Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

Abstract

We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.

1. Introduction

Many methods to solve inverse problems (e.g., the Landweber iteration, conjugate gradient method) use the Fréchet derivatives of the cost functionals of these problems [1]. The explicit formula for the Fréchet derivative in terms of the variation of the unknowns of the inverse problem contains the solution of an adjoint problem.

The derivation of the explicit formula for such a Fréchet derivative includes testing the direct problem with the solution of the adjoint problem and vice versa: testing the adjoint problem with the solution of the direct problem. In the case of the parabolic weak problem, such a procedure is cumbersome, because of the asymmetry of the properties of the solution and the test function. In the classical formulation of the parabolic weak problem (see, e.g., [2] and also (19) below), the test function must have higher time regularity than the weak solution. This means that in case of nonsmooth coefficients neither the solution of the direct problem nor the solution of the adjoint problem can be used as a test function. Another formulation of the weak parabolic problem consists in reducing the problem to an abstract Cauchy problem over the time variable (see, e.g., [3]). In such a case, a partial integration over the time has to be implemented within singular distributions in the derivation procedure.

In this paper, we present a new method that enables the deduction of the formulas for the Fréchet derivatives for cost functionals of inverse problems related to weak solutions of parabolic problems. The method is based on an integrated convolutional form of the weak direct problem. The requirements to the test function are weaker than in the classical case and coincide with the properties of the solution of the direct problem. All computations in the deduction procedure can be implemented within usual regular Sobolev spaces.

More precisely, we will consider inverse problems related to a parabolic integrodifferential equation that occur in heat flow with memory [4–6]. This equation contains a time convolution. Therefore, the convolutional form of the weak problem is especially suitable. Supposedly, the proposed method can be generalised to parabolic systems, as well.

2. Formal Direct Problem: Notation

Let Ω be an 𝑛-dimensional domain, where 𝑛≥1, and Γ be the boundary of Ω. Let Γ=Γ₁∪Γ₂ where either Γ₁ or Γ₂ is allowed to be an empty set. In case 𝑛≥2, we assume that Γ is sufficiently smooth, meas Γ₁∩Γ₂=0, and for any 𝑗∈{1;2} it holds either Γ_𝑗=∅ or meas Γ_𝑗>0. Denote Ω_𝑡=Ω×(0,𝑡),Γ_1,𝑡=Γ₁×(0,𝑡),Γ_2,𝑡=Γ₂×(0,𝑡), (1) for 𝑡≥0. Consider the problem (direct problem) to find 𝑢(𝑥,𝑡)∶Ω_𝑇→ℝ such that 𝑢_𝑡=𝐴𝑢−𝑚∗𝐴𝑢+𝑓+∇⋅𝜙inΩ_𝑇,𝑢=𝑢₀inΩ×{0},𝑢=𝑔inΓ_1,𝑇,−𝜈_𝐴⋅∇𝑢+𝑚∗𝜈_𝐴⋅∇𝑢=𝜗𝑢+ℎ+𝜈⋅𝜙inΓ_2,𝑇, (2)(3)(4)(5) where 𝑇>0 is a fixed number, 𝐴𝑣=^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝑣_{𝑥𝑗𝑥𝑖}𝜈+𝑎𝑣,_𝐴=^𝑛_𝑗=1𝑎_𝑖𝑗𝜈_𝑗∣_{𝑖=1,…,𝑛},𝜈=𝜈₁,…,𝜈_𝑛−outernormalofΓ₂, (6)𝑎_𝑖𝑗,𝑎,𝑢₀∶Ω→ℝ, 𝑓∶Ω_𝑇→ℝ, 𝜙∶Ω_𝑇→ℝ^𝑛, 𝑔∶Γ_1,𝑇→ℝ, 𝜗∶Γ₂→ℝ, ℎ∶Γ_2,𝑇→ℝ, 𝑚∶(0,𝑇)→ℝ are given functions, the subscripts 𝑡,  𝑥_𝑗,  𝑥_𝑖 denote the partial derivatives and 𝑚∗𝑤(𝑡)=^𝑡₀𝑚(𝑡−𝜏)𝑤(𝜏)𝑑𝜏 (7) denotes the time convolution. In case Γ₁=∅ (Γ₂=∅), the boundary condition (4) and (5) is dropped.

The problem (2)–(5) describes the heat flow in the body Ω with the thermal memory. Concerning the physical background, we refer the reader to [4, 6, 7]. The solution 𝑢 is the temperature of the body and 𝑚 is the heat flux relaxation (or memory) kernel. The boundary condition (5) is of the third kind where the term −𝜈_𝐴⋅∇𝑢+𝑚∗𝜈_𝐴⋅∇𝑢 equals the heat flux in the direction of the conormal vector.

Let us introduce some additional notations. Let 𝑡>0. We use the Sobolev spaces 𝑊^𝑙₂(Ω)=𝑣∶Ω→ℝ∶‖𝑣‖_𝑊^𝑙_2(Ω)∶=_{|𝛼|≤𝑙}𝐷^𝛼_𝑥𝑣²_𝐿2(Ω)^1/2<∞. (8) Here, 𝑙=0,1,2,…, 𝛼=(𝛼₁,…,𝛼_𝑛) is the multiindex, |𝛼|=𝛼₁+⋯+𝛼_𝑛 and 𝐷^𝛼_𝑥𝑣=𝜕^|𝛼|𝑣/𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑛_𝑛. Further, let 𝑋 be a Banach space. We denote by 𝐶([0,𝑡];𝑋) the space of abstract continuous functions from [0,𝑡] to 𝑋 endowed with the usual maximum norm ‖𝑣‖_{𝐶([0,𝑡];𝑋)}∶=max_{𝑠∈[0,𝑡]}‖𝑣(𝑠)‖. Moreover, let 𝐿^𝑝((0,𝑡);𝑋)∶=𝑣∶(0,𝑡)→𝑋∶‖𝑣‖_𝐿^𝑝_{((0,𝑡);𝑋)}≥^𝑡₀‖𝑣(𝑠)‖^𝑝𝑑𝑠^1/𝑝𝐿<∞for1<𝑝<∞,_∞((0,𝑡);𝑋)∶=𝑣∶(0,𝑡)→𝑋∶‖𝑣‖_𝐿^∞_{((0,𝑡);𝑋)}=esssup_{()𝑡∈0,𝑇}‖𝑣(𝑠)‖<∞. (9) By means of these spaces, we define the following important functional spaces: 𝒰Ω_𝑡=𝐶[0,𝑡];𝐿²(Ω)∩𝐿²(0,𝑡);𝑊¹₂(Ω,𝒰₀)Ω_𝑡=𝜂∈𝒰Ω_𝑡∶𝜂|_Γ1,𝑡=0incaseΓ₁≠∅. (10)

Convention. In case 𝑛=1, the integrals ∫_Γ𝑗𝑣(𝑥)𝑑Γ, 𝑗=1,2 are equal to ∑^𝐾_𝑘=1𝑣(𝑥_𝑘), where 𝑥_𝑘∈Γ_𝑗 and 𝐾 is the number of points in Γ_𝑗, and 𝐿_𝑝(Γ_𝑗) is simply ℝ^𝐾.

3. Weak Direct Problem and Its Convolutional Form

Let us return to the direct problem (2)–(5). Throughout the paper we assume the following basic regularity conditions on the coefficients, the kernel, and the initial and boundary functions: 𝑎_𝑖𝑗∈𝐿^∞(Ω),𝑎_𝑖𝑗=𝑎_𝑗𝑖,𝜗∈𝐶Γ₂,𝜗≥0,𝑎∈𝐿^𝑞1(Ω),where𝑞₁=1if𝑛=1,𝑞₁>𝑛2if𝑛≥2𝑚∈𝐿¹(0,𝑇),𝑔∈𝐿²(0,𝑇);𝑊¹₂(Ω),𝑔_𝑡∈𝐿²Ω_𝑇,𝑓∈𝐿²(0,𝑇);𝐿^𝑞2(Ω),where𝑞₂𝑞=1if𝑛=1,₂∈1,𝑞₁if𝑛=2,𝑞₂=2𝑛𝜙𝑛+2if𝑛≥3,𝜙=₁,…,𝜙_𝑛∈𝐿²Ω_𝑇^𝑛𝑢₀∈𝐿²(Ω),ℎ∈𝐿²Γ_2,𝑇 (11)(12)(13)(14)(15)(16) and the ellipticity condition ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜆_𝑖𝜆_𝑖≥𝜖|𝜆|²,𝑥∈Ω,𝜆∈ℝ^𝑛withsome𝜖>0, (17) (for the sake of simplicity we introduced an assumption for the extension of 𝑔 onto Ω_𝑇).

