On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems

We consider the inverse Stefan type free boundary problem, where the coefficients, boundary heat flux, and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue an optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov-Holder spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov-Holder spaces for the numerical solution of the inverse Stefan problem.

1. Inverse Stefan Problem (ISP). The general one-phase Stefan problem is to find the temperature u and boundary s satisfying a(0, t)u x (0, t) = g(t), 0 ≤ t ≤ T a(s(t), t)u x (s(t), t) + γ(s(t), t)s (t) = χ(s(t), t), 0 ≤ t ≤ T (4) where where a, b, c, f, φ, g, γ, χ, µ are given functions. Assume now that a, b, c, f , and g are not known, where a is the coefficient of diffusion, b is the coefficient of advection, c is the coefficient of reaction, f is the density of heat sources, and g is the heat flux at x = 0. In order to find a, b, c, f , and g along with u and s, we must have • The solution of ISP does not depend continuously on the phase transition temperature. A small perturbation of the phase transition temperature may imply significant change of the solution to the ISP. • In the existing formulation, at each step of the iterative method a Stefan problem must be solved which incurs a high computational cost. A new method developed in [1,2] addresses both issues with a new variational formulation. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems to the continuous optimal control problem was proved in [1,2]. In [3], the Frechet differentiability of the variational formulation as well as existence were established under minimal conditions on the data, using precise estimates in Besov spaces. The previous work focused on the identification of the boundary s, the heat flux g, and the density of sources f ; our goal in this work is to extend the Frechet differentiability framework used in [3] to the problem with unknown coefficients.
The structure of the remainder of the paper is as follows: in Section 2 we define all the functional spaces. Section 3 formulates optimal control problem. Section 4 describes the main results: Theorem 4.4 states the Frechet differentiability result and presents the formula for the Frechet differential; in Corollary 1 we present the necessary condition for the optimal control in the form of the variational inequality. Section 5 gives a heuristic derivation of the Frechet gradient which suggests the form of the adjoint problem and the Frechet differential. Section 6 describes important preliminary results. In Section 6.1 we recall the existence, uniqueness and energy estimates in Besov spaces for the Neumann problem to the second order linear parabolic PDEs. In Section 6.2 we formulate optimal trace embedding results for the Besov spaces. In Section 6.3 we give three important technical lemmas. Lemmas 6.4 and 6.5 are on the estimation of the Neumann problem, and adjoint PDE problem in respective Besov space norm. In Lemma 6.6 we prove an estimate on the increment of the state vector with respect to the control vector in a Besov space norm. By applying Lemmas 6.4-6.6 in Section 7 we complete the proof of the main results. Finally, conclusions are presented in Section 8.

2.
Notation. We will use the notation 1 I (x) = 1, x ∈ I 0, x ∈ I for the indicator function of the set I, and [r] for the integer part of the real number r.
We will require the notions of Sobolev-Slobodeckij or Besov spaces [6,25,24,28,33,34]. In this section, assume U is a domain in R and denote by Q T = (0, 1) × (0, T ]. • For ∈ Z + , W p (U ) is the Banach space of measurable functions with finite norm dx dy By [7, §18, thm. 9], it follows that for p = 2 and ∈ Z + , the B p (U ) norm is equivalent to the W p (U ) norm (i.e. the two spaces coincide.) p,x,t (Q T ) is defined as the closure of the set of smooth functions under the norm When p = 2, if either 1 or 2 is an integer, the Besov seminorm may be replaced with the corresponding Sobolev seminorm due to equivalence of the norms and the corresponding Sobolev-Besov space may be denoted by W 1, 2 2 . • The Hölder space C α,α/2 x,t (Q T ) is the set of continuous functions with [α] xderivatives and [α/2] t-derivatives, and for which the highest order x-and t-derivatives satisfy Hölder conditions of order α − [α] and α/2 − [α/2], respectively.
3. Optimal control problem. Fix any α, α * such that 0 < α < α * . Consider the minimization of the functional on the control set where β 0 , β 1 , β 2 ≥ 0 and a 0 , δ, R > 0 are given, where = (R) > 0 is chosen such that for any control v ∈ V R , its component s satisfies s(t) ≤ . This follows from Morrey inequality [25,6]: For a given control vector v ∈ V R , the state vector u(x, t; v) is a solution to the Neumann problem (1)-(4). The formulated optimal control problem (8)-(9) will be called Problem I.
Since the data appearing in the Neumann problem (1)-(4) are in general nonsmooth, the solutions may not exist in the classical sense. The notion of solution must be understood in a weak sense, i.e. for a fixed control vector v ∈ V R , u ∈ W 2,1 2 (Ω) is called a solution of the Neumann problem (1)-(4) if it satisfies the equation (1) and conditions (2)-(4) pointwise almost everywhere.
Remark 1. Note that Problem I imposes additional requirements on ISP. The requirements , and 0 < δ ≤ s(t) are necessary to arrange well-posedness of the Neumann problem (1)-(4) in respective Besov space which is essential for the proof of the Frechet differentiability, derivation of the Frechet gradient and necessary condition for the optimality. The constant bound R provides for weak compactness of the control set V R in respective Hilbert space and is necessary for the existence result. If ISP has a solution in the class of described data, then the constant R is chosen large enough to guarantee that the solution of the ISP is contained in V R . Otherwise, in practical applications Problem I is numerically solved for the increasing sequence of bounds R.

