Optimal Control of the Ill-Posed Cauchy Elliptic Problem

HAL Id: hal-01891254

https://hal.univ-guyane.fr/hal-01891254

Submitted on 15 Oct 2018

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Problem

A. Berhail, A. Omrane

To cite this version:

A. Berhail, A. Omrane.

Optimal Control of the Ill-Posed Cauchy Elliptic Problem.

Interna-tional Journal of Differential Equations, Hindawi Publishing Corporation, 2015, 2015, pp.468918.

�10.1155/2015/468918�. �hal-01891254�

Research Article

Optimal Control of the Ill-Posed Cauchy Elliptic Problem

A. Berhail

and A. Omrane

1_{University of 08 May 1945, 24000 Guelma, Algeria}

2_{UMR 228 Espce-Dev, Universit´e de Guyane, UR, UM2, UNC, Campus de Troubiran, 97337 Cayenne, French Guiana}

Correspondence should be addressed to A. Omrane; aomrane@gmail.com Received 24 July 2015; Revised 17 October 2015; Accepted 22 October 2015 Academic Editor: Jingxue Yin

Copyright © 2015 A. Berhail and A. Omrane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give a characterization of the control for ill-posed problems of oscillating solutions. More precisely, we study the control of Cauchy elliptic problems via a regularization approach which generates incomplete information. We obtain a singular optimality system characterizing the no-regret control for the Cauchy problem.

1. Introduction

Cauchy problems for partial differential equations of elliptic type are present in many physical systems such as plasma physics [1], mechanical engineering [2], or electrocardiog-raphy [3]. One of the important examples is the Helmholtz equation and its applications in acoustic, wave propagation and scattering, vibration of the structure, and electromag-netic scattering (see [4–6] and the references therein). Here, we investigate a Cauchy problem for the Laplacian elliptic operator:

Δ𝑧 = 0, (1)

where 𝑧 = 𝑧(𝑥), on an open set Ω ⊂ R3 of class C2, of boundary𝜕Ω = Γ. Dirichlet and Neumann conditions are prescribed on a part of the boundaryΓ₀⊂ Γ, Γ₀ ̸= Γ:

𝑧 = V₀, 𝜕𝑧 𝜕] = V1

onΓ₀.

(2)

The goal is to reconstruct the solution onΩ and its trace on Γ1 = Γ \ Γ0from a perturbed system, under the assumption

that the solution exists for the exact data V₀ and V₁ which here are control variables. This problem is a classical example

of ill-posed problems. So, regularization methods may be considered.

Theoretical concepts and also computational implemen-tation related to the Cauchy problem of the elliptic equation have been discussed by many authors, and a lot of methods were provided (see, e.g., Qian et al. [7] for the 4th-order regularization method or Xiong and Fu [8]). Generally, it is assumed that instead of exact data some noisy boundary conditionsV𝜀₀andV𝜀₁are given with the error bound.

But, in this paper, we consider another regularization method of the Laplacian, where we introduce a new data:

𝑧 = 𝑔₀, 𝜕𝑧 𝜕] = 𝑔1

onΓ₁

(3)

with𝑔₀ and 𝑔₁ being unknown functions. Hence, we have a regularized problem but with incomplete data. A special optimal control method of problems of incomplete data should then be applied. We here use a method that we find well adapted: the low-regret control concept, introduced by Lions in the late 80s (see, e.g., [9, 10] and the references therein).

In [11, 12], the control of distributed system with incom-plete data is performed. The proof of the existence and

Hindawi Publishing Corporation

International Journal of Differential Equations Volume 2015, Article ID 468918, 9 pages http://dx.doi.org/10.1155/2015/468918

characterization of the no-regret control is obtained as the limit of the low-regret control. Here, we admit the possibility of making a choice of controlsV slightly worse than by doing

better thanV = 0 (i.e., better than a noncontrolled system),

with respect to certain criteria (cost function): 𝐽 (𝑢𝛾, 𝑔) ≤ 𝐽 (0, 𝑔) + 𝛾 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2_𝑌

∀𝑔 in a Hilbert space 𝑌, (4) where𝛾 is the small positive parameter that tends to 0, with 𝑔 being the pollution or the incomplete data.

This method is previously introduced by Savage [13] in statistics. Lions was the first to use it to control distributed systems of incomplete data, motivated by a number of applications in economics and ecology. In this paper, we generalize the method to ill-posed problems of elliptic type.

It seems that the control of Cauchy system for elliptic operators is globally an open problem. Lions in [14] proposed a method of approximation by penalization and obtained a singular optimality system, under a supplementary hypothe-sis of Slater type. In [15], Sougalo and Nakoulima analyzed the Cauchy problem using a regularization method, consisting in viewing a singular problem as a limit of a family of well-posed problems. They have obtained a singular optimality system for the considered control problem, also assuming the Slater condition. Unfortunately, the recent paper by Massengo Mophou and Nakoulima [16] is the same as the one by Sougalo and Nakoulima (1998) using the same old references, and nothing new is brought.

In the present paper, we use another approximation method which consists in considering the elliptic Cauchy problem as a singular limit of sequence of well-posed elliptic problems, where the Slater condition is not used and where we apply the low-regret control notion. The same analysis can be generalized to the Helmholtz equation with no difficulty.

The paper is organized as follows. In Section 2, we present the regularization method. In Section 3, the optimal control of the regularized system is discussed and the approximated optimality system is presented. We pass to the limit in the last section; we show that we obtain a singular optimality system for the low-regret and no-regret controls to the original problem of Laplacian.

