# Optimal Control of the Ill-Posed Cauchy Elliptic Problem

Authors: A. Berhail; A. Omrane
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

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.


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## Body

## 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 electrocardiography [3]. One of the important examples is the Helmholtz equation and its applications in acoustic, wave propagation and scattering, vibration of the structure, and electromagnetic scattering (see [4–6] and the references therein). Here, we investigate a Cauchy problem for the Laplacian elliptic operator:(1)Δz=0,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 boundary Γ0⊂Γ, Γ0≠Γ:(2)z=v0,∂z∂ν=v1onΓ0.The goal is to reconstruct the solution on Ω and its trace on Γ1=Γ∖Γ0 from 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 implementation 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 conditions v0ε and v1ε are given with the error bound.But, in this paper, we consider another regularization method of the Laplacian, where we introduce a new data:(3)z=g0,∂z∂ν=g1onΓ1with g0 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 incomplete data is performed. The proof of the existence and 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 controls vslightly worse than by doing better thanv=0  (i.e., better than a noncontrolled system), with respect to certain criteria (cost function):(4)Juγ,g≤J0,g+γgY2∀ginaHilbertspaceY,where γ 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 hypothesis 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 Section2, 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 boundary Γ of class C2, Γ=Γ0∪Γ1 with Γ0∩Γ1=∅. The boundaries Γ0 and Γ1 are nonempty and are of positive measure. We consider here the problem:(5)Δz=0inΩ,z=v0,∂z∂ν=v1onΓ0,with z∈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(6)A=v0,v1,z∈L2Γ02×L2Ω,Δz=0inΩ,zΓ0=v0,∂z∂νΓ0=v1,and suppose that A≠∅. We will call any control-state pair (v0,v1,z)∈Aadmissible couple.LetJ be a strictly convex cost functional, defined for all admissible control-state couples (v0,v1,z) by(7)Jv0,v1,z=z-zdL2Ω2+N0v0L2Γ02+N1v1L2Γ02,where zd and (N0,N1) are, respectively, given in L2(Ω) and in R+∖{0}2. We want to find the couple control-state solution of(8)inf⁡Jv0,v1,z,v0,v1,z∈A.According to the structure of J, 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:(9)The  admissible  set  of  controls  has  a  nonempty  interior.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 incomplete data. For any ε>0, we consider the regularized problem:(10)Δ2zε+εzε=0inΩ,zε-∂Δzε∂ν=v0,∂zε∂ν+Δzε=v1onΓ0,εzε-∂Δzε∂ν=εg0,ε∂zε∂ν+Δzε=εg1onΓ1.We denote v≔(v0,v1) and g≔(g0,g1) for simplicity.We begin by an important remark.Remark 1.
For every fixedεg0 and εg1 we assume the existence of a unique solution to (10). Indeed, in the following subsections, εg0 and εg1 are 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 put ε=0 and we do the change of variables η=Δz, we obtain(11)Δη=0inΩ,∂η∂ν=0,η=0onΓ1,(12)z-∂η∂ν=v0,∂z∂ν+η=v1onΓ0.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(13)∂η∂ν=η=0onΓ0.Hence, conditions (12) become(14)z=v0,∂z∂ν=v1onΓ0,that is, the same conditions of the original problem (5).