The first aim is to reformulate the problem (2)–(5) in a weak form. Let us suppose that 𝑎_𝑖𝑗∈𝑊¹₂(Ω), (𝜕/𝜕𝑥_𝑖)𝜙_𝑖∈𝐿²(Ω_𝑇), 𝑖=1,…,𝑛 and (2)–(5) has a classical solution 𝑢∈𝑊₂^2,1(Ω_𝑇). Then, we multiply (2) with a test function 𝜂 from the space 𝒯Ω_𝑇=𝜂∈𝐿²(0,𝑇);𝑊¹₂(Ω∶𝜂_𝑡)∈𝐿²(0,𝑇);𝐿²(Ω),𝜂|_Γ1,𝑇=0incaseΓ₁≠∅ (18) and integrate by parts with respect to time and space variables. We obtain the following relation: 0=_Ω𝑢(𝑥,𝑇)𝜂(𝑥,𝑇)−𝑢₀(𝑥)𝜂(𝑥,0)𝑑𝑥−_Ω𝑇𝑢𝜂_𝑡+𝑑𝑥𝑑𝑡_{Ω𝑇𝑛𝑖,𝑗=1}𝑎_𝑖𝑗𝑢_𝑥𝑗−𝑚∗𝑢_𝑥𝑗𝜂_𝑥𝑖()𝜂+−𝑎𝑢−𝑚∗𝑢𝑑𝑥𝑑𝑡_Γ2,𝑇(𝜗𝑢+ℎ)𝜂𝑑Γ𝑑𝑡−_Ω𝑇𝑓𝜂−𝜙⋅∇𝜂𝑑𝑥𝑑𝑡. (19) This relation makes sense also in a more general case when 𝑎_𝑖𝑗, 𝜙 satisfies only (11) and (15) and 𝑢 does not have regular first-order time and second-order spatial derivatives.

We call a weak solution of the problem (2)–(5) a function from the space 𝒰(Ω_𝑇) that satisfies the relation (19) for any 𝜂∈𝒯(Ω_𝑇) and in case Γ₁≠∅ fulfills the boundary condition (4).

Lemma 1. The following assertions are valid: (i)𝒰(Ω_𝑇)↪𝐿²((0,𝑇);𝐿^𝑞3(Ω)) where 𝑞₃=∞ if 𝑛=1, 𝑞₃∈(𝑞₁𝑞₂/(𝑞₁−𝑞₂),∞) if 𝑛=2 and 𝑞₃=2𝑛/(𝑛−2) if 𝑛>2, where 𝑞₁ and 𝑞₂ are given in (12) and (14), respectively;(ii)for any 𝑢∈𝐿²((0,𝑇);𝐿^𝑞3(Ω)) it holds 𝑎𝑢∈𝐿²((0,𝑇);𝐿^𝑞2(Ω)) and ‖𝑎𝑢‖_𝐿²_{((0,𝑇);𝐿}^𝑞_2(Ω))≤𝐶‖𝑎‖_𝐿^𝑞_1(Ω)‖𝑢‖_𝐿²_{((0,𝑇);𝐿}^𝑞_3(Ω)), where 𝐶 is a constant.

Proof. Since 𝒰(Ω_𝑇)↪𝐿²((0,𝑇);𝑊¹₂(Ω)), the assertion (i) follows from the continuous embedding of 𝑊¹₂(Ω) in 𝐿^𝑞3(Ω). The assertion (ii) can be proved by means of Hölder's inequality.

Theorem 2. The problem (2)–(5) has a unique weak solution. This solution satisfies the estimate ‖𝑢‖_𝒰(Ω𝑇)≤𝐶₀𝑢_0𝐿2(Ω)+𝑓_{𝐿2((0,𝑇);𝐿𝑞2(Ω))}+𝜙_{(𝐿2(Ω𝑇))}^𝑛+𝜃₁𝑔_{𝐿2((0,𝑇);𝑊12(Ω))}+𝑔_{𝑡𝐿2(Ω𝑇)}+𝜃₂‖ℎ‖_𝐿²_(Γ2,𝑇), (20) where 𝜃₁=0 in case Γ₁=∅, 𝜃₂=0 in case Γ₂=∅ and 𝐶₀ is a constant depending on Ω, Γ_𝑗, 𝑎_𝑖𝑗, 𝑎, 𝜗 and 𝑚.