2. Existence of Solutions to Cauchy

Elliptic Problems

LetΩ be an open bounded subset of 𝑅𝑛, with a boundaryΓ of class𝐶2,Γ = Γ₀∪Γ₁withΓ₀∩Γ₁= 0. The boundaries Γ₀andΓ₁ are nonempty and are of positive measure. We consider here the problem: Δ𝑧 = 0 in Ω, 𝑧 = V₀, 𝜕𝑧 𝜕] = V1 onΓ₀, (5)

with𝑧 ∈ 𝐿2(Ω) and (V₀, V₁) ∈ 𝐿2(Γ₀) × 𝐿2(Γ₀). Problem (5) is a Cauchy problem for the Laplacian operator. It is well known that it is ill-posed in the sense that it does not admit a solution in general and that existing solutions (if any) are unstable. This problem is present in many applications, so it is important to control the Cauchy data.

Denote by𝐴 the closed subset of (𝐿2(Γ₀))2×𝐿2(Ω) defined by 𝐴 = {((V₀, V₁) , 𝑧) ∈ (𝐿2(Γ₀))2× 𝐿2(Ω) , Δ𝑧 = 0 in Ω, 𝑧|Γ0= V0, 𝜕𝑧_𝜕]󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 Γ0 = V₁} , (6)

and suppose that𝐴 ̸= 0. We will call any control-state pair (V₀, V₁, 𝑧) ∈ 𝐴 admissible couple.

Let𝐽 be a strictly convex cost functional, defined for all admissible control-state couples(V₀, V₁, 𝑧) by

𝐽 (V₀, V_{1, 𝑧) = 󵄩󵄩󵄩󵄩𝑧 − 𝑧𝑑}󵄩󵄩󵄩󵄩2_𝐿2_(Ω)+ 𝑁₀󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩2_𝐿2_(Γ

0)

+ 𝑁₁󵄩󵄩󵄩󵄩V₁󵄩󵄩󵄩󵄩2_𝐿2_(Γ0),

(7)

where𝑧_𝑑and (𝑁₀, 𝑁₁) are, respectively, given in𝐿2(Ω) and in (𝑅₊\ {0})2_{. We want to find the couple control-state solution}

of

inf𝐽 (V₀, V₁, 𝑧) , (V₀, V₁, 𝑧) ∈ 𝐴. (8)

According to the structure of𝐽, problem (8) admits a unique solution(𝑢₀, 𝑢₁, 𝑦) that we should characterize. To obtain a singular optimality system (SOS) associated with(𝑢₀, 𝑢₁, 𝑦), Lions [14] has proposed a method of approximation by penalization. He obtained SOS, under the supplementary hypothesis of Slater type:

The admissible set of controls has a nonempty interior. (9)

Here, we do not consider Slater hypothesis, but instead we consider the no-regret and low-regret techniques.

3. The Low-Regret and No-Regret Control

Due to the ill-posedness of the Cauchy elliptic problem, it is impossible to solve it directly [17]. This requires special techniques as the technique of regularization. Our method consists in regularizing (5) into an elliptic problem of

International Journal of Differential Equations 3

incomplete data. For any𝜀 > 0, we consider the regularized problem: Δ2𝑧_𝜀+ 𝜀𝑧_𝜀= 0 in Ω, 𝑧_𝜀−𝜕Δ𝑧𝜀 𝜕] = V0, 𝜕𝑧_𝜀 𝜕] + Δ𝑧𝜀= V1 onΓ₀, 𝜀𝑧_𝜀−𝜕Δ𝑧_𝜕]𝜀 = 𝜀𝑔₀, 𝜀𝜕𝑧_𝜕]𝜀 + Δ𝑧_𝜀= 𝜀𝑔₁ onΓ₁. (10)

We denoteV fl (V₀, V₁) and 𝑔 fl (𝑔₀, 𝑔₁) for simplicity. We begin by an important remark.

Remark 1. For every fixed 𝜀𝑔₀ and 𝜀𝑔₁ we assume the

existence of a unique solution to (10). Indeed, in the following subsections,𝜀𝑔₀and𝜀𝑔₁are considered as data perturbations, and the solution of (10) may not exist in the sense of Hadamard [18].

3.1. Back to the Original Problem. We show how to come back

to the original Cauchy problem, starting from (10). Indeed, if we put𝜀 = 0 and we do the change of variables 𝜂 = Δ𝑧, we obtain Δ𝜂 = 0 in Ω, 𝜕𝜂 𝜕] = 0, 𝜂 = 0 onΓ₁, (11) 𝑧 −𝜕𝜂 𝜕] = V0, 𝜕𝑧 𝜕] + 𝜂 = V1 onΓ₀. (12)

Using the uniqueness property of the solution of the Laplace equation and the unique continuation theorem of Mizohata [19], we deduce from (11) that we also have

𝜕𝜂

𝜕] = 𝜂 = 0 on Γ0. (13)

Hence, conditions (12) become

𝑧 = V₀, 𝜕𝑧 𝜕] = V1

onΓ₀,

(14)

that is, the same conditions of the original problem (5).

3.2. Cost Function and Low-Regret Control. Consider the cost

functional

𝐽_{𝜀(V, 𝑔) = 󵄩󵄩󵄩󵄩𝑧𝜀}(V, 𝑔) − 𝑧_𝑑󵄩󵄩󵄩󵄩2_𝐿2_(Ω)+ 𝑁₀󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩2_𝐿2_(Γ0)

+ 𝑁₁󵄩󵄩󵄩󵄩V₁󵄩󵄩󵄩󵄩2_𝐿2_(Γ0)

(15)

that we want to minimize in the context of no-regret control due to the presence of the incomplete data𝑔.