### 3.2. Cost Function and Low-Regret Control

Consider the cost functional(15)Jεv,g=zεv,g-zdL2Ω2+N0v0L2Γ02+N1v1L2Γ02that we want to minimize in the context of no-regret control due to the presence of the incomplete data g.Definition 2.
We say thatu∈L2Γ02 is a no-regret control for (5)–(15), if u is a solution to the following problem:(16)infv∈L2Γ02⁡supg∈L2Γ12⁡Jεv,g-Jε0,g.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 pollution g, 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:(17)infv∈L2Γ02⁡supg∈L2Γ12⁡Jεv,g-Jε0,g-γg0L2Γ12-γg1L2Γ12,where γ is a strictly positive parameter. The solution to problem (17), if it exists, will be the low-regret control.Now, we introduceξε≔ξε(v,0) solution to the adjoint problem(18)Δ2ξε+εξ=zεv,0inΩ,ξε-∂∂νΔξε=0,∂ξε∂ν+Δξε=0onΓ0,εξε-∂∂νΔξε=0,ε∂ξε∂ν+Δξε=0onΓ1.Then we have the following result.Proposition 3.
The low-regret control is the solution to the classical optimal control problem(19)infv∈L2Γ02⁡Jεγvwith(20)Jεγv≔Jεv,0-Jε0,0+ε2γξεL2Γ12+∂ξε∂νL2Γ12.Proof.
After simple computations we have(21)Jεv,g-Jε0,g=Jεv,0-Jε0,0+2zεv,0,zε0,g,where ·,· is the inner product in L2(Ω). To estimate the integral zεv,0,zε0,g in (21), we use the Green formula:(22)Δ2z,ψ=z,Δ2ψ+∂∂νΔz,ψΓ-Δz,∂ψ∂νΓ+∂z∂ν,ΔψΓ-z,∂∂νΔψΓ,together with (18). We have(23)zεv,0,zε0,g=Δ2ξε+εξε,zε0,g=0+∂∂νΔξε,zεΓ-Δξε,∂zε∂νΓ+∂ξε∂ν,ΔzεΓ-ξε,∂∂νΔzεΓ=-εξε,zεΓ1-ξε,∂∂νΔzεΓ1+ε∂ξε∂ν,∂zε∂νΓ1+∂ξε∂ν,ΔzεΓ1=ξε,εg0Γ1+∂ξε∂ν,εg1Γ1,where zε=zε(0,g). Then(24)supg∈L2Γ12⁡2zεv,0,zε0,g-γg0L2Γ12-γg1L2Γ12=supg∈L2Γ12⁡ξε,εg0Γ1+∂ξε∂ν,εg1Γ1-γg0L2Γ12-γg1L2Γ12=ε2γξεL2Γ12+ε2γ∂ξε∂νL2Γ12thanks 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 parameterγ: 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 uεγ called the low-regret control.Proof.