Proof. The assertion of the theorem in case 𝑚=0 is well known from the theory of parabolic equations (see, e.g., [2]). Let 𝒵 be the operator that assigns to the data vector 𝑑∶=(𝑢₀,𝑓,𝜙,𝑔,ℎ) the weak solution of the problem (2)–(5) in case 𝑚=0. Then it holds ‖𝒵𝑑‖_𝒰(Ω𝑇)≤RHS, (21) where RHS is the right-hand side of (20).
Further, let us formulate the problem for the difference 𝑣=𝑢−𝒵𝑑. Introducing the linear operator 𝒜 by the formula 𝒜𝑤=𝒵0,−𝑎𝑚∗𝑤,−^𝑛_𝑗=1𝑎_𝑖𝑗𝑚∗𝑤_𝑥𝑗,0,0, (22) the weak problem (2)–(5) for the solution 𝑢∈𝒰(Ω_𝑇) equivalent to the following operator equation for the quantity 𝑣: 𝑣=𝒜𝑣+𝒜𝒵𝑑. (23) We have to estimate 𝒜. For this purpose, we firstly prove the following auxiliary inequality: 𝑚∗𝑦_{𝐿2((0,𝑡);𝐿𝑝(Ω))}≤^𝑡₀|𝑚(𝑡−𝜏)|𝑦_{𝐿2((0,𝜏);𝐿𝑝(Ω))}[],𝑑𝜏,𝑡∈0,𝑇 (24) for any 𝑝≥1 and 𝑦∈𝐿²((0,𝑡);𝐿^𝑝(Ω)).
Denoting 𝑦(𝑡)=𝑦(⋅,𝑡), ‖𝑦(𝑡)‖=‖𝑦(𝑡)‖_𝐿𝑝(Ω), making use of the following property of the Bochner integral: ‖∫^𝑠₀𝑤(𝑠,𝜏)𝑑𝜏‖_𝐿𝑝(Ω)≤∫^𝑠₀‖𝑤(𝑠,𝜏)‖_𝐿^𝑝_(Ω)𝑑𝜏 for functions 𝑤∈𝐿¹((0,𝑇);𝐿^𝑝(Ω)) and the Cauchy's inequality, the relation (24) can be deduced by means of the following computations: 𝑚∗𝑦_{𝐿2((0,𝑡);𝐿𝑝(Ω))}=^𝑡₀^𝑠₀𝑚(𝜏)𝑦(𝑠−𝜏)𝑑𝜏²_𝐿𝑝(Ω)𝑑𝑠^1/2≤𝐼,where𝐼=^𝑡₀^𝑠₀|𝑚(𝜏)|𝑦(𝑠−𝜏)𝑑𝜏²𝑑𝑠^1/2=^𝑡₀|𝑚(𝜏)|^𝑡_𝜏𝑦(𝑠−𝜏)^𝑠₀|𝑚(𝑧)|𝑦(𝑠−𝑧)𝑑𝑧𝑑𝑠𝑑𝜏^1/2≤^𝑡₀|𝑚(𝜏)|^𝑡_𝜏𝑦(𝑠−𝜏)²𝑑𝑠^1/2×^𝑡_𝜏^𝑠₀|𝑚(𝑧)|𝑦(𝑠−𝑧)𝑑𝑧²𝑑𝑠^1/2𝑑𝜏^1/2≤^𝑡₀|𝑚(𝜏)|₀^𝑡−𝜏𝑦(𝑠)²𝑑𝑠^1/2𝑑𝜏^1/2×^𝑡₀^𝑠₀|𝑚(𝑧)|𝑦(𝑠−𝑧)𝑑𝑧²𝑑𝑠^1/4=^𝑡₀|𝑚(𝑡−𝜏)|𝑦_{𝐿2((0,𝜏);𝐿𝑝(Ω))}𝑑𝜏^1/2×𝐼^1/2. (25)
Next, let 𝑡∈[0,𝑇] and introduce the operator𝑃_𝑡𝑤=𝑤inΩ_𝑡0inΩ_𝑇⧵Ω_𝑡. (26) Due to the causality we have 𝒵(0,𝑃_𝑡𝑓,𝑃_𝑡𝜙,0,0)(𝑥,𝑡)=𝒵(0,𝑓,𝜙,0,0)(𝑥,𝑡) for any (𝑥,𝑡)∈Ω_𝑡. Using these relations, the continuity of the linear operator 𝒵, the inequality (24), and the boundedness of 𝑎_𝑖𝑗, we compute the following: ‖𝒜𝑤‖_𝒰(Ω𝑡)=𝒵0,−𝑎𝑚∗𝑤,−^𝑛_𝑗=1𝑎_𝑖𝑗𝑚∗𝑤_𝑥𝑗,0,0_𝒰(Ω𝑡)=𝒵0,−𝑃_𝑡[𝑎𝑚∗𝑤],−𝑃_𝑡^𝑛_𝑗=1𝑎_𝑖𝑗𝑚∗𝑤_𝑥𝑗,0,0_𝒰(Ω𝑡)≤𝒵0,−𝑃_𝑡[𝑎𝑚∗𝑤],−𝑃_𝑡^𝑛_𝑗=1𝑎_𝑖𝑗𝑚∗𝑤_𝑥𝑗,0,0_𝒰(Ω𝑇)≤𝐶𝑃_𝑡[𝑎𝑚∗𝑤]_{𝐿2((0,𝑇);𝐿𝑞2(Ω))}+^𝑛_𝑖=1𝑃_𝑡𝑎_𝑖𝑗𝑚∗𝑤_{𝑥𝑗𝐿2(Ω𝑇)}=𝐶‖𝑎𝑚∗𝑤‖_𝐿²_{((0,𝑡);𝐿}^𝑞_2(Ω))+^𝑛_𝑖=1𝑎_𝑖𝑗𝑚∗𝑤_{𝑥𝑗𝐿2(Ω𝑡)}≤𝐶^𝑡₀|𝑚(𝑡−𝜏)|‖𝑎𝑤‖_𝐿²_{((0,𝜏);𝐿}^𝑞_2(Ω))+‖∇𝑤‖_𝐿²_(Ω𝜏)𝑑𝜏, (27) with some constants 𝐶 and 𝐶 depending on Ω, Γ_𝑗, 𝑎_𝑖𝑗, 𝑎, 𝜗. Using Lemma 1, we obtain ‖𝑎𝑤‖_𝐿²_{((0,𝜏);𝐿}^𝑞_2(Ω))≤𝐶₁‖𝑎‖_𝐿^𝑞_1(Ω)‖𝑤‖_𝐿²_{((0,𝜏);𝑊}¹_2(Ω)). (28) Using this relation in (27), we arrive at the following basic estimate for 𝒜: ‖𝒜𝑤‖_𝒰(Ω𝑡)≤𝐶₂^𝑡₀|𝑚(𝑡−𝜏)|‖𝑤‖_𝒰(Ω𝜏)𝑑𝜏,𝑡∈[0,𝑇], (29) where 𝐶₂ is a constant depending on Ω, Γ_𝑗, 𝑎_𝑖𝑗, 𝑎, 𝜗. Let us define the weighted norms in 𝒰(Ω_𝑇): ‖𝑣‖_𝜎=sup_0<𝑡<𝑇𝑒^−𝜎𝑡‖𝑣‖_𝒰(Ω𝑡) where 𝜎≥0. The estimate (29) implies the further estimate ‖𝒜𝑤‖_𝜎≤𝐶₂sup_0<𝑡<𝑇𝑒^{−𝜎𝑡𝑡}₀|𝑚(𝑡−𝜏)|‖𝑤‖_𝒰(Ω𝜏)𝑑𝜏=𝐶₂sup_0<𝑡<𝑇^𝑡₀𝑒^{−𝜎(𝑡−𝜏)}|𝑚(𝑡−𝜏)|𝑒^−𝜎𝜏‖𝑤‖_𝒰(Ω𝜏)𝑑𝜏≤𝐶₂^𝑇₀𝑒^−𝜎𝑠|𝑚(𝑠)|𝑑𝑠‖𝑤‖_𝜎. (30) Since ∫^𝑇₀𝑒^−𝜎𝑠|𝑚(𝑠)|𝑑𝑠→0 as 𝜎→∞, there exists 𝜎₀, depending on 𝐶₂ and 𝑚, such that 𝐶₂∫^𝑇₀𝑒^−𝜎0𝑠|𝑚(𝑠)|𝑑𝑠≤1/2. Thus, ‖𝒜𝑤‖_𝜎0≤(1/2)‖𝑤‖_𝜎0. The operator 𝒜 is a contraction in 𝒰(Ω_𝑇). This implies the existence and uniqueness assertions of the theorem.
To prove the estimate (20), we firstly deduce from (23) the inequality ‖𝑣‖_𝜎0≤‖𝒜𝑣‖_𝜎0+‖𝒜𝒵𝑑‖_𝜎0≤(1/2)[‖𝑣‖_𝜎0+‖𝒵𝑑‖_𝜎0]. This implies ‖𝑣‖_𝜎0≤‖𝒵𝑑‖_𝜎0. Using the equivalence relations 𝑒^−𝜎0𝑇‖⋅‖₀≤‖⋅‖_𝜎0≤‖⋅‖₀, where ‖⋅‖₀=‖⋅‖_𝒰(Ω𝑇) and (21), we reach (20).

We note the upper integration bound 𝑇 in (19) can be released to be any number 𝑡 from the interval [0,𝑇]. Indeed, (19) is equivalent to the following problem: 0=_Ω𝑢(𝑥,𝑡)𝜂(𝑥,𝑡)−𝑢₀(𝑥)𝜂(𝑥,0)𝑑𝑥−_Ω𝑡𝑢𝜂_𝑡𝑑𝑥𝑑𝑡+_Ω𝑡^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝑢_𝑥𝑗−𝑚∗𝑢_𝑥𝑗𝜂_𝑥𝑖()𝜂+−𝑎𝑢−𝑚∗𝑢𝑑𝑥𝑑𝑡_Γ2,𝑡()−𝜗𝑢+ℎ𝜂𝑑Γ𝑑𝑡_Ω𝑡𝑓𝜂−𝜙⋅∇𝜂𝑑𝑥𝑑𝑡,𝑡∈[0,𝑇], (31) for any 𝜂∈𝒯(Ω_𝑇). This assertion can be proved using the standard technique defining the test function as follows: 𝜂^𝜖()=𝜂()[],𝑥,𝑡𝜂()𝑥,𝜏for𝜏∈0,𝑡𝑥,𝜏1−𝜏−𝑡𝜖(),[],for𝜏∈𝑡,𝑡+𝜖0for𝜏∈𝑡+𝜖,𝑇 (32) and letting the parameter 𝜖 to approach 0.

Next we transform the weak direct problem (31) to a form that does not contain a time derivative of the test function 𝜂. This form enables the extension of the test space. This is useful for treatment of problems for adjoint states of quasisolutions of inverse problems in next sections.

Theorem 3. The function 𝑢∈𝒰(Ω_𝑇) satisfies the relation (19) for any 𝜂∈𝒯(Ω_𝑇) if and only if it satisfies the following relation: 0=_Ω𝑢∗𝜂𝑑𝑥−_Ω^𝑡₀𝑢₀(𝑥)𝜂()+𝑥,𝜏𝑑𝜏𝑑𝑥_Ω1∗^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝑢_𝑥𝑗−𝑚∗𝑢_𝑥𝑗∗𝜂_𝑥𝑖()+−𝑎𝑢−𝑚∗𝑢∗𝜂𝑑𝑥_Γ2()−1∗𝜗𝑢+ℎ∗𝜂𝑑Γ_Ω1∗𝑓∗𝜂−^𝑛_𝑖=1𝜙_𝑖∗𝜂_𝑥𝑖𝑑𝑥,𝑡∈[0,𝑇], (33) for any 𝜂∈𝒰₀(Ω_𝑇).