Definition 2. We say that𝑢 ∈ (𝐿2(Γ₀))2is a no-regret control

for (5)–(15), if𝑢 is a solution to the following problem:

inf

V∈(𝐿2_(Γ

0))2

( sup

𝑔∈(𝐿2_(Γ1))2(𝐽𝜀(V, 𝑔) − 𝐽𝜀(0, 𝑔))) . (16)

As seen in [11], the no-regret control is difficult to characterize directly. Below, we define the low-regret control which tends to the no-regret control when the parameter of penalization tends to zero. In the case of no pollution𝑔, the no-regret control and the classical control are the same.

3.2.1. The Low-Regret Control. As in [20], we define the

low-regret control as the solution to the following MinMax problem: inf V∈(𝐿2_(Γ0))2( sup 𝑔∈(𝐿2_(Γ 1))2 [𝐽_𝜀(V, 𝑔) − 𝐽_{𝜀(0, 𝑔) − 𝛾 󵄩󵄩󵄩󵄩𝑔0}󵄩󵄩󵄩󵄩2_𝐿2_(Γ1)− 𝛾 󵄩󵄩󵄩󵄩𝑔₁󵄩󵄩󵄩󵄩2_𝐿2_(Γ1)]) , (17)

where 𝛾 is a strictly positive parameter. The solution to problem (17), if it exists, will be the low-regret control.

Now, we introduce𝜉_𝜀 fl 𝜉_𝜀(V, 0) solution to the adjoint problem

Δ2𝜉_𝜀+ 𝜀𝜉 = 𝑧_𝜀(V, 0) in Ω, 𝜉_𝜀− 𝜕 𝜕](Δ𝜉𝜀) = 0, 𝜕𝜉_𝜀 𝜕] + Δ𝜉𝜀= 0 onΓ₀, 𝜀𝜉_𝜀− 𝜕 𝜕](Δ𝜉𝜀) = 0, 𝜀𝜕𝜉𝜀 𝜕] + Δ𝜉𝜀= 0 onΓ₁. (18)

Then we have the following result.

Proposition 3. The low-regret control is the solution to the classical optimal control problem

inf V∈(𝐿2_(Γ0))2J 𝛾 𝜀(V) (19) with J𝛾_𝜀(V) fl 𝐽_𝜀(V, 0) − 𝐽_𝜀(0, 0) +𝜀_𝛾2_{(󵄩󵄩󵄩󵄩𝜉𝜀}󵄩󵄩󵄩󵄩2_𝐿2_(Γ1)+󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝜉𝜀 𝜕]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 𝐿2_(Γ 1) ) . (20)

Proof. After simple computations we have

𝐽_𝜀(V, 𝑔) − 𝐽_𝜀(0, 𝑔) = 𝐽_𝜀(V, 0) − 𝐽_𝜀(0, 0)

+ 2 ⟨𝑧_𝜀(V, 0) , 𝑧_𝜀(0, 𝑔)⟩ , (21) where ⟨⋅, ⋅⟩ is the inner product in 𝐿2(Ω). To estimate the integral⟨𝑧_𝜀(V, 0), 𝑧_𝜀(0, 𝑔)⟩ in (21), we use the Green formula:

⟨Δ2𝑧, 𝜓⟩ = ⟨𝑧, Δ2𝜓⟩ + ⟨_𝜕]𝜕 (Δ𝑧) , 𝜓⟩ Γ − ⟨Δ𝑧,𝜕𝜓_𝜕]⟩ Γ+ ⟨ 𝜕𝑧 𝜕], Δ𝜓⟩_Γ − ⟨𝑧,_𝜕]𝜕 (Δ𝜓)⟩ Γ, (22)

together with (18). We have

⟨𝑧_𝜀(V, 0) , 𝑧_𝜀(0, 𝑔)⟩ = ⟨Δ2𝜉_𝜀+ 𝜀𝜉_𝜀, 𝑧_𝜀(0, 𝑔)⟩ = 0 + ⟨_𝜕]𝜕 (Δ𝜉_𝜀) , 𝑧_𝜀⟩ Γ − ⟨Δ𝜉_𝜀,𝜕𝑧_𝜕]𝜀⟩ Γ + ⟨𝜕𝜉𝜀 𝜕], Δ𝑧𝜀⟩Γ − ⟨𝜉_𝜀,_𝜕]𝜕 (Δ𝑧_𝜀)⟩ Γ = −𝜀 ⟨𝜉_𝜀, 𝑧_𝜀⟩_Γ1 − ⟨𝜉_𝜀,_𝜕]𝜕 (Δ𝑧_𝜀)⟩ Γ1 + ⟨𝜀𝜕𝜉_𝜕]𝜀,𝜕𝑧_𝜕]𝜀⟩ Γ1 + ⟨𝜕𝜉𝜀 𝜕], Δ𝑧𝜀⟩Γ1 = ⟨𝜉_𝜀, 𝜀𝑔₀⟩_Γ 1+ ⟨ 𝜕𝜉_𝜀 𝜕], 𝜀𝑔1⟩Γ1, (23) where𝑧_𝜀= 𝑧_𝜀(0, 𝑔). Then sup 𝑔∈(𝐿2_(Γ1))2(2 ⟨𝑧𝜀(V, 0) , 𝑧𝜀(0, 𝑔)⟩ − 𝛾 󵄩󵄩󵄩󵄩𝑔0󵄩󵄩󵄩󵄩 2 𝐿2_(Γ1) − 𝛾 󵄩󵄩󵄩󵄩𝑔1󵄩󵄩󵄩󵄩2𝐿2_(Γ1)) = sup 𝑔∈(𝐿2_(Γ1))2(⟨𝜉𝜀, 𝜀𝑔0⟩Γ1 + ⟨𝜕𝜉_𝜕]𝜀, 𝜀𝑔1⟩ Γ1− 𝛾 󵄩󵄩󵄩󵄩𝑔 0󵄩󵄩󵄩󵄩2𝐿2_(Γ1)− 𝛾 󵄩󵄩󵄩󵄩𝑔₁󵄩󵄩󵄩󵄩2_𝐿2_(Γ1)) =𝜀2 𝛾 󵄩󵄩󵄩󵄩𝜉𝜀󵄩󵄩󵄩󵄩 2 𝐿2_(Γ 1)+ 𝜀2 𝛾 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝜕𝜉_𝜀 𝜕]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 𝐿2_(Γ1) (24)

thanks to the conjugate formula. Combining (17) and (21), we obtain the desired result.

