (D) the following inequality is valid where C 3 is independent of γ, χ and n. Hence, from (3.26) and (3.27) it follows the estimation with C being independent of n. If (3.25) is not satisfied, then we can partition [0, T ] into finitely many segments [t n j−1 , t n j ], j = 1, q with t n 0 = 0, t nq = T in such a way that by replacing [0, T ] with any of the subsegments [t n j−1 , t n j ] (3.20) will be satisfied with C * small enough to obey (3.25). Hence, we divide D into finitely many subsets such that every norm u τ 2 V 2 (D j ) is uniformly bounded through the right-hand side of (3.28). Summation with j = 1, . . . , q implies (3.28).
From (3.28) it follows that the sequence {u τ } is weakly precompact in W 1,0 2 (D). Let u ∈ W 1,0 2 (D) be a weak limit point of u τ in W 1,0 2 (D), and assume that whole sequence {u τ } converges to u weakly in W 1,0 2 (D). Let us prove that in fact u satisfies the integral identity (1.10) for arbitrary test function Φ ∈ W 1,1 2 (Ω) such that Φ| t=T = 0. Due to density of C 1 (Ω) in W 1,1 2 (Ω) it is enough to assume Φ ∈ C 1 (Ω). Without loss of generality we can also assume that Φ ∈ C 1 (D T +τ ), Φ ≡ 0, for T ≤ t ≤ T + τ , where Otherwise, we can continue Φ to D T +τ with the described properties. Let τ As before, we construct piecewise constant interpolations Φ τ , Φ τ t . Obviously, the sequences {Φ τ }, { ∂Φ τ ∂x } and {Φ τ t } converge as τ → 0 uniformly in D to Φ, ∂Φ ∂x and ∂Φ ∂t respectively. By choosing in (1.13) η(x) = τ Φ(x; k), after summation with respect to k = 1, n and transformation of the time difference term as follows and all of the integrands are uniformly bounded in L 1 (D), it follows that the first term in the expression of R converges to zero as τ → 0. In a similar way one can see that the second and third terms also converge to zero as τ → 0. The last term in the expression of R converges to zero due to Corollary 2.1 and uniform convergence of {Φ τ } in D. Hence, we have Due to weak convergence of u τ to u in W 1,0 2 (D) and uniform convergence of the sequences {Φ τ }, { ∂Φ τ ∂x } and {Φ τ t } to Φ, ∂Φ ∂x and ∂Φ ∂t respectively, passing to limit as τ → 0, it follows that first, second and fourth integrals on the left-hand side of (3.30) converge to similar integrals with u τ , Φ τ , Φ τ t , Φ τ (x, τ ) and Φ τ (0, t) replaced by u,Φ, ∂Φ ∂t , Φ(x, 0) and Φ(0, t) respectively. Since s n converges to s strongly in W 1 2 [0, T ], the traces γ(s n (t), (t)), χ(s n (t), t) converge strongly in L 2 [0, T ] to traces γ(s(t), (t)), χ(s(t), t) respectively. Since Φ τ (s n (t), t) converge uniformly on [0, T ] to Φ(s(t), t), passing to the limit as τ → 0, the last integral on the left-hand side of (3.30) converge to similar integral with s n and Φ τ replaced by s and Φ.

Second Energy Estimate and its Consequences
The main goal of this section to prove the following energy estimation for the discrete state vector.

Theorem 3.3
For all sufficiently small τ discrete state vector [u ([v] n )] n satisfies the following stability estimation: We split the proof into two lemmas. is defined asũ Then for all sufficiently small τ , [ũ([v] n )] n satisfies the following estimation:

36)
Proof: By choosing η(x) = 2τũ t (x; k) in (1.13) and by using the following identity By adding inequalities (3.38) with respect to k from 1 to arbitrary m ≤ n we derive where C is independent of n. Note that we replacedũ with u in first two integrals on the right-hand side of (3.40). Since γ, χ ∈ W 1,1 2 (D) we have γ(s n (t), t), χ(s n (t), t) ∈ W 1 4 2 [0, T ] ( [27,4,23]) and γ(s n (t), t) where C is independent of n. According to Lemma 2.3 P n ([v] n ) ∈ V R+1 . By applying Morrey inequality to (s n ) ′ we easily deduce that γ(s n (t), t)(s n ) ′ (t) ∈ W where C is independent of n. Let w(x, t) be a function in W 2,1 2 (D) such that and The existence of w follows from the result on traces of Sobolev functions [4,27]. For example, w can be constructed as a solution from W 2,1 2 (Ω n ) of the heat equation in under initial-boundary conditions (3.43),(3.44)with subsequent continuation to W 2,1 2 (D) with norm preservation [32,33].
Hence, by replacing in the original problem (1.1)-(1.4) u with u−w we can derive modified (3.40) without the last three terms on the right-hand side and with f , replaced by