We haveJεγ(v)≥-Jε(0,0), ∀v∈L2Γ02. Then(25)dεγ=infv∈L2Γ02⁡Jεγvexists. Let vn=vn(ε,γ) be a minimizing sequence such that dεγ=limn→∞⁡Jεγ(vn). Then we have(26)-Jε0,0≤Jεvn,0-Jε0,0+ε2γξεvnL2Γ12+∂ξε∂νvnL2Γ12≤dεγ+1.And we deduce the bounds(27)vnL2Γ02≤cεγ,zεvn,0-zdL2Ω≤cεγ,εγξεvnL2Γ1≤cεγ,εγ∂ξε∂νvnL2Γ1≤cεγ,where the constant cεγ (independent of n) is not the same each time.
Hence, there existsuεγ∈L2Γ02 such that vn(ε,γ)⇀uεγ weakly in the Hilbert space L2Γ02. Also, zε(vn,0)→zε(uεγ,0) (continuity with respect to the data). We also deduce from the strict convexity of the cost function Jεγ that uεγ is unique.Now we give the optimality system for the approximate low-regret controluεγ. We denote yεγ≔zε(uεγ,0). Then, we proceed as in [20]. We first have the following.Proposition 6.
The approached low-regret controluεγ≔(u0εγ,u1εγ) solution to (19)-(20) is characterized by the unique solution yεγ,ξεγ,ρεγ,pεγ of the optimality system(28)Δ2yεγ+εyεγ=0,Δ2ξεγ+εξεγ=yεγ,Δ2ρεγ+ερεγ=0,Δ2pεγ+εpεγ=yεγ-zd-ρεγinΩ,yεγ-∂∂νΔyεγ=u0εγ,∂yεγ∂ν+Δyεγ=u1εγ,ξε-∂∂νΔξε=0,∂ξε∂ν+Δξε=0,ρεγ-∂∂νΔρεγ=0,∂ρεγ∂ν+Δρεγ=0,pεγ-∂∂νΔpεγ=0,∂pεγ∂ν+Δpεγ=0onΓ0,εyεγ-∂∂νΔyεγ=0,ε∂yεγ∂ν+Δyεγ=0,εξε-∂∂νΔξε=0,ε∂ξε∂ν+Δξε=0,ερεγ-∂∂νΔρεγ=ε2γξε,ε∂ρεγ∂ν+Δρεγ=-ε2γ∂ξε∂νεpεγ-∂∂νΔpεγ=0,ε∂pεγ∂ν+Δpεγ=0onΓ1,with the adjoint equation(29)pεγ+N0u0εγ+N1u1εγ=0inL2Γ0.Proof.
Letuεγ be the solution of (19)-(20) on L2(Γ0). The Euler-Lagrange necessary condition gives for every w≔(w0,w1)∈L2Γ02(30)yεγ-zd,zεw,0+N0u0εγ,w0Γ0+N1u1εγ,w1Γ0+ε2γξεγ,ξεwΓ1+ε2γ∂ξεγ∂ν,∂ξε∂νwΓ1=0,where ξεγ≔ξε(uεγ,0). Denoting ρεγ=ρ(uεγ,0) as the unique solution to(31)Δ2ρεγ+ερεγ=0inΩ,ρεγ-∂∂νΔρεγ=0,∂ρεγ∂ν+Δρεγ=0onΓ0,ερεγ-∂∂νΔρεγ=ε2γξεγ,ε∂ρεγ∂ν+Δρεγ=-ε2γ∂ξεγ∂νonΓ1,we have by the Green formula(32)0=Δ2ρεγ+ερεγ,ξεw,0=ρεγ,zεw,0+ε2γξεγ,ξεw,0Γ1+ε2γ∂ξεγ∂ν,∂ξε∂νw,0Γ1.And as it is classical, we introduce the adjoint state pεγ≔p(uεγ,0) defined by(33)Δ2pεγ+εpεγ=yεγ-zd-ρεγinΩ,pεγ-∂∂νΔpεγ=0,∂pεγ∂ν+Δpεγ=0onΓ0,εpεγ-∂∂νΔpεγ=0,ε∂pεγ∂ν+Δpεγ=0onΓ1,and, using again the Green formula, we obtain(34)yεγ-zd-ρεγ,zεw,0=pεγ,wΓ0,∀w∈L2Γ02.And then (30) becomes(35)pεγ+N0u0εγ+N1u1εγ,wΓ0=0,∀w∈L2Γ02which is (29).