Here, according to the definition of the time convolution in the previous section, 1∗𝑤(𝑡)=∫^𝑡₀𝑤(𝜏)𝑑𝜏.

Proof. It is sufficient to prove that 𝑢∈𝒰(Ω_𝑇) satisfies (31) for any 𝜂∈𝒯(Ω_𝑇) if and only if it satisfies (33) for any 𝜂∈𝒰₀(Ω_𝑇). Suppose that 𝑢∈𝒰(Ω_𝑇) satisfies (31) and choose an arbitrary 𝜂∈𝒯(Ω_𝑇). Let 𝑡₁ be an arbitrary number on the interval [0,𝑇] and choose some function 𝜉^𝑡1∈𝒯(Ω_𝑇) such that the relation 𝜉^𝑡1(𝑥,𝑡)=𝜂𝑥,𝑡₁−𝑡for𝑡∈0,𝑡₁ (34) is valid. For instance, it is possible to define 𝜉^𝑡1 as a periodic function with respect to 𝑡, that is, 𝜉^𝑡1(𝑥,𝑡)=𝜂(𝑥,𝑡₁−𝑡) for 𝑡∈[0,𝑡₁], 𝜉^𝑡1(𝑥,𝑡)=𝜂(𝑥,𝑡−𝑡₁) for 𝑡∈[𝑡₁,2𝑡₁], 𝜉^𝑡1(𝑥,𝑡)=𝜂(𝑥,3𝑡₁−𝑡) for 𝑡∈[2𝑡₁,3𝑡₁] and so on. Using the relation (31) with 𝜂 replaced by 𝜉^𝑡1 and setting there 𝑡=𝑡₁ we obtain the equality 0=𝐾₁𝑡₁+𝐾₂𝑡₁, (35) where 𝐾₁(𝑡)=_Ω𝑢(𝑥,𝑡)𝜂(𝑥,0)−𝑢₀(𝑥)𝜂(+𝑥,𝑡)𝑑𝑥_Ω𝑡0𝑢(𝑥,𝜏)𝜂_𝑡()𝐾𝑥,𝑡−𝜏𝑑𝜏𝑑𝑥,₂(𝑡)=_Ω^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝑢_𝑥𝑗−𝑚∗𝑢_𝑥𝑗∗𝜂_𝑥𝑖−𝑎(𝑢−𝑚∗𝑢)∗𝜂𝑑𝑥+_Γ2()−𝜗𝑢+ℎ∗𝜂𝑑Γ_Ω𝑓∗𝜂−^𝑛_𝑖=1𝜙_𝑖∗𝜂_𝑥𝑖𝑑𝑥. (36)(37) Note that the time derivative of 𝜂 can be removed from 𝐾₁ by integration. Indeed, let 𝑡₂∈[0,𝑇]. Then ^𝑡2₀𝐾₁𝑡₁𝑑𝑡₁=^𝑡2_0Ω𝑢𝑥,𝑡₁𝜂(𝑥,0)𝑑𝑥𝑑𝑡₁−^𝑡2_0Ω𝑢₀(𝑥)𝜂𝑥,𝑡₁𝑑𝑥𝑑𝑡₁+^𝑡2_0Ω^𝑡1₀𝑢(𝑥,𝜏)𝜂_𝑡1𝑥,𝑡₁−𝜏𝑑𝜏𝑑𝑥𝑑𝑡₁. (38) Changing the order of the integrals over 𝜏 and 𝑡₁ in the last term, we easily obtain ^𝑡2₀𝐾₁𝑡₁𝑑𝑡₁=_Ω^𝑡2₀𝑢(𝑥,𝜏)𝜂𝑥,𝑡₂−−𝜏𝑑𝜏𝑑𝑥_𝑡20Ω𝑢₀(𝑥)𝜂𝑥,𝑡₁𝑑𝑥𝑑𝑡₁. (39) Integrating now the whole equality (35) over 𝑡₁ from 0 to 𝑡₂, observing (37) and (39), and finally redenoting 𝑡₂ by 𝑡, we reach the desired relation (33). Summing up, we have proved that (33) holds for any 𝜂∈𝒯(Ω_𝑇). But all terms in the right-hand side of (33) are well defined for 𝜂∈𝒰₀(Ω_𝑇), too. Since 𝒯(Ω_𝑇) is densely embedded in 𝒰₀(Ω_𝑇), we conclude that (33) holds for any 𝜂∈𝒰₀(Ω_𝑇).
It remains to show that (33) implies (31). Suppose that 𝑢∈𝒰(Ω_𝑇) satisfies (33) and choose an arbitrary 𝜂∈𝒯(Ω_𝑇) and 𝑡₁∈[0,𝑇]. Again, let 𝜉^𝑡1 be a function from 𝒯(Ω_𝑇) such (34) is valid. Inserting 𝜉^𝑡1 instead of 𝜂 into (33), differentiating with respect to 𝑡 and setting 𝑡=𝑡₁ we come to the relation (31). Theorem is proved.

Corollary 4. A function 𝑢∈𝒰(Ω_𝑇) is a weak solution of (2)–(5) if and only if it satisfies the relation (33) for any 𝜂∈𝒰₀(Ω_𝑇) and in case Γ₁≠∅ fulfills the boundary condition (4).

4. Inverse Problems and Quasisolutions

In the sequel, let us pose some inverse problems for the weak solution of (2)–(5). These problems are selected in order to demonstrate the wide possibilities of the method that we will introduce in Section 5.

Firstly, we suppose that (2)–(5) has the following specific form: 𝑢_𝑡=𝐴𝑢−𝑚∗𝐴𝑢+𝑓₀+∇⋅𝜙+^𝑁_𝑗=1𝛾_𝑗(𝑡)𝜔_𝑗(𝑥)inΩ_𝑇,𝑢=𝑢₀inΩ×{0},𝑢=𝑔inΓ_1,𝑇,−𝜈_𝐴⋅∇𝑢+𝑚∗𝜈_𝐴⋅∇𝑢=𝜗𝑢+ℎ+𝜈⋅𝜙inΓ_2,𝑇, (40) where 𝜔=(𝜔₁,…,𝜔_𝑁) is unknown. The coefficients and other given functions 𝑓₀, 𝜙, 𝑢₀, 𝑔, ℎ are assumed to satisfy (11)–(17). Moreover, 𝛾∈(𝐿²(0,𝑇))^𝑁 is prescribed.

IP1. Find the vector 𝜔∈(𝐿²(Ω))^𝑁 such that the weak solution of (40) satisfies the following instant additional conditions: 𝑢𝑥,𝑇_𝑖=𝑢_𝑇𝑖(𝑥),𝑥∈Ω,𝑖=1,2,…,𝑁, (41) where 0<𝑇₁<𝑇₂<⋯<𝑇_𝑁≤𝑇 and 𝑢_𝑇𝑖∈𝐿²(Ω), 𝑖=1,…,𝑁 are given functions (observations of 𝑢).

Since ∑^𝑁_𝑖=1𝛾_𝑗𝜔_𝑗∈𝐿²((0,𝑇);𝐿²(Ω))⊂𝐿²((0,𝑇);𝐿^𝑞2(Ω)) for 𝜔∈(𝐿²(Ω))^𝑁, the weak solution 𝑢 of (40) exists in 𝒰(Ω_𝑇); hence, it has traces 𝑢(⋅,𝑇_𝑖)∈𝐿²(Ω), 𝑖=1,…,𝑁. In practice, the term ∑^𝑁_𝑗=1𝛾_𝑗𝜔_𝑗 may represent an approximation of a more general function 𝐹(𝑥,𝑡)∈𝐿²(Ω_𝑇), where 𝛾_𝑗, 𝑗=1,2,… form a basis in 𝐿²(0,𝑇).

Further, let 𝑢₀ also be unknown.