47)
Proof: Obviously, we can equivalently replaceũ with u in the first term on the left-hand side of (3.36). We can do so also in the second term provided s k−1 ≥ s k for all k = 1, m. Hence, we only need to estimate By using (2.28) we have Hence, for sufficiently small τ we have  Hence, {û τ } is weakly precompact in W 1,1 2 (D). It follows that it is strongly precompact in L 2 (D). Let u be a weak limit point of {û τ } in W 1,1 2 (D), and therefore a strong limit point in L 2 (D). From (3.35) it follows that Therefore, u is a strong limit point of the sequence {u τ } in L 2 (D). By Theorem 3. (Ω n ) and W 1,1 2 (Ω) respectively. By Corollary 3.2, both satisfy energy estimation (3.50) with g n and g on the right-hand side respectively. Since v n ∈ V R , u n W 1,1 2 (D) is uniformly bounded. Hence, the sequence ∆u = u n − u it satisfies uniformly with respect to n. Accordingly, {∆u} is weakly precompact in W 1,1 2 (D). Without loss of generality assume that the whole sequence u n −u converges weakly in W 1,1 2 (D) to some function v ∈ W 1,1 2 (D). Let us subtract integral identities (1.9) for u n and u, by assuming that the fixed test function Φ belongs to W 1,1 2 (D). Indeed, otherwise Φ can be continued to D as an element of W 1,1 2 (D).
By using energy estimate (3.50), and continuity of traces γ(s(t), t), χ(s(t), t) of elements γ, χ ∈ W 1,1 2 (D), strongly in L 2 [0, T ] with respect to s ∈ W 1 2 [0, T ], passing to the limit as n → +∞, from (3.53) it follows that the weak limit function v satisfies the integral identity for arbitrary Φ ∈ W 1,1 2 (D). Since, any element Φ ∈ W 1,1 2 (Ω) can be continued to D as element of W 1,1 2 (D), (3.54) is valid for arbitrary Φ ∈ W 1,1 2 (Ω). Hence, v is a weak solution from W 1,1 2 (Ω) of the problem (1.1)-(1.4) with f = g = γ = χ = 0. From (3.50) and uniqueness it follows that v = 0. Thus u n converges to u weakly in W 1,1 2 (D). From Sobolev trace theorem ( [4,27]) it follows that and v is a solution of the Problem I. Theorem is proved. Remark: By applying first and second energy estimates we proved that functional J (v) is weakly continuous in W 2 Since V R is weakly compact existence of the optimal control follows from Weierstrass theorem in weak topology. u(0, t) and u(s(t), t) respectively. By Sobolev embedding theorem ( [4,27]) it is enough to prove that the sequences {u τ } and {û τ } are equivalent in strong topology of W 1,0 2 (Ω). In Theorem 3.4 it is proved that they are equivalent in strong topology of L 2 (D). It remains only to demonstrate that the sequences of derivatives {u τ x } and {û τ x } are equivalent in strong topology of L 2 (Ω). We have where s k = s n (t k ), s n is the first component of P n ([v] n ) and Γ n = ∪ n k=1 {t k−1 < t ≤ t k , min(s k−1 ; s k ) < x < s(t)} Since s n converges to s uniformly on [0, T ], it follows that the Lebesgue measure of Γ n converges to zero as n → +∞. By Theorems 3.2 and 3.4 the integrand is uniformly bounded in L 2 (D). Therefore, the second term on the right-hand side of (3.59) converges to zero as n → +∞. First term on the right-hand side of (3.59) converges to zero due to stability estimation (3.36) and the claim is proved. Let We estimate the first term in I n (Q n (v)) as follows We estimate the second term in I n (Q n (v)) as follows We have are corresponding discrete state vectors according to Definition 1.3. Let s k = s n (t k ),s k = s(t k ) and ∆u(x; k) = u n (x; k) −ũ(x; k) We have By the Morrey inequality we have where C is independent of n. Let us subtract integral identities (1.13) for u n (x; k) and u(x; k), by assuming that the fixed test function η belongs to W 1 2 [0, l]. Indeed, otherwise η can be continued to [0, l] as a element of W 1 2 [0, l]: dũ dx η − c k (x)ũη + f k (x)η +ũtη dx+ + (γ s n (s n ) ′ ) k − (γss ′ ) k η(s k ) + (γss ′ ) k [η(s k ) − η(s k )] − χ k s n − χ k s η(s k ) = 0 (3.75) Our goal now is to derive from (3.75) that the right-hand side of (3.74) converges to zero as n → +∞. The proof goes along the same lines as the derivation of the first energy estimate in Lemma 3.1. By choosing η(x) = 2τ ∆u(x; k) in (3.75), and by using (1.6), Cauchy inequalities with appropriately chosen ǫ > 0, and Morrey inequality (2.10) we derive similar to (3.5): where C is independent of τ and +ũt∆u(x; j) dx + where C 1 is independent of τ . Our next goal is to absorb the first term on the right-hand side of (3.79) into the left-hand side. We apply the same method used in the proof of Theorem 3.2 (see (3.20)-(3.26)). The only difference is that in the estimations (3.23) and (3.24) we replace D with Ω 1 n = {0 < t < T, 0 < x <s n 1 (t)} Let us also introduce the region Ω n = n k=1 {t k−1 < t ≤ t k , 0 < x < s k } Note that ∆u τ 2 where C 2 is independent of τ . Since the sequences {s n 1 } and {s n } are equivalent in strong topology of W 1 2 [0, T ], the first term on the right-hand side of (3.80) converges to zero as n → +∞. It only remains to prove that lim n→+∞ n j=1 |L j | = 0. {(x, t) : t j−1 < t < t j , min(s j ,s j ) < x < max(s j ,s j )} From (3.68) it follows that the Lebesgue measure of∆ converges to zero as n → ∞. Since by the first energy estimate W 1,0 2 (D) norm ofũ τ and ∆u τ are uniformly bounded, the righthand side of (3.82) converges to zero as n → ∞. For the same reason, the next three terms in the expression of n j=1 |L j | also converge to zero as n → ∞. We have By the second energy estimate W 1,1 2 (D) norm ofû τ is uniformly bounded. Accordingly, the right-hand side of (3.83) converges to zero as n → +∞. We have n j=1 t j t j−1 γ(s n (t), t)(s n ) ′ (t) − γ(s(t), t)s ′ (t) ∆u(s j , j) dt ≤  and all three terms on the right-hand side converge to zero as n → +∞. Similarly one can prove that all the last three terms in the expression of n j=1 |L j | converges to zero as n → ∞. Hence, (3.81) is proved. From (3.80) and (3.81), (3.73) follows. Lemma is proved.