## 3.1. Back to the Original Problem

We show how to come back to the original Cauchy problem, starting from (10). Indeed, if we put ε=0 and we do the change of variables η=Δz, we obtain(11)Δη=0inΩ,∂η∂ν=0,η=0onΓ1,(12)z-∂η∂ν=v0,∂z∂ν+η=v1onΓ0.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(13)∂η∂ν=η=0onΓ0.Hence, conditions (12) become(14)z=v0,∂z∂ν=v1onΓ0,that is, the same conditions of the original problem (5).


## 3.2. Cost Function and Low-Regret Control

Consider the cost functional(15)Jεv,g=zεv,g-zdL2Ω2+N0v0L2Γ02+N1v1L2Γ02that we want to minimize in the context of no-regret control due to the presence of the incomplete data g.Definition 2.
We say thatu∈L2Γ02 is a no-regret control for (5)–(15), if u is a solution to the following problem:(16)infv∈L2Γ02⁡supg∈L2Γ12⁡Jεv,g-Jε0,g.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 pollution g, 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:(17)infv∈L2Γ02⁡supg∈L2Γ12⁡Jεv,g-Jε0,g-γg0L2Γ12-γg1L2Γ12,where γ is a strictly positive parameter. The solution to problem (17), if it exists, will be the low-regret control.Now, we introduceξε≔ξε(v,0) solution to the adjoint problem(18)Δ2ξε+εξ=zεv,0inΩ,ξε-∂∂νΔξε=0,∂ξε∂ν+Δξε=0onΓ0,εξε-∂∂νΔξε=0,ε∂ξε∂ν+Δξε=0onΓ1.Then we have the following result.Proposition 3.
The low-regret control is the solution to the classical optimal control problem(19)infv∈L2Γ02⁡Jεγvwith(20)Jεγv≔Jεv,0-Jε0,0+ε2γξεL2Γ12+∂ξε∂νL2Γ12.Proof.
After simple computations we have(21)Jεv,g-Jε0,g=Jεv,0-Jε0,0+2zεv,0,zε0,g,where ·,· is the inner product in L2(Ω). To estimate the integral zεv,0,zε0,g in (21), we use the Green formula:(22)Δ2z,ψ=z,Δ2ψ+∂∂νΔz,ψΓ-Δz,∂ψ∂νΓ+∂z∂ν,ΔψΓ-z,∂∂νΔψΓ,together with (18). We have(23)zεv,0,zε0,g=Δ2ξε+εξε,zε0,g=0+∂∂νΔξε,zεΓ-Δξε,∂zε∂νΓ+∂ξε∂ν,ΔzεΓ-ξε,∂∂νΔzεΓ=-εξε,zεΓ1-ξε,∂∂νΔzεΓ1+ε∂ξε∂ν,∂zε∂νΓ1+∂ξε∂ν,ΔzεΓ1=ξε,εg0Γ1+∂ξε∂ν,εg1Γ1,where zε=zε(0,g). Then(24)supg∈L2Γ12⁡2zεv,0,zε0,g-γg0L2Γ12-γg1L2Γ12=supg∈L2Γ12⁡ξε,εg0Γ1+∂ξε∂ν,εg1Γ1-γg0L2Γ12-γg1L2Γ12=ε2γξεL2Γ12+ε2γ∂ξε∂νL2Γ12thanks 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 parameterγ: 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.