IP2. Find the vector 𝜔∈(𝐿²(Ω))^𝑁 and 𝑢₀∈𝐿²(Ω) such that the weak solution of (40) satisfies the following integral additional conditions: ^𝑇₀𝜅_𝑖(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑡=𝑣_𝑖(𝑥),𝑥∈Ω,𝑖=1,2,…,𝑁+1, (42) where 𝑣_𝑖∈𝐿²(Ω), 𝑖=1,…,𝑁+1 are given observation functions and 𝜅_𝑖, 𝑖=1,…,𝑁 are given weights that satisfy the following condition: 𝜅_𝑖(𝑥,𝑡)≤𝜅(𝑡)inΩ_𝑇,𝑖=1,…,𝑁+1withsome𝜅∈𝐿²(0,𝑇). (43) Note that the integral ∫^𝑇₀𝜅_𝑖(⋅,𝑡)𝑢(⋅,𝑡)𝑑𝑡 in (42) belongs to 𝐿²(Ω) for any 𝜔∈(𝐿²(Ω))^𝑁 and 𝑢₀∈𝐿²(Ω). Indeed, for such 𝜔 and 𝑢₀ it holds 𝑢∈𝒰(Ω_𝑇)⊂𝐿²(Ω_𝑇), which implies ^𝑇₀𝜅_𝑖(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑡_𝐿2(Ω)≤‖𝜅‖_𝐿²_(0,𝑇)‖𝑢‖_𝐿²_(Ω𝑇)<∞. (44) In practice, the weights 𝜅_𝑖 are usually concentrated in neighborhoods of some fixed values of time 𝑡=𝑇_𝑖.

Finally, let us pose a nonlinear inverse problem for the coefficient 𝑎 and the kernel 𝑚. Assume that 𝑛∈{1;2;3}. Then any coefficient 𝑎 that belongs to 𝐿²(Ω) satisfies (12). Moreover, let us set 𝑞₁=2 if 𝑛=2 and Γ₂≢∅. The other coefficients and the given functions 𝑢₀, 𝑓, 𝜙, 𝑔, ℎ are assumed to satisfy (11)–(17).

IP3. Find 𝑎∈𝐿²(Ω) and 𝑚∈𝐿¹(0,𝑇) such that the weak solution of (2)–(5) satisfies the following integral additional conditions: 𝑢(𝑥,𝑇)=𝑢_𝑇(𝑥),𝑥∈Ω,_Γ2𝜅(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑Γ=𝑣(𝑡),𝑡∈(0,𝑇), (45) where 𝑢_𝑇∈𝐿²(Ω),   𝑣∈𝐿²(0,𝑇) are given observation functions and 𝜅 is a given weight function such that 𝜅∈𝐿^∞((0,𝑇);𝐿²(Γ₂)).

As in IP1, we can show that the trace 𝑢(⋅,𝑇) belongs to 𝐿²(Ω). Moreover, using the property 𝑢∈𝒰(Ω_𝑇), the embedding of 𝑊¹₂(Ω) in 𝐿²(Γ₂) and Hölder’s inequality, one can immediately check that the term ∫_Γ2𝜅(𝑥,⋅)𝑢(𝑥,⋅)𝑑Γ in (45) belongs to 𝐿²(0,𝑇).

Available existence, uniqueness, and stability results for IP1–IP3 require stronger smoothness of the data than imposed in the present paper. Let us cite some of these results.

In case 𝑁=1, the well posedness of IP1 was proved in [8]. Partial results were deduced earlier in [9]. A more general problem involving both IP1 and IP2 without the unknown 𝑢₀ in case 𝑁=1 was studied in [10] by means of different techniques. IP1 and IP2 in case 𝑚=0 and 𝑁=1 were treated in many papers, for example, [11–14]. The case 𝑁>1 is open even if 𝑚=0. Inverse problems to determine 𝑚 with given 𝑎 were studied in a number of papers, for example, [7, 15–23]. The problem for 𝑎 with given 𝑚 was treated in [8].

In the present paper, we will deal with quasisolutions of IP1–IP3 and related cost functionals. Denote 𝒵₁=(𝐿²(Ω))^𝑁. Let 𝑀⊆𝒵₁. The quasi-solution of IP1 in the set 𝑀 is an element 𝜔^∗∈argmin_𝜔∈𝑀𝐽₁(𝜔), where 𝐽₁ is the following cost functional 𝐽₁(𝜔)=^𝑁_𝑖=1𝑢(𝑥,𝑇_𝑖;𝜔)−𝑢_𝑇𝑖(𝑥)²_𝐿2(Ω), (46) and 𝑢(𝑥,𝑡;𝜔) is the solution of (40) that corresponds to a fixed element 𝜔.

Similarly, let 𝑀⊆𝒵₂∶=(𝐿²(Ω))^𝑁+1. The quasi-solution of IP2 in the set 𝑀 is 𝑧^∗∈argmin_𝑧∈𝑀𝐽₂(𝑧), where 𝐽₂ is the cost functional 𝐽₂(𝑧)=^𝑁+1_𝑖=1^𝑇₀𝜅_𝑖(𝑥,𝑡)𝑢(𝑥,𝑡;𝑧)𝑑𝑡−𝑣_𝑖(𝑥)²_𝐿2(Ω), (47) and 𝑢(𝑥,𝑡;𝑧) is the weak solution of (40) that corresponds to a given vector 𝑧=(𝜔,𝑢₀).

Finally, defining 𝑀⊆𝒵₃∶=𝐿²(Ω)×𝐿²(0,𝑇), the quasi-solution of IP3 in 𝑀 is an element 𝑧^∗∈argmin_𝑧∈𝑀𝐽₃(𝑧), where 𝐽₃ is the cost functional 𝐽₃(𝑧)=𝑢(𝑥,𝑇;𝑧)−𝑢_𝑇(𝑥)²_𝐿2(Ω)+_Γ2𝜅(𝑥,𝑡)𝑢(𝑥,𝑡;𝑧)𝑑Γ−𝑣(𝑡)²_{𝐿2()0,𝑇}, (48) and 𝑢(𝑥,𝑡;𝑧) is the weak solution of the direct problem (2)–(5) corresponding to given 𝑧=(𝑎,𝑚). Here, we restricted the space for the unknown 𝑚 to 𝐿²(0,𝑇), because we will seek for the Fréchet derivative of 𝐽₃ in a Hilbert space. Moreover, the kernel of the second addend corresponding to 𝑚 in the representation formula of 𝐽^′₃ (90) is an element of 𝐿²(0,𝑇) and in general does not belong to the adjoint space 𝐿^∞(0,𝑇).

According to the above-mentioned arguments, the functionals 𝐽_𝑘, 𝑘=1,2,3, are well-defined in 𝒵₁, 𝒵₂, and 𝒵₃, respectively.

5. The Fréchet Derivatives of Cost Functionals of Inverse Problems

5.1. General Procedure

Suppose that the solution 𝑢 of the direct problem depends on a vector of parameters 𝑝 that has to be determined in an inverse problem making use of certain measurements of 𝑢. Let the quasi-solution of the inverse problem be sought by a method involving the Fréchet derivative of the cost functional (i.e., some gradient-type method). Usually in practice, a solution of a proper adjoint problem is used to represent the Fréchet derivative.

We introduce a general procedure to deduce such adjoint problems. Assume that Δ𝑢 is the difference of solutions of the direct problem corresponding to a difference of the vector of the parameters Δ𝑝. More precisely, we suppose that Δ𝑢 is a solution of the following problem: Δ𝑢_𝑡=𝐴Δ𝑢−𝑚∗𝐴Δ𝑢+𝑓^†+∇⋅𝜙^†inΩ_𝑇,Δ𝑢=Δ𝑢₀inΩ×{0},Δ𝑢=0inΓ_1,𝑇,−𝜈_𝐴⋅∇Δ𝑢+𝑚∗𝜈_𝐴⋅∇Δ𝑢=𝜗Δ𝑢+ℎ^†+𝜈⋅𝜙^†inΓ_2,𝑇, (49)(50)(51)(52) with some data 𝑓^†, 𝜙^†, Δ𝑢₀, ℎ^† depending on Δ𝑝. We restrict ourselves to the case when the Dirichlet boundary condition of the state 𝑢 is independent of 𝑝. Therefore, the condition (51) for Δ𝑢 is homogeneous.

In practice, the adjoint parabolic problems are usually formulated as backward problems. In our context, it is better to pose adjoint problems in the forward form. The involved memory term with 𝑚 is defined via a forward convolution and from the practical viewpoint, it is preferable to have the direct and adjoint problems in a similar form (e.g., to simplify parallelisation of computations).