Remark 4. The no-regret control is obtained by the passage

to the limit in the positive parameter 𝛾: it is the weak convergence of the control-state variables of the perturbed system, which corresponds to the limit of the standard low-regret control sequence.

3.3. Approached Optimality System. For the general theory of

the characterization of the low-regret optimal control see [10– 12]. In this paper, we generalize to the ill-posed problems of elliptic type (5).

Proposition 5. Problem (19)-(20) admits a unique solution 𝑢𝛾_𝜀

called the low-regret control.

Proof. We haveJ𝛾_𝜀(V) ≥ −𝐽_𝜀(0, 0), ∀V ∈ (𝐿2(Γ₀))2. Then

𝑑𝛾_𝜀 = inf

V∈(𝐿2_(Γ0))2J

𝛾

𝜀(V) (25)

exists. LetV_𝑛 = V_𝑛(𝜀, 𝛾) be a minimizing sequence such that 𝑑𝛾

𝜀 = lim𝑛 → ∞J𝛾𝜀(V𝑛). Then we have

−𝐽_𝜀(0, 0) ≤ 𝐽_𝜀(V_𝑛, 0) − 𝐽_𝜀(0, 0) +𝜀2 𝛾 (󵄩󵄩󵄩󵄩𝜉𝜀(V𝑛)󵄩󵄩󵄩󵄩 2 𝐿2_(Γ1)+󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝜉𝜀 𝜕] (V𝑛)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 𝐿2_(Γ1)) ≤ 𝑑𝛾 𝜀+ 1. (26)

International Journal of Differential Equations 5

And we deduce the bounds 󵄩󵄩󵄩󵄩V𝑛󵄩󵄩󵄩󵄩(𝐿2_(Γ 0))2 ≤ 𝑐 𝛾 𝜀, 󵄩󵄩󵄩󵄩𝑧𝜀(V𝑛, 0) − 𝑧𝑑󵄩󵄩󵄩󵄩𝐿2_(Ω)≤ 𝑐_𝜀𝛾, 𝜀 √𝛾󵄩󵄩󵄩󵄩𝜉𝜀(V𝑛)󵄩󵄩󵄩󵄩𝐿2(Γ1)≤ 𝑐 𝛾 𝜀, 𝜀 √𝛾󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝜕𝜉_𝜀 𝜕] (V𝑛)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝐿2_(Γ 1) ≤ 𝑐_𝜀𝛾, (27)

where the constant𝑐_𝜀𝛾(independent of𝑛) is not the same each time.

Hence, there exists𝑢𝛾_𝜀 ∈ (𝐿2(Γ₀))2such thatV_𝑛(𝜀, 𝛾) ⇀ 𝑢𝛾

𝜀 weakly in the Hilbert space(𝐿2(Γ0))2. Also,𝑧𝜀(V𝑛, 0) →

𝑧_𝜀(𝑢𝛾

𝜀, 0) (continuity with respect to the data). We also deduce

from the strict convexity of the cost functionJ𝛾_𝜀 that 𝑢𝛾_𝜀 is unique.

Now we give the optimality system for the approximate low-regret control𝑢𝛾_𝜀. We denote𝑦_𝜀𝛾 fl 𝑧_𝜀(𝑢𝛾_𝜀, 0). Then, we proceed as in [20]. We first have the following.

Proposition 6. The approached low-regret control 𝑢𝛾_𝜀 fl

(𝑢𝛾_0𝜀, 𝑢𝛾_1𝜀) solution to (19)-(20) is characterized by the unique

solution{𝑦_𝜀𝛾, 𝜉𝛾_𝜀, 𝜌_𝜀𝛾, 𝑝𝛾_𝜀} of the optimality system

Δ2𝑦_𝜀𝛾+ 𝜀𝑦_𝜀𝛾= 0, Δ2𝜉𝛾_𝜀 + 𝜀𝜉_𝜀𝛾= 𝑦_𝜀𝛾, Δ2𝜌_𝜀𝛾+ 𝜀𝜌_𝜀𝛾= 0, Δ2𝑝𝛾_𝜀 + 𝜀𝑝_𝜀𝛾= 𝑦_𝜀𝛾− 𝑧_𝑑− 𝜌_𝜀𝛾 𝑖𝑛 Ω, 𝑦𝛾_𝜀 − 𝜕 𝜕](Δ𝑦𝜀𝛾) = 𝑢𝛾0𝜀, 𝜕𝑦𝛾_𝜀 𝜕] + Δ𝑦𝜀𝛾= 𝑢𝛾1𝜀, 𝜉_𝜀− 𝜕 𝜕](Δ𝜉𝜀) = 0, 𝜕𝜉_𝜀 𝜕] + Δ𝜉𝜀= 0, 𝜌_𝜀𝛾− 𝜕 𝜕](Δ𝜌𝜀𝛾) = 0, 𝜕𝜌_𝜀𝛾 𝜕] + Δ𝜌𝜀𝛾= 0, 𝑝𝛾_𝜀 − 𝜕 𝜕](Δ𝑝𝛾𝜀) = 0, 𝜕𝑝𝛾_𝜀 𝜕] + Δ𝑝𝜀𝛾= 0 𝑜𝑛 Γ0, 𝜀𝑦𝛾_𝜀 −_𝜕]𝜕 (Δ𝑦_𝜀𝛾) = 0, 𝜀𝜕𝑦𝛾𝜀 𝜕] + Δ𝑦𝜀𝛾= 0, 𝜀𝜉𝜀−_𝜕]𝜕 (Δ𝜉𝜀) = 0, 𝜀𝜕𝜉𝜀 𝜕] + Δ𝜉𝜀= 0, 𝜀𝜌_𝜀𝛾− 𝜕 𝜕](Δ𝜌𝜀𝛾) = 𝜀 2 𝛾𝜉𝜀, 𝜀𝜕𝜌𝜀𝛾 𝜕] + Δ𝜌𝜀𝛾= −𝜀 2 𝛾 𝜕𝜉_𝜀 𝜕] 𝜀𝑝𝛾_𝜀 − 𝜕 𝜕](Δ𝑝𝛾𝜀) = 0, 𝜀𝜕𝑝_𝜕]𝛾𝜀 + Δ𝑝𝛾_𝜀 = 0 𝑜𝑛 Γ₁, (28)