## 3.2.1. The Low-Regret Control

As in [20], we define the low-regret control as the solution to the following MinMax problem:(17)infv∈L2Γ02⁡supg∈L2Γ12⁡Jεv,g-Jε0,g-γg0L2Γ12-γg1L2Γ12,where γ is a strictly positive parameter. The solution to problem (17), if it exists, will be the low-regret control.Now, we introduceξε≔ξε(v,0) solution to the adjoint problem(18)Δ2ξε+εξ=zεv,0inΩ,ξε-∂∂νΔξε=0,∂ξε∂ν+Δξε=0onΓ0,εξε-∂∂νΔξε=0,ε∂ξε∂ν+Δξε=0onΓ1.Then we have the following result.Proposition 3.
The low-regret control is the solution to the classical optimal control problem(19)infv∈L2Γ02⁡Jεγvwith(20)Jεγv≔Jεv,0-Jε0,0+ε2γξεL2Γ12+∂ξε∂νL2Γ12.Proof.
After simple computations we have(21)Jεv,g-Jε0,g=Jεv,0-Jε0,0+2zεv,0,zε0,g,where ·,· is the inner product in L2(Ω). To estimate the integral zεv,0,zε0,g in (21), we use the Green formula:(22)Δ2z,ψ=z,Δ2ψ+∂∂νΔz,ψΓ-Δz,∂ψ∂νΓ+∂z∂ν,ΔψΓ-z,∂∂νΔψΓ,together with (18). We have(23)zεv,0,zε0,g=Δ2ξε+εξε,zε0,g=0+∂∂νΔξε,zεΓ-Δξε,∂zε∂νΓ+∂ξε∂ν,ΔzεΓ-ξε,∂∂νΔzεΓ=-εξε,zεΓ1-ξε,∂∂νΔzεΓ1+ε∂ξε∂ν,∂zε∂νΓ1+∂ξε∂ν,ΔzεΓ1=ξε,εg0Γ1+∂ξε∂ν,εg1Γ1,where zε=zε(0,g). Then(24)supg∈L2Γ12⁡2zεv,0,zε0,g-γg0L2Γ12-γg1L2Γ12=supg∈L2Γ12⁡ξε,εg0Γ1+∂ξε∂ν,εg1Γ1-γg0L2Γ12-γg1L2Γ12=ε2γξεL2Γ12+ε2γ∂ξε∂νL2Γ12thanks 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 parameterγ: 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 uεγ called the low-regret control.Proof.
We haveJεγ(v)≥-Jε(0,0), ∀v∈L2Γ02. Then(25)dεγ=infv∈L2Γ02⁡Jεγvexists. Let vn=vn(ε,γ) be a minimizing sequence such that dεγ=limn→∞⁡Jεγ(vn). Then we have(26)-Jε0,0≤Jεvn,0-Jε0,0+ε2γξεvnL2Γ12+∂ξε∂νvnL2Γ12≤dεγ+1.And we deduce the bounds(27)vnL2Γ02≤cεγ,zεvn,0-zdL2Ω≤cεγ,εγξεvnL2Γ1≤cεγ,εγ∂ξε∂νvnL2Γ1≤cεγ,where the constant cεγ (independent of n) is not the same each time.
Hence, there existsuεγ∈L2Γ02 such that vn(ε,γ)⇀uεγ weakly in the Hilbert space L2Γ02. Also, zε(vn,0)→zε(uεγ,0) (continuity with respect to the data). We also deduce from the strict convexity of the cost function Jεγ that uεγ is unique.Now we give the optimality system for the approximate low-regret controluεγ. We denote yεγ≔zε(uεγ,0). Then, we proceed as in [20]. We first have the following.Proposition 6.