More precisely, let the adjoint state 𝜓 be a solution of the following problem: 𝜓_𝑡=𝐴𝜓−𝑚∗𝐴𝜓+𝑓^∘+∇⋅𝜙^∘inΩ_𝑇,𝜓=𝑢^∘inΩ×{0},𝜓=0inΓ_1,𝑇,−𝜈_𝐴⋅∇𝜓+𝑚∗𝜈_𝐴⋅∇𝜓=𝜗𝜓+ℎ^∘+𝜈⋅𝜙^∘inΓ_2,𝑇, (53) where 𝑓^∘, 𝜙^∘, 𝑢^∘, and ℎ^∘ are some data depending on Δ𝑢 and the cost functional under consideration.

Assume that the quadruplets 𝑓^†, 𝜙^†, Δ𝑢₀, ℎ^†, and 𝑓^∘, 𝜙^∘, 𝑢^∘, ℎ^∘ satisfy the conditions (14)–(16). Then, due to Theorem 2, the problems (49)–(52) and (53) have unique weak solutions in the space 𝒰(Ω_𝑇). Actually, we have Δ𝑢,𝜓∈𝒰₀(Ω_𝑇) because of the homogeneous boundary conditions on Γ_1,𝑇.

Let us write the relation (33) for Δ𝑢 and use the test function 𝜂=𝜓. Then we obtain for any 𝑡∈[0,𝑇]0=_ΩΔ𝑢∗𝜓𝑑𝑥−_Ω^𝑡₀Δ𝑢₀(𝑥)𝜓()+𝑥,𝜏𝑑𝜏𝑑𝑥_Ω1∗^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗Δ𝑢_𝑥𝑗−𝑚∗Δ𝑢_𝑥𝑗∗𝜓_𝑥𝑖()+−𝑎Δ𝑢−𝑚∗Δ𝑢∗𝜓𝜂𝑑𝑥_Γ21∗𝜗Δ𝑢+ℎ^†−∗𝜓𝑑Γ_Ω1∗𝑓^†∗𝜓−^𝑛_𝑖=1𝜙^†_𝑖∗𝜓_𝑥𝑖𝑑𝑥. (54) Secondly, let us write this relation for 𝜓 and use the test function 𝜂=Δ𝑢. Then we have for any 𝑡∈[0,𝑇]0=_Ω𝜓∗Δ𝑢𝑑𝑥−_Ω^𝑡₀𝑢^∘(𝑥)()+Δ𝑢𝑥,𝜏𝑑𝜏𝑑𝑥_Ω1∗^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜓_𝑥𝑗−𝑚∗𝜓_𝑥𝑗∗Δ𝑢_𝑥𝑖+−𝑎𝜓−𝑚∗𝜓∗Δ𝑢𝑑𝑥_Γ21∗𝜗𝜓+ℎ^∘−∗Δ𝑢𝑑Γ_Ω1∗𝑓^∘∗Δ𝑢−^𝑛_𝑖=1𝜙^∘_𝑖∗Δ𝑢_𝑥𝑖𝑑𝑥. (55) Subtracting (54) from (55), using the commutativity of the convolution, the symmetricity relations 𝑎_𝑖𝑗=𝑎_𝑗𝑖 and differentiating with respect to 𝑡, we arrive at the following basic equality that can be used in various inverse problems: _Ω𝑢^∘(𝑥)Δ𝑢(𝑥,𝑡)𝑑𝑥−_Γ2ℎ^∘+∗Δ𝑢𝑑Γ_Ω𝑓^∘∗Δ𝑢−^𝑛_𝑖=1𝜙^∘_𝑖∗Δ𝑢_𝑥𝑖=𝑑𝑥_ΩΔ𝑢₀(𝑥)𝜓(𝑥,𝑡)𝑑𝑥−_Γ2ℎ^†+∗𝜓𝑑Γ_Ω𝑓^†∗𝜓−^𝑛_𝑖=1𝜙^†_𝑖∗𝜓_𝑥𝑖𝑑𝑥,𝑡∈[0,𝑇]. (56)

5.2. Derivative of 𝐽₁

Theorem 5. The functional 𝐽₁ is the Fréchet differentiable in (𝐿²(Ω))^𝑁 and 𝐽^′₁(𝜔)=Δ𝜔_{𝑁𝑗=1Ω𝑁𝑖=1𝑇𝑖0}𝛾_𝑗(𝑡)𝜓_𝑖𝑥,𝑇_𝑖−𝑡;𝜔𝑑𝑡Δ𝜔_𝑗(𝑥)𝑑𝑥, (57) where 𝜓_𝑖∈𝒰(Ω_𝑇𝑖), 𝑖=1,…,𝑁, are the unique 𝜔-dependent weak solutions of the following problems: 𝜓_𝑖,𝑡=𝐴𝜓_𝑖−𝑚∗𝐴𝜓_𝑖𝑖𝑛Ω_𝑇𝑖,𝜓_𝑖=2𝑢𝑥,𝑇_𝑖;𝜔−𝑢_𝑇𝑖(𝑥{0},𝜓)𝑖𝑛Ω×_𝑖=0𝑖𝑛Γ_1,𝑇𝑖,−𝜈_𝐴⋅∇𝜓_𝑖+𝑚∗𝜈_𝐴⋅∇𝜓_𝑖=𝜗𝜓_𝑖𝑖𝑛Γ_2,𝑇𝑖, (58)𝑖=1,…,𝑁.

Proof. Let us fix some 𝜔,Δ𝜔∈(𝐿²(Ω))^𝑁. One can immediately check that it holds 𝐽₁(𝜔+Δ𝜔)−𝐽₁(𝜔)=2^𝑁_𝑖=1Ω𝑢𝑥,𝑇_𝑖;𝜔−𝑢_𝑇𝑖(𝑥)Δ𝑢𝑥,𝑇_𝑖+;𝜔𝑑𝑥_𝑁𝑖=1ΩΔ𝑢𝑥,𝑇_𝑖;𝜔²𝑑𝑥, (59) where Δ𝑢(𝑥,𝑡;𝜔)=𝑢(𝑥,𝑡;𝜔+Δ𝜔)−𝑢(𝑥,𝑡;𝜔)∈𝒰₀(Ω_𝑇) is the weak solution of the following problem: Δ𝑢_𝑡=𝐴Δ𝑢−𝑚∗𝐴Δ𝑢+^𝑁_𝑗=1𝛾_𝑗Δ𝜔_𝑗inΩ_𝑇,Δ𝑢=0inΩ×{0},Δ𝑢=0inΓ_1,𝑇,−𝜈_𝐴⋅∇Δ𝑢+𝑚∗𝜈_𝐴⋅∇Δ𝑢=𝜗Δ𝑢inΓ_2,𝑇. (60) Applying the estimate (20) to the solution of this problem we deduce the following estimate for the second term in the right-hand side of (59): ^𝑁_𝑖=1ΩΔ𝑢𝑥,𝑇_𝑖;𝜔²𝑑𝑥≤𝑛‖Δ𝑢‖²_𝒰Ω𝑇≤𝐶²₀𝑛^𝑁_𝑗=1𝛾_𝑗Δ𝜔_𝑗²_{𝐿2)((0,𝑇;𝐿𝑞2(Ω))}≤𝐶₄‖𝜔‖²_{𝐿2(Ω)𝑁}, (61) with some constant 𝐶₄. This implies that 𝐽₁ is the Fréchet differentiable and the first term in the right-hand side of (59) represents the Fréchet derivative, that is, 𝐽^′₁(𝜔)Δ𝜔=^𝑁_𝑖=1𝜎_𝑖with𝜎_𝑖=2_Ω𝑢𝑥,𝑇_𝑖;𝜔−𝑢_𝑇𝑖(𝑥)Δ𝑢𝑥,𝑇_𝑖;𝜔𝑑𝑥. (62)
Further, let us use the method presented in Section 5.1 to deduce the proper adjoint problems. Comparing (60) with (49)–(52) we see that 𝑓^†=∑^𝑁_𝑗=1𝛾_𝑗Δ𝜔_𝑗, 𝜙^†=Δ𝑢₀=ℎ^†=0. Therefore, the relation (56) has the form _Ω𝑢^∘(𝑥)Δ𝑢(𝑥,𝑡)𝑑𝑥−_Γ2ℎ^∘+∗Δ𝑢𝑑Γ_Ω𝑓^∘∗Δ𝑢−^𝑛_𝑖=1𝜙^∘_𝑖∗Δ𝑢_𝑥𝑖=𝑑𝑥_𝑁𝑗=1Ω𝛾_𝑗Δ𝜔_𝑗∗𝜓𝑑𝑥,𝑡∈[0,𝑇]. (63) In order to deduce a formula for the component 𝜎_𝑖 in the quantity 𝐽^′₁(𝜔)Δ𝜔, we set 𝑢^∘=𝑢^∘_𝑖=2[𝑢(𝑥,𝑇_𝑖;𝜔)−𝑢_𝑇𝑖(𝑥)], ℎ^∘=𝑓^∘=𝜙^∘=0 and 𝑡=𝑇_𝑖 in (63). Then we immediately have 𝜎_𝑖=^𝑁_𝑗=1Ω𝛾_𝑗Δ𝜔_𝑗∗𝜓_𝑖𝑑𝑥∣_{𝑡=𝑇𝑖}, (64) where according to (53) and the definition of 𝑢^∘_𝑖,ℎ^∘,𝜙^∘, the function 𝜓_𝑖 is the weak solution of the problem (58) in the domain Ω_𝑇 instead of Ω_𝑇𝑖. Due to Theorem 2, this problem has a unique solution. From (62) and (64) we obtain (57). The latter formula contains the values of 𝜓_𝑖 in Ω_𝑇𝑖. Therefore, we can restrict the problem (58) from Ω_𝑇 to Ω_𝑇𝑖.