4.
Main results. Let α, α * be fixed as in (9). The main results are established under the assumptions Given a control vector v ∈ V R , under the conditions (11) there exists a unique pointwise a.e. solution u ∈ W 2,1 2 (Ω) of the Neumann problem (1)-(4)( [25,33]). Definition 4.1. Let V be a convex and closed subset of the Banach space H. We say that the functional J : V → R is differentiable in the sense of Frechet at the point v ∈ V if there exists an element J (v) ∈ H of the dual space such that where v + h ∈ V ∩ {u : |u| < γ} for some γ > 0; ·, · H is a pairing between H and its dual H , and o(h, v) |h| → 0, as |h| → 0.
The expression dJ(v) = J (v), · H is called a Frechet differential of J at v ∈ V , and the element J (v) ∈ H is called Frechet derivative or gradient of J at v ∈ V .
Note that if Frechet gradient J (v) exists at v ∈ V , then the Frechet differential dJ(v) is uniquely defined on a convex cone Indeed, let J 1 (v) and J 2 (v) be two Frechet gradients of J at v ∈ V . Choose e ∈ H v , |e| = 1. For some δ > 0 we have From (12) it follows that Dividing by t and passing to limit as t ↓ 0 we have which proves the assertion.
Theorem 4.4. The functional J (v) is differentiable in the sense of Frechet, and the Frechet differential is ∆s(t) dt where J (v) ∈ H is the Frechet derivative, ψ is a solution to the adjoint problem in the sense of definition 4.2, and ∆v = (∆a, ∆b, ∆c, ∆f , ∆g, ∆s) is a variation of the control vector v ∈ V R such that v + ∆v ∈ V R .

Corollary 1 (Optimality Condition).
If v is an optimal control, then the following variational inequality is satisfied: Note that V R is a closed, bounded, and convex subset of H, so the left hand side of the optimality condition (18) is uniquely defined for Frechet gradient J (v) defined in the sense of definition 4.1.

5.
Heuristic derivation of Frechet gradient. To give a first indication of the form of the gradient, we apply the heuristic method of Lagrange-type multipliers; the rigorous proof follows in Section 7. Consider the functional . We will also denote bys(t) = s(t) + θ(t)∆s(t) where 0 ≤ θ(t) ≤ 1 standing for all functions arising from application of mean value theorem in the region between s(t) and s(t). Define In what follows, all terms of higher than linear order with respect to ∆v will be absorbed into the expression R. Partition the time domain as [0 Transforming ∆J as in [3], we derive Each term in I 1 is transformed in a similar way; for example,