with the adjoint equation

𝑝_𝜀𝛾+ 𝑁₀𝑢𝛾_0𝜀+ 𝑁₁𝑢𝛾_1𝜀= 0 𝑖𝑛 𝐿2(Γ₀) . (29)

Proof. Let𝑢𝛾_𝜀be the solution of (19)-(20) on𝐿2(Γ₀). The

Euler-Lagrange necessary condition gives for every𝑤 fl (𝑤₀, 𝑤₁) ∈ (𝐿2_(Γ 0))2 ⟨𝑦𝛾_𝜀 − 𝑧𝑑, 𝑧𝜀(𝑤, 0)⟩ + 𝑁0⟨𝑢𝛾0𝜀, 𝑤0⟩Γ₀+ 𝑁1⟨𝑢𝛾1𝜀, 𝑤1⟩Γ₀ + ⟨𝜀_𝛾2𝜉𝜀𝛾, 𝜉𝜀(𝑤)⟩ Γ1 + ⟨𝜀_𝛾2𝜕𝜉_𝜕]𝛾𝜀,𝜕𝜉_𝜕]𝜀 (𝑤)⟩ Γ1 = 0, (30)

where𝜉𝛾_𝜀 fl 𝜉_𝜀(𝑢𝛾_𝜀, 0). Denoting 𝜌_𝜀𝛾 = 𝜌(𝑢𝛾_𝜀, 0) as the unique solution to Δ2𝜌_𝜀𝛾+ 𝜀𝜌_𝜀𝛾= 0 in Ω, 𝜌_𝜀𝛾−_𝜕]𝜕 (Δ𝜌_𝜀𝛾) = 0, 𝜕𝜌𝛾 𝜀 𝜕] + Δ𝜌𝜀𝛾= 0 onΓ₀, 𝜀𝜌_𝜀𝛾−_𝜕]𝜕 (Δ𝜌_𝜀𝛾) = 𝜀_𝛾2𝜉𝛾_𝜀, 𝜀𝜕𝜌_𝜕]𝜀𝛾 + Δ𝜌_𝜀𝛾= −𝜀_𝛾2𝜕𝜉_𝜕]𝛾𝜀 onΓ₁, (31)

we have by the Green formula 0 = ⟨Δ2𝜌_𝜀𝛾+ 𝜀𝜌_𝜀𝛾, 𝜉_𝜀(𝑤, 0)⟩ = ⟨𝜌_𝜀𝛾, 𝑧_𝜀(𝑤, 0)⟩ + ⟨𝜀2 𝛾𝜉𝛾𝜀, 𝜉𝜀(𝑤, 0)⟩ Γ1 + ⟨𝜀_𝛾2𝜕𝜉𝜀𝛾 𝜕], 𝜕𝜉_𝜀 𝜕] (𝑤, 0)⟩_Γ1. (32)

And as it is classical, we introduce the adjoint state𝑝𝛾_𝜀 fl 𝑝(𝑢𝛾 𝜀, 0) defined by Δ2_𝑝𝛾 𝜀 + 𝜀𝑝𝛾𝜀 = 𝑦𝜀𝛾− 𝑧𝑑− 𝜌𝜀𝛾 inΩ, 𝑝𝛾_𝜀 − 𝜕 𝜕](Δ𝑝𝛾𝜀) = 0, 𝜕𝑝𝛾 𝜀 𝜕] + Δ𝑝𝛾𝜀 = 0 onΓ₀, 𝜀𝑝𝛾_𝜀 −_𝜕]𝜕 (Δ𝑝𝛾_𝜀) = 0, 𝜀𝜕𝑝𝛾𝜀 𝜕] + Δ𝑝𝛾𝜀 = 0 onΓ₁, (33)

and, using again the Green formula, we obtain ⟨𝑦𝛾_𝜀 − 𝑧𝑑− 𝜌𝜀𝛾, 𝑧𝜀(𝑤, 0)⟩ = ⟨𝑝𝜀𝛾, 𝑤⟩Γ0,

∀𝑤 ∈ (𝐿2(Γ₀))2. (34) And then (30) becomes

⟨𝑝_𝜀𝛾+ 𝑁₀𝑢𝛾_0𝜀+ 𝑁₁𝑢𝛾_1𝜀, 𝑤⟩_Γ0= 0, ∀𝑤 ∈ (𝐿2(Γ₀))2 (35) which is (29).