To use the formula (57) one has to solve 𝑁 weak problems for the functions 𝜓_𝑖 in domains Ω_𝑇𝑖. In the following theorem, we will show that computational work related to the evaluation of the Fréchet derivative can be considerably reduced. Actually, it is sufficient to solve 𝑁 weak problems in the smaller domains Ω_{𝑇𝑖−𝑇𝑖−1}, 𝑖=1,…,𝑁. Here, 𝑇₀=0.

Theorem 6. The Fréchet derivative of the functional 𝐽₁ can also be written in the form 𝐽^′₁(𝜔)=Δ𝜔_{𝑁𝑗=1Ω𝑁𝑙=1𝑇𝑙𝑇𝑙−1}𝛾_𝑗(𝑡)𝛽_𝑙𝑥,𝑇_𝑙−𝑡,𝜔𝑑𝑡Δ𝜔_𝑗(𝑥)𝑑𝑥, (65) where 𝛽_𝑙∈𝒰₀(Ω_{𝑇𝑙−𝑇𝑙−1}) are the unique 𝜔-dependent weak solutions of the following sequence of recursive problems in the domains Ω_{𝑇𝑙−𝑇𝑙−1}: 𝛽_𝑙,𝑡=𝐴𝛽_𝑙−𝑚∗𝐴𝛽_𝑙−𝑎𝑓^𝑙−∇⋅Φ^𝑙𝑖𝑛Ω_{𝑇𝑙−𝑇𝑙−1},𝛽_𝑙=𝑢^𝑙₀{0},𝛽𝑖𝑛Ω×_𝑙=0𝑖𝑛Γ_{1,𝑇𝑙−𝑇𝑙−1},−𝜈_𝐴⋅∇𝛽_𝑙+𝑚∗𝜈_𝐴⋅∇𝛽_𝑙=𝜗𝛽_𝑙−𝜈⋅Φ^𝑙𝑖𝑛Γ_{2,𝑇𝑙−𝑇𝑙−1}, (66) where 𝑙=𝑁,𝑁−1,…,2,1. Here, 𝑢^𝑙₀(𝑥)=2𝑢𝑥,𝑇_𝑙;𝜔−𝑢_𝑇𝑙(𝑥)+Θ_𝑙𝛽_𝑙+1𝑥,𝑇_𝑙+1−𝑇_𝑙 (67) and the function 𝑓^𝑙 and the vector Φ^𝑙 are defined via 𝛽_𝑁,𝛽_𝑁−1,…,𝛽_𝑙+1 as follows: 𝑓^𝑙=Θ_𝑙^𝑁−1_𝑘=𝑙^{𝑇𝑘+1−𝑇𝑘}₀𝑚𝑇_𝑘−𝑇_𝑙+𝑡+𝜏𝛽_𝑘+1×𝑥,𝑇_𝑘+1−𝑇_𝑘−𝜏𝑑𝜏, (68)Φ^𝑙=(Φ^𝑙₁,…,Φ^𝑙_𝑛), Φ^𝑙_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝜕/𝜕𝑥_𝑗)𝑓^𝑙 and Θ_𝑁=0, Θ_𝑙=1 for 𝑙<𝑁.