UGUR G. ABDULLA, EVAN COSGROVE AND JONATHAN GOLDFARB
Treating the other terms similarly, we derive Transform I 2 using integration by parts to derive Using the boundary condition (4) , it follows that for t ∈ T 1 , Similarly, boundary condition (4) implies that for t ∈ T 2 , So it follows that Combining (21), (23), (25), it follows that Combining (19), (20), (26) it follows that ∆s dt Due to arbitrariness of ∆u, its coefficients in the 1st, 4th, 8th, and 10th terms in (27) must be zero, and hence ψ is a solution of (13)-(16) and the Frechet differential ∆J is as in (17) if R = o(∆v).
and the consistency condition of order k = − 3 2p − 1 2 holds; that is,

EVAN COSGROVE AND JONATHAN GOLDFARB
Then the solution u of (28)-(31) satisfies the energy estimate In particular, energy estimates (32), (33) imply the existence and uniqueness of the solution in respective spaces 6.2. Traces and embeddings of Besov functions. For functions u ∈ W 2,1 2 (Ω), the applicability of the boundary conditions are justified by the following trace and regularity results.

OPTIMAL CONTROL OF PARABOLIC FREE BOUNDARY PROBLEMS 331
Estimate (47) implies that I 1 → 0 as ∆v → 0. By using (46) and the Minkowski's inequality we estimate I 2 as follows: I 21 is easily estimated using Minkowski's inequality as By condition (9), it follows that Using the boundedness of ∆a and membership of ∆a in C 1/2+2α * ,0 x,t , it follows that Estimation of I 23 is completed as in [3, eq. 45]. Therefore, from (48), (49) it follows that I 2 → 0 as ∆v → 0. Consider the last term I 3 ; by (46) and Minkowski's inequality, The estimation of I 31 and I 33 coincide with the estimation of the same term in the proof of Lemma 6.6 in [3]; I 32 will be estimated in a similar way as well. Write where Estimate in I 1 32 using condition (9), mean value theorem, and Morrey's inequality, it follows that Similarly, Estimate in I 3 32 using condition (9) to derive Applying mean value theorem and Morrey's inequality, it follows that The last term, I 4 34 requires more careful treatment. Estimating as for the other terms, we derive For arbitrary u ∈ B 5/2+2α,5/4+α 2,x,t (Q T ), the highest order space derivative does not have the required time regularity. However, the fact that u is a pointwise a.e. solution of (34) does imply the required regularity through the transformation of this term. Indeed, it follows that Each term is now handled in a relatively easy way using elementary estimates and Sobolev embedding as in [3, lem. 6.6, pp. 19-20]. Having (50)-(53), it follows that I 3 → 0 as ∆v → 0, and hence Γ 2 → 0 as ∆v → 0. We would next like to show dx dτ dt 1 2 Estimate I 1 as where I 11 = and I 12 is bounded by By (54)-(55), it follows that I 2 → 0 as ∆v → 0. By Minkowski's inequality, Estimate I 2,1 using boundedness of ∆c as I 2,2 is estimated similarly, using boundedness ofũ: Estimate N 1 as We show N 2 goes to zero by applying Lebesgue's Dominated Convergence Theorem.
Since the integrand converges pointwise to zero, we must show it is bounded by an integrable function. Estimate the integrand as follows: which is integrable since 1 − 4(α * − α) < 1. Therefore, Lebesgue's Dominated Convergence theorem implies that N 2 → 0 as ∆v → 0. Having (56)-(58), it follows that I 2 → 0 as ∆v → 0. Considering I 3 , applying Minkowski's inequality again we derive where I 3,1 is estimated in a similar way to I 2,1 : Lastly, I 3,2 is estimated through the boundedness ofũ: By condition (9), it follows that Considering N 4 , by letting u = xs(t), derive We show N 5 goes to zero by applying Lebesgue's Dominated Convergence Theorem.