4. Singular Optimality System (SOS)

In this section, we give the SOS for the low-regret control for the Cauchy problem (5). We first show the following estimates.

Lemma 7. There is a positive constant 𝐶 such that

󵄩󵄩󵄩󵄩𝑢𝛾 𝜀0󵄩󵄩󵄩󵄩𝐿2_(Γ0)≤ 𝐶, 󵄩󵄩󵄩󵄩𝑢𝛾 𝜀1󵄩󵄩󵄩󵄩𝐿2_(Γ0)≤ 𝐶, 󵄩󵄩󵄩󵄩𝑦𝛾 𝜀󵄩󵄩󵄩󵄩𝐿2_(Ω)≤ 𝐶, 𝜀 √𝛾󵄩󵄩󵄩󵄩𝜉 𝛾 𝜀󵄩󵄩󵄩󵄩𝐿2_(Γ1)≤ 𝐶, 𝜀 √𝛾󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 𝜕𝜉𝛾 𝜀 𝜕]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩𝐿2_(Γ 1) ≤ 𝐶. (36)

Proof. Since𝑢𝛾_𝜀is the approximate low-regret control, we have

𝐽_𝜀𝛾(𝑢𝛾_𝜀) ≤ 𝐽_𝜀𝛾(V) , ∀V ∈ (𝐿2(Γ₀))2. (37) In the particular case whereV = 0, we obtain

󵄩󵄩󵄩󵄩𝑦𝛾 𝜀 − 𝑧𝑑󵄩󵄩󵄩󵄩2𝐿2_(Ω)+ 𝑁₀󵄩󵄩󵄩󵄩𝑢𝛾_𝜀0󵄩󵄩󵄩󵄩2_𝐿2_(Γ0)+ 𝑁₁󵄩󵄩󵄩󵄩𝑢𝛾_𝜀1󵄩󵄩󵄩󵄩2_𝐿2_(Γ0) +𝜀_𝛾2_{(󵄩󵄩󵄩󵄩𝜉𝜀}𝛾󵄩󵄩󵄩󵄩2_𝐿2_(Γ1)+󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩 𝜕𝜉𝛾 𝜀 𝜕]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 2 𝐿2_(Γ 1) ) ≤ 󵄩󵄩󵄩󵄩𝑧𝜀(0, 0) − 𝑧𝑑󵄩󵄩󵄩󵄩2𝐿2_(Ω) +𝜀2 𝛾 (󵄩󵄩󵄩󵄩𝜉𝜀(0, 0)󵄩󵄩󵄩󵄩 2 𝐿2_(Γ 1)+󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝜕𝜉_𝜀 𝜕] (0, 0)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 𝐿2_(Γ1)) . (38) But, 𝑧𝜀(0, 0) = 0 in Ω, 𝜉_𝜀(0, 0) = 𝜕𝜉_𝜕]𝜀 (0, 0) = 0 on Γ₁. (39) Then 󵄩󵄩󵄩󵄩𝑦𝛾 𝜀 − 𝑧𝑑󵄩󵄩󵄩󵄩2𝐿2_(Ω)+ 𝑁₀󵄩󵄩󵄩󵄩𝑢𝛾_𝜀0󵄩󵄩󵄩󵄩2_𝐿2_(Γ0)+ 𝑁₁󵄩󵄩󵄩󵄩𝑢𝛾_𝜀1󵄩󵄩󵄩󵄩2_𝐿2_(Γ0) +𝜀2 𝛾 (󵄩󵄩󵄩󵄩𝜉𝛾𝜀󵄩󵄩󵄩󵄩2𝐿2_(Γ 1)+ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 𝜕𝜉𝛾 𝜀 𝜕]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 2 𝐿2_(Γ1)) ≤ 󵄩󵄩󵄩󵄩𝑧𝑑󵄩󵄩󵄩󵄩 2 𝐿2_(Ω)= 𝐶. (40)

Remark 8. From Section 3.1, we showed how to come back

from the bi-Laplacian problem to the original one. We will use these techniques in this last part.

Theorem 9. The low-regret control 𝑢𝛾 _{for problem (5) is}

characterized by the unique solution {𝑦𝛾, 𝜉𝛾, 𝜌𝛾, 𝑝𝛾} of the

optimality system: Δ𝑦𝛾 = 0, Δ𝜉𝛾 = 0, Δ𝜌𝛾 = 0, Δ𝑝𝛾 = 𝑦𝛾− 𝑧𝑑+ 𝜌𝛾 𝑖𝑛 Ω, 𝑦𝛾 = 𝑢𝛾₀, 𝜉𝛾 = 0, 𝜌𝛾 = 0, 𝑝𝛾 = 0, 𝜕𝑦𝛾 𝜕] = 𝑢𝛾1, 𝜕𝜉𝛾 𝜕] = 0,

International Journal of Differential Equations 7 𝜕𝜌𝛾 𝜕] = 0, 𝜕𝑝𝛾 𝜕] = 0 𝑜𝑛 Γ₀, (41)

with the adjoint equation

𝑝𝛾+ 𝑁₀𝑢𝛾₀+ 𝑁₁𝑢𝛾₁= 0 𝑖𝑛 𝐿2(Γ₀) . (42)

Proof. From the optimality system (28) in Proposition 6, we

deduce that𝑦𝛾_𝜀 is solution of the system Δ2𝑦_𝜀𝛾+ 𝜀𝑦_𝜀𝛾= 0, in Ω, 𝑦𝛾_𝜀 −_𝜕]𝜕 (Δ𝑦𝛾_𝜀) = 𝑢𝛾_0𝜀, 𝜕𝑦𝛾 𝜀 𝜕] + Δ𝑦𝜀𝛾= 𝑢𝛾1𝜀 onΓ₀, 𝜀𝑦𝛾_𝜀 − 𝜕 𝜕](Δ𝑦𝛾𝜀) = 0, 𝜀𝜕𝑦_𝜕]𝛾𝜀 + Δ𝑦_𝜀𝛾= 0, onΓ₁. (43) As in Section 3.1, we denote 𝜂𝛾_𝜀 = Δ𝑦_𝜀𝛾. (44) Then system (43) is written as