Proof. Firstly, let us check that (66) indeed have unique weak solutions 𝛽_𝑙 in 𝒰₀(Ω_{𝑇𝑙−𝑇𝑙−1}). To this end we can use Theorem 2. For the problem 𝛽_𝑁 this is immediate, because the initial condition of the problem for 𝛽_𝑁 belongs to 𝐿²(Ω) and other equations in this problem are homogeneous. Further, we use the induction. Choose some 𝑙 in the range 𝑁>𝑙≥1 and suppose that 𝛽_𝑘+1∈𝒰₀(Ω_{𝑇𝑘+1−𝑇𝑘}) for all 𝑘 such that 𝑁−1≥𝑘≥𝑙. The aim is to us to show that then the problem for 𝛽_𝑙 has a unique weak solution in 𝒰₀(Ω_{𝑇𝑙−𝑇𝑙−1}). Let us represent the 𝑘th addend in (68) in the form 𝐼_𝑘=^{𝑇𝑘+1−𝑇𝑘}₀𝑚𝑇_𝑘−𝑇_𝑙+𝑡+𝜏𝛽_𝑘+1𝑥,𝑇_𝑘+1−𝑇_𝑘=−𝜏𝑑𝜏_{𝑇𝑘+1−𝑇𝑘𝑡+𝑡}𝑚𝑇_𝑘−𝑇_𝑙+𝜏𝛽_𝑘+1𝑥,𝑇_𝑘+1−𝑇_𝑘+𝑡−𝜏𝑑𝜏. (69) For any 𝑘 in the range 𝑁−1≥𝑘≥𝑙 we have 𝐼_𝑘²_{𝐿20,𝑇𝑙−𝑇𝑙−1;𝑊12(Ω)}≤_|𝛼|≤1^{𝑇𝑙−𝑇𝑙−1}₀^{𝑇𝑘+1−𝑇𝑘}_𝑡^+𝑡𝑚𝑇_𝑘−𝑇_𝑙⋅𝐷+𝜏_𝛼𝑥𝛽_𝑘+1⋅,𝑇_𝑘+1−𝑇_𝑘+𝑡−𝜏_𝐿2(Ω)𝑑𝜏²=𝑑𝑡_{|𝛼|≤1𝑇𝑙−𝑇𝑙−10𝑇𝑘+1−𝑇𝑘𝑡+𝑡𝑇𝑘+1−𝑇𝑘0+𝑡}𝑚𝑇_𝑘−𝑇_𝑙+𝜏𝑧_𝑘,𝛼⋅𝑇_𝑘+1−𝑇_𝑘+𝑡−𝜏𝑑𝜏²𝑑𝑡=_|𝛼|≤1^{𝑇𝑙−𝑇𝑙−1+𝑇𝑘+1−𝑇𝑘}_{𝑇𝑘+1−𝑇𝑘}^𝑡₀𝑚_𝑘(𝜏)𝑧_𝑘,𝛼(𝑡−𝜏)𝑑𝜏²𝑑𝑡, (70) where 𝑧_𝑘,𝛼(𝑡)=‖𝐷^𝛼_𝑥𝛽_𝑘+1(⋅,𝑡)‖_𝐿2(Ω) for 𝑡∈[0,𝑇_𝑘+1−𝑇_𝑘], 𝑧_𝑘,𝛼(𝑡)=0 for 𝑡∉[0,𝑇_𝑘+1−𝑇_𝑘] and 𝑚_𝑘(𝑡)=|𝑚(𝑇_𝑘−𝑇_𝑙+𝑡)|. Since 𝑚∈𝐿¹(0,𝑇) and 𝛽_𝑘+1∈𝐿²((0,𝑇_𝑘+1−𝑇_𝑘);𝑊¹₂(Ω)), we have 𝑚_𝑘∈𝐿¹(0,𝑇_𝑙−𝑇_𝑙−1+𝑇_𝑘+1−𝑇_𝑘) and 𝑧_𝑘,𝛼∈𝐿²(0,𝑇_𝑙−𝑇_𝑙−1+𝑇_𝑘+1−𝑇_𝑘). Due to the Young’s theorem for convolutions, we get 𝑚_𝑘∗𝑧_𝑘,𝛼∈𝐿²(0,𝑇_𝑙−𝑇_𝑙−1+𝑇_𝑘+1−𝑇_𝑘). Therefore, ‖𝐼_𝑘‖²_{𝐿2((0,𝑇𝑙−𝑇𝑙−1);𝑊12(Ω))}<∞. This implies that 𝑓_𝑙 belongs to 𝐿²((0,𝑇_𝑙−𝑇_𝑙−1);𝑊¹₂(Ω)). From the latter relation and 𝑎_𝑖𝑗∈𝐿^∞(Ω) we immediately have Φ^𝑙∈(𝐿²(Ω_{𝑇𝑙−𝑇𝑙−1}))^𝑛. Using the embedding theorem and Lemma 1 we see that 𝑎𝑓^𝑙 satisfies the property (14). Finally, the initial condition 𝑢^𝑙₀ belongs to 𝐿²(Ω), because 𝑢−𝑢_𝑇𝑙,  𝛽_𝑙+1∈𝐶([0,𝑇_𝑙+1−𝑇_𝑙],𝐿²(Ω)). All assumptions of Theorem 2 are satisfied for the problem for 𝛽_𝑙. Consequently, it possesses a unique weak solution in 𝒰₀(Ω_{𝑇𝑙−𝑇𝑙−1}).
Secondly, let us define the functions 𝛽^∗_𝑙(𝑥,𝑡)=^𝑁_𝑖=𝑙𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙+𝑡for(𝑥,𝑡)∈Ω_{𝑇𝑙−𝑇𝑙−1}, (71) where 𝑙=1…,𝑁 and 𝜓_𝑖 are the solutions of (58). We are going to show that 𝛽_𝑙=𝛽^∗_𝑙, 𝑙=1,…,𝑁. From the definition of 𝛽^∗_𝑙 using the value of 𝜓_𝑙(𝑥,0) and simple computations, we immediately get 𝛽^∗_𝑙(𝑥,0)=2𝑢𝑥,𝑇_𝑙;𝜔−𝑢_𝑇𝑙(𝑥)+Θ_𝑙𝛽^∗_𝑙+1𝑥,𝑇_𝑙+1−𝑇_𝑙,𝑙=1,…,𝑁. (72) Let us fix 𝑙=1,…,𝑁 and choose some 𝜂∈𝒯(Ω_{𝑇𝑙−𝑇𝑙−1}). We continue 𝜂 by the formulae 𝜂(𝑥,𝑡)=𝜂(𝑥,𝑇_𝑙−𝑇_𝑙−1) for 𝑡>𝑇_𝑙−𝑇_𝑙−1 and 𝜂(𝑥,𝑡)=𝜂(𝑥,0) for 𝑡<0. Further, let us define 𝜂_𝑖(𝑥,𝑡)=𝜂(𝑥,𝑇_𝑙−𝑇_𝑖+𝑡) where 𝑖=𝑙,…,𝑁. By the definition, it holds 𝜂_𝑖∈𝒯(Ω_𝑇𝑖).
Let us write down the weak form (31) for the problem for 𝜓_𝑖 (58) with the test function 𝜂_𝑖. We fix some 𝑡∈[0,𝑇_𝑙−𝑇_𝑙−1] and compute the difference of this weak problem with 𝑡 replaced by 𝑇_𝑖−𝑇_𝑙+𝑡 and 𝑡 replaced by 𝑇_𝑖−𝑇_𝑙 and take the sum over 𝑖=𝑙,…,𝑁. This results in the following expression: 0=𝑍₁+𝑍₂+𝑍₃+𝑍₄, (73) where 𝑍₁=^𝑁_𝑖=𝑙Ω𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙+𝑡𝜂_𝑖𝑥,𝑇_𝑖−𝑇_𝑙+𝑡−𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙𝜂_𝑖𝑥,𝑇_𝑖−𝑇_𝑙𝑍𝑑𝑥,₂=^𝑁_{𝑖=𝑙Ω𝑇𝑖𝑙−𝑇+𝑡⧵Ω𝑇𝑖𝑙−𝑇}−𝜓_𝑖𝜂_𝑖,𝑡+^𝑛_𝑠,𝑗=1𝑎_𝑠𝑗𝜓_{𝑖,𝑥𝑗}𝜂_{𝑖,𝑥𝑠}−𝑎𝜓_𝑖𝜂_𝑖+𝑑𝑥𝑑𝑡_{𝑁𝑖=𝑙Γ𝑖𝑙2,𝑇−𝑇+𝑡⧵Γ𝑖𝑙2,𝑇−𝑇}𝜗𝜓_𝑖𝜂_𝑖𝑍𝑑Γ𝑑𝑡,₃=^𝑁_{𝑖=𝑙Ω𝑇𝑖𝑙−𝑇+𝑡⧵Ω𝑇𝑖𝑙−𝑇}𝑎𝑚∗𝜓_𝑖𝜂_𝑖𝑍𝑑𝑥𝑑𝑡,₄=−^𝑁_{𝑖=𝑙Ω𝑇𝑖𝑙−𝑇+𝑡⧵Ω𝑇𝑖𝑙−𝑇}^𝑛_𝑠,𝑗=1𝑎_𝑠𝑗𝑚∗𝜓_{𝑖,𝑥𝑗}𝜂_{𝑖,𝑥𝑠}𝑑𝑥𝑑𝑡. (74) Using the definitions of 𝜂 and 𝛽^∗_𝑙 and the formula (72), we have 𝑍₁=_Ω𝛽^∗_𝑙(𝑥,𝑡)𝜂(𝑥,𝑡)−𝛽^∗_𝑙()𝜂(=𝑥,0𝑥,0)𝑑𝑥_Ω𝛽^∗_𝑙()𝜂()−2𝑢𝑥,𝑡𝑥,𝑡𝑥,𝑇_𝑙;𝜔−𝑢_𝑇𝑙(𝑥)+Θ_𝑙𝛽^∗_𝑙+1𝑥,𝑇_𝑙+1−𝑇_𝑙×𝜂(𝑥,0)𝑑𝑥. (75) Similarly, using the definitions of 𝜂 and 𝛽^∗_𝑙 and changing the variable of integration in 𝑍₂, we deduce 𝑍₂=_Ω𝑡−𝛽^∗_𝑙𝜂_𝑡+^𝑛_𝑠,𝑗=1𝑎_𝑠𝑗𝛽^∗_𝑙𝜂_𝑥𝑠−𝑎𝛽^∗_𝑙𝜂+𝑑𝑥𝑑𝑡_Γ2,𝑡𝜗𝛽^∗_𝑙𝜂𝑑Γ𝑑𝑡. (76) By the change of variable, the quantity 𝑍₃ is transformed to 𝑍₃=_Ω𝑡𝑎(𝑥)^𝑁_𝑖=𝑙𝑚∗𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙+𝑡𝜂(𝑥,𝑡)𝑑𝑥𝑑𝑡. (77) Let us consider the term ∑^𝑁_𝑖=𝑙(𝑚∗𝜓_𝑖)(𝑥,𝑇_𝑖−𝑇_𝑙+𝑡) in the latter formula. We compute ^𝑁_𝑖=𝑙𝑚∗𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙=+𝑡_{𝑁𝑖=𝑙𝑇𝑖−𝑇𝑙0+𝑡}𝑚(𝜏)𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙=+𝑡−𝜏𝑑𝜏_𝑡0𝑚(𝜏)^𝑁_𝑖=𝑙𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙++𝑡−𝜏𝑑𝜏_{𝑁𝑖=𝑙𝑇𝑖−𝑇𝑙𝑡+𝑡}𝑚(𝜏)𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑙=+𝑡−𝜏𝑑𝜏_𝑡0𝑚(𝜏)𝛽^∗_𝑙()+𝑥,𝑡−𝜏𝑑𝜏_{𝑁𝑖=𝑙𝑖−1𝑘=𝑙𝑇𝑘+1−𝑇𝑘0}𝑚𝑇_𝑘−𝑇_𝑙+𝑡+𝜏𝜓_𝑖𝑥,𝑇_𝑖−𝑇_𝑘=−𝜏𝑑𝜏𝑚∗𝛽^∗_𝑙()+𝑥