Proofs of the main results.
Proof of Theorem 4.3. Let {v n = (a n , b n , c n , f n , g n , s n )} ⊆ V R be a minimizing sequence, i.e. J (v n ) → J * Since the components (a n , b n , c n ) are members of a bounded subset of from Arzela-Ascoli theorem [24, thm. 1.5.10], it follows that there is a subsequence (a n k , b n k , c n k ) which converges to an element (a * , b * , c * ) in Assume the whole sequence (a n , b n , c n ) converges inH. Similarly, (f n , g n , s n ) are members of a bounded and closed subset of an Hilbert space B , the sequence is weakly precompact (see e.g. [37, ch. V, § 2, p.126]); that is, there exists a subsequence (f n k , g n k , s n k ) which converges weakly. Assume the whole sequence converges weakly to (f * , g * , s * ). Note that in fact, from the condition (9) it follows that (a * , b * , c * ) ∈ C . The corresponding solutions u n ∈ W 2,1 2 (Ω) and transformed solutions u n ∈ W 2,1 2 (Q T ) are uniformly bounded by (38); that is, there exists C > 0 such that u n W 2,1 2 (Q T ) ≤ C It follows that there exists an elementṽ ∈ W 2,1 2 (Q T ) such that for some subsequence n k , u n k →ṽ weakly in W 2,1 2 (Q T ) Multiply (1) written for u n k by an arbitrary test function Φ ∈ L 2 (Q T ) to derive 0 = Q T L n kũ n k −f n k Φ dx dt = I 1 + I 2 whereL n := 1 s 2 n ã n u y y + 1 s n b n + ys n u y +c n u − u t , Uniform convergence of (a n k , b n k , c n k ) and boundedness ofũ n k and Φ in L 2 (Q T ) imply that I 1 → 0 as n k → ∞; weak convergence ofũ n k toṽ implies thatṽ is a W 2,1 2 (Q T ) weak solution of (34)- (37). Since all weak limit points of { u n } are W 2,1 2 (Q T ) weak solutions of the same equation, by uniqueness of the weak solution, it follows thatṽ = u * and the whole sequence u n → u * weakly in W 2,1 2 (Q T ). Sobolev trace theorem [6] then implies that Together with the convergence of s n (t) → s * (t) uniformly on [0, T ], it follows that The other two terms in J (v) are handled similarly, so it follows that Proof of Theorem 4.4. Consider the increment We have On applying mean value theorem to u(·, t) we havē wheres(t) = s(t) + θ(t)∆s(t), 0 ≤ θ(t) ≤ 1. It follows that , t)∆s(t) + ∆u(s(t), t)| 2 dt, x=s(t) ∆s dt, and x=s(t) ∆s dt, Hence from (65), (66), (67), (68), it follows that Each term in I 1 is transformed in a similar way; for example,

EVAN COSGROVE AND JONATHAN GOLDFARB
Treating the other terms similarly, we derive ∆cuψ dx dt, ∆f ψ dx dt Transform I 2 using integration by parts to derive ∆u(s(t), t) dt, From (22), it follows that Where the main linear part of the increment is as in (17). It remains to show that 37 i=1 R i = o(∆v). All of the remainder terms R 1 -R 10 , and R 17 , R 19 -R 28 are common with the remainder terms derived in [3]; in particular, R 24 of this paper is shown to satisfy R 24 = o(∆v). Therefore, we demonstrate that fact here for the new terms R 11 and R 12 , as the other new terms are estimated similarly. By condition (9) and CBS inequality, x,t (D) Ω [|∆u x | + |∆u xx |] |ψ| dx dt ≤ ∆a C 1,0 x,t (D) ∆u W 2,0 2 ( Ω) ψ L2(Ω) Uniform boundedness of ψ in L 2 (Ω) follows from Lemma 6.5, hence it follows that R 11 → 0 from (40). Similarly, By CBS and Morrey's inequalities, it follows that By Sobolev embedding theorem and Lemma 6.5, it follows that R 12 = o(∆v). All of the other terms are proven in a similar way.

8.
Conclusions. The new variational formulation of the inverse Stefan problem introduced in [1,2], and the Frechet differentiability result established within a Besov spaces framework in [3], are extended to the ISP with unknown coefficients. Existence of an optimal control and Frechet differentiability is proved under minimal regularity assumptions on the data. The result implies a necessary condition for the optimality in the form of variational inequality, and opens a way for the implementation of an effective numerical method for identification of the unknown coefficients based on the projective gradient method in Besov-Hölder space framework. The main idea of the new variational formulation is optimal control setting, where the free boundary is the component of the control vector. This allows for the development of an iterative gradient type numerical method of low computational cost. It also creates a framework for the regularization of the error existing in the information on the phase transition temperature.