Δ𝜂𝛾_𝜀 + 𝜀𝑦𝛾_𝜀 = 0, in Ω, 𝑦_𝜀𝛾−𝜕𝜂𝜀𝛾 𝜕] = 𝑢 𝛾 0𝜀, 𝜕𝑦_𝜀𝛾 𝜕] + 𝜂𝛾𝜀 = 𝑢𝛾1𝜀 onΓ₀, 𝜀𝑦𝛾 𝜀 −𝜕𝜂 𝛾 𝜀 𝜕] = 0, 𝜀𝜕𝑦𝜀𝛾 𝜕] + 𝜂𝛾𝜀 = 0, onΓ₁. (45)

From estimates (36) of Lemma 7, sequence(𝑦𝛾_𝜀) is bounded in𝐿2(Ω) by constant 𝑀. Hence, there exists a subsequence still denoted by(𝑦𝛾_𝜀), such that

𝑦𝛾

𝜀 ⇀ 𝑦𝛾 weakly in𝐿2(Ω) (46)

as𝜀 → 0. We deduce from the first equation in (45) that 󵄩󵄩󵄩󵄩Δ𝜂𝛾

𝜀󵄩󵄩󵄩󵄩 ≤ 𝜀󵄩󵄩󵄩󵄩𝑦𝜀𝛾󵄩󵄩󵄩󵄩 ≤ 𝜀𝑀 󳨀→ 0. (47)

That is,

Δ𝜂_𝜀𝛾⇀ 0 weakly in 𝐿2(Ω) . (48) Using the same arguments and from the last two equations in (45) we have 𝜕𝜂𝛾 𝜀 𝜕] ⇀ 0, 𝜂𝛾 𝜀 ⇀ 0 in 𝐿2(Γ₁) (49) when𝜀 → 0. We resume by Δ𝜂𝛾= 0 in Ω, 𝜕𝜂𝛾 𝜕] = 0, 𝜂𝛾= 0 onΓ₁. (50)

Using the unique continuation theorem of Mizohata [19], we deduce from (50) that we also have

𝜂𝛾 ≡ 0 everywhere on Ω. (51) Then,

𝜕𝜂𝛾

𝜕] = 𝜂𝛾 = 0 on Γ0. (52) From another side, estimates (36) also give

(𝑢𝛾_𝜀0, 𝑢𝛾_𝜀1) ⇀ (𝑢𝛾₀, 𝑢𝛾₁) weakly in 𝐿2(Γ₀) × 𝐿2(Γ₀) . (53) We come back to the notation 𝜂𝛾 = Δ𝑦𝛾. System (45) transforms to Δ𝑦𝛾= 0, in Ω, 𝑦𝛾_{= 𝑢}𝛾 0, 𝜕𝑦𝛾 𝜕] = 𝑢𝛾1 onΓ₀. (54)

Again, we use the estimates of Lemma 7 and from (36) we deduce the following limits:

𝜀 √𝛾𝜉 𝛾 𝜀 ⇀ 𝜆𝛾0 weakly in 𝐿2(Γ1) , 𝜀 √𝛾 𝜕𝜉𝛾 𝜀 𝜕] ⇀ 𝜆𝛾1 weakly in 𝐿2(Γ1) . (55)

Hence, from the optimality system (28) we deduce that both 𝜀2 𝛾𝜉𝛾𝜀, 𝜀 2 𝛾 𝜕𝜉𝛾 𝜀 𝜕] tend to0 when 𝜀 󳨀→ 0. (56) Now, as above we use the same arguments and we obtain𝜌_𝜀 ⇀ 𝜌𝛾in𝐿2(Ω) solution to (41). Finally we have

Δ𝜉𝛾= 𝑦𝛾, in Ω, 𝜉𝛾= 0, 𝜕𝜉𝛾 𝜕] = 0 onΓ₀. (57)

From another side, we deduce from (29)

𝑝𝛾_𝜀 = −𝑁₀𝑢𝛾_0𝜀− 𝑁₁𝑢𝛾_1𝜀 in𝐿2(Γ₀) (58) and finally, from (36) and (53), that

𝑝𝛾_𝜀 ⇀ 𝑝𝛾= −𝑁₀𝑢𝛾₀− 𝑁₁𝑢𝛾₁ weakly in 𝐿2(Γ₀) . (59)

We now easily deduce the singular optimality system of the no-regret control for (5) by the following.

Corollary 10. The no-regret control 𝑢 for problem (5) is

char-acterized by the unique solution{𝑦, 𝜉, 𝜌, 𝑝} of the optimality

system: Δ𝑦 = 0, Δ𝜉 = 0, Δ𝜌 = 0, Δ𝑝 = 𝑦 − 𝑧_𝑑+ 𝜌 𝑖𝑛 Ω, 𝑦 = 𝑢₀, 𝜉 = 0, 𝜌 = 0, 𝑝 = 0, 𝜕𝑦 𝜕] = 𝑢1, 𝜕𝜉 𝜕] = 0, 𝜕𝜌 𝜕] = 0, 𝜕𝑝 𝜕] = 0 𝑜𝑛 Γ₀, (60)

with the adjoint equation

𝑝 + 𝑁₀𝑢₀+ 𝑁₁𝑢₁= 0 𝑖𝑛 𝐿2(Γ₀) . (61)

Proof. We here pass to the limit in𝛾 → 0. From the equalities

in (41) we easily deduce the limits: 𝜉𝛾⇀ 𝜉 = 0,

𝜌𝛾⇀ 𝜌 = 0, 𝑝𝛾⇀ 𝑝 = 0

onΓ₀. (62)

Then from the adjoint equation (42) we obtain

(𝑢𝛾₀, 𝑢𝛾₁) ⇀ (𝑢0, 𝑢1) weakly in 𝐿2(Γ0) × 𝐿2(Γ0) . (63)

Hence sequences(𝑦𝛾)_𝛾and(𝜕𝑦𝛾/𝜕])_𝛾are bounded in𝐿2(Γ₀) and we obtain the weak convergence:

𝑦𝛾 ⇀ 𝑦 = 𝑢₀, 𝜕𝑦𝛾 𝜕] ⇀ 𝜕𝑦 𝜕] = 𝑢1 onΓ₀. (64)

The same applies to the other limits.

5. Conclusion

In this work, we obtain a characterization of the control for the ill-posed Laplacian Cauchy problem, using the no-regret concept. The method consists in considering the elliptic Cauchy problem as a singular limit of sequence of well-posed elliptic problems.

The regularization approach generates incomplete infor-mation which implies the use of the low-regret approach. In case of no perturbation, the no-regret optimal control function is the same as the classical control one.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] J. Blum, Numerical Simulation and Optimal Control in Plasma

Physics: With Applications to Tokamaks, Modern Applied

Math-ematics, Wiley, Cambridge, UK; Gauthier-Villars, Paris, France, 1989.

[2] L. Bourgeois, Contrˆole optimal et probl`emes inverses en plasticit´e

[Th`ese de Doctorat], Ecole Polytechnique, 1998.

[3] P. Colli Franzone, L. Guerri, S. Tentoni et al., “A mathematical procedure for solving the inverse potential problem of electro-cardiography. Analysis of the time-space accuracy from in vitro experimental data,” Mathematical Biosciences, vol. 77, no. 1-2, pp. 353–396, 1985.

International Journal of Differential Equations 9

[4] D. Fasino and G. Inglese, “An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods,”

Inverse Problems, vol. 15, no. 1, pp. 41–48, 1999.

[5] T. Regi´ska and K. Regi´ski, “Approximate solution of a Cauchy problem for the Helmholtz equation,” Inverse Problems, vol. 22, no. 3, pp. 975–989, 2006.

[6] G. Bal and T. Zhou, “Hybrid inverse problems for a system of Maxwell’s equations,” Inverse Problems, vol. 30, no. 5, 2014. [7] Z. Qian, C.-L. Fu, and X.-T. Xiong, “Fourth-order modified

method for the Cauchy problem for the Laplace equation,”

Journal of Computational and Applied Mathematics, vol. 192, no.

2, pp. 205–218, 2006.

[8] X.-T. Xiong and C.-L. Fu, “Central difference regularization method for the Cauchy problem of the Laplace’s equation,”

Applied Mathematics and Computation, vol. 181, no. 1, pp. 675–

684, 2006.

[9] J. L. Lions, “Contrˆole `a moindres regrets des syst`emes dis-tribu´es,” Comptes Rendus de l’Acad´emie des Sciences de Paris.

Series I—Mathematics, vol. 315, pp. 1253–1257, 1992.

[10] J. L. Lions, No-Regret and Low-Regret Control. Environment,

Economics and their Mathematical Models, Masson, Paris,

France, 1994.

[11] O. Nakoulima, A. Omrane, and J. Velin, “Perturbations `a moindres regrets dans les syst`emes distribu´es `a donn´ees man-quantes,” Comptes Rendus de l’Acad´emie des Sciences: Series I:

Mathematics, vol. 330, no. 9, pp. 801–806, 2000.

[12] O. Nakoulima, A. Omrane, and J. Velin, “On the Pareto control and no-regret control for distributed systems with incomplete data,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1167–1184, 2003.

[13] L. J. Savage, The Foundations of Statistics, Dover Publications, 2nd edition, 1972.

[14] J. L. Lions, Contrˆole optimal pour les syst`emes distribu´es

sin-guliers, Gauthiers-Villard, Paris, France, 1983.

[15] S. Sougalo and O. Nakoulima, Contrˆole Optimal pour le

Probl`eme de Cauchy pour un Op´erateur Elliptique,

Pr´ebublica-tion du D´epartement de Math´ematiques et Informatique DMI, Universit´e des Antilles et de la Guyane, Guadeloupe, France, 1998.

[16] G. Massengo Mophou and O. Nakoulima, “Control of cauchy system for an elliptic operator,” Acta Mathematica Sinica,

English Series, vol. 25, no. 11, pp. 1819–1834, 2009.

[17] H. Han and H.-J. Reinhardt, “Some stability estimates for Cauchy problems for elliptic equations,” Journal of Inverse and

Ill-Posed Problems, vol. 5, no. 5, pp. 437–454, 1997.

[18] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial

Differential Equations, Yale University Press, New Haven, Conn,

USA, 1923.

[19] S. Mizohata, “Unicit´e du prolongement des solutions des ´equations ellipliques du quatri´eme ordre,” Proceedings of the

Japan Academy, vol. 34, pp. 687–692, 1958.

[20] R. Dorville, O. Nakoulima, and A. Omrane, “Low-regret control of singular distributed systems: the ill-posed backwards heat problem,” Applied Mathematics Letters, vol. 17, no. 5, pp. 549– 552, 2004.

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematical PhysicsAdvances in

Complex Analysis

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Optimization

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Combinatorics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic Analysis