The approached low-regret controluεγ≔(u0εγ,u1εγ) solution to (19)-(20) is characterized by the unique solution yεγ,ξεγ,ρεγ,pεγ of the optimality system(28)Δ2yεγ+εyεγ=0,Δ2ξεγ+εξεγ=yεγ,Δ2ρεγ+ερεγ=0,Δ2pεγ+εpεγ=yεγ-zd-ρεγinΩ,yεγ-∂∂νΔyεγ=u0εγ,∂yεγ∂ν+Δyεγ=u1εγ,ξε-∂∂νΔξε=0,∂ξε∂ν+Δξε=0,ρεγ-∂∂νΔρεγ=0,∂ρεγ∂ν+Δρεγ=0,pεγ-∂∂νΔpεγ=0,∂pεγ∂ν+Δpεγ=0onΓ0,εyεγ-∂∂νΔyεγ=0,ε∂yεγ∂ν+Δyεγ=0,εξε-∂∂νΔξε=0,ε∂ξε∂ν+Δξε=0,ερεγ-∂∂νΔρεγ=ε2γξε,ε∂ρεγ∂ν+Δρεγ=-ε2γ∂ξε∂νεpεγ-∂∂νΔpεγ=0,ε∂pεγ∂ν+Δpεγ=0onΓ1,with the adjoint equation(29)pεγ+N0u0εγ+N1u1εγ=0inL2Γ0.Proof.
Letuεγ be the solution of (19)-(20) on L2(Γ0). The Euler-Lagrange necessary condition gives for every w≔(w0,w1)∈L2Γ02(30)yεγ-zd,zεw,0+N0u0εγ,w0Γ0+N1u1εγ,w1Γ0+ε2γξεγ,ξεwΓ1+ε2γ∂ξεγ∂ν,∂ξε∂νwΓ1=0,where ξεγ≔ξε(uεγ,0). Denoting ρεγ=ρ(uεγ,0) as the unique solution to(31)Δ2ρεγ+ερεγ=0inΩ,ρεγ-∂∂νΔρεγ=0,∂ρεγ∂ν+Δρεγ=0onΓ0,ερεγ-∂∂νΔρεγ=ε2γξεγ,ε∂ρεγ∂ν+Δρεγ=-ε2γ∂ξεγ∂νonΓ1,we have by the Green formula(32)0=Δ2ρεγ+ερεγ,ξεw,0=ρεγ,zεw,0+ε2γξεγ,ξεw,0Γ1+ε2γ∂ξεγ∂ν,∂ξε∂νw,0Γ1.And as it is classical, we introduce the adjoint state pεγ≔p(uεγ,0) defined by(33)Δ2pεγ+εpεγ=yεγ-zd-ρεγinΩ,pεγ-∂∂νΔpεγ=0,∂pεγ∂ν+Δpεγ=0onΓ0,εpεγ-∂∂νΔpεγ=0,ε∂pεγ∂ν+Δpεγ=0onΓ1,and, using again the Green formula, we obtain(34)yεγ-zd-ρεγ,zεw,0=pεγ,wΓ0,∀w∈L2Γ02.And then (30) becomes(35)pεγ+N0u0εγ+N1u1εγ,wΓ0=0,∀w∈L2Γ02which 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 constantC such that(36)uε0γL2Γ0≤C,uε1γL2Γ0≤C,yεγL2Ω≤C,εγξεγL2Γ1≤C,εγ∂ξεγ∂νL2Γ1≤C.Proof.
Sinceuεγ is the approximate low-regret control, we have(37)Jεγuεγ≤Jεγv,∀v∈L2Γ02.In the particular case where v=0, we obtain(38)yεγ-zdL2Ω2+N0uε0γL2Γ02+N1uε1γL2Γ02+ε2γξεγL2Γ12+∂ξεγ∂νL2Γ12≤zε0,0-zdL2Ω2+ε2γξε0,0L2Γ12+∂ξε∂ν0,0L2Γ12.But,(39)zε0,0=0inΩ,ξε0,0=∂ξε∂ν0,0=0onΓ1.Then(40)yεγ-zdL2Ω2+N0uε0γL2Γ02+N1uε1γL2Γ02+ε2γξεγL2Γ12+∂ξεγ∂νL2Γ12≤zdL2Ω2=C.Remark 8.
From Section3.1, 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 controluγ for problem (5) is characterized by the unique solution {yγ,ξγ,ργ,pγ} of the optimality system:(41)Δyγ=0,Δξγ=0,Δργ=0,Δpγ=yγ-zd+ργinΩ,yγ=u0γ,ξγ=0,ργ=0,pγ=0,∂yγ∂ν=u1γ,∂ξγ∂ν=0,∂ργ∂ν=0,∂pγ∂ν=0onΓ0,with the adjoint equation(42)pγ+N0u0γ+N1u1γ=0inL2Γ0.Proof.
From the optimality system (28) in Proposition 6, we deduce that yεγ is solution of the system(43)Δ2yεγ+εyεγ=0,inΩ,yεγ-∂∂νΔyεγ=u0εγ,∂yεγ∂ν+Δyεγ=u1εγonΓ0,εyεγ-∂∂νΔyεγ=0,ε∂yεγ∂ν+Δyεγ=0,onΓ1.As in Section 3.1, we denote(44)ηεγ=Δyεγ.Then system (43) is written as(45)Δηεγ+εyεγ=0,inΩ,yεγ-∂ηεγ∂ν=u0εγ,∂yεγ∂ν+ηεγ=u1εγonΓ0,εyεγ-∂ηεγ∂ν=0,ε∂yεγ∂ν+ηεγ=0,onΓ1.From estimates (36) of Lemma 7, sequence (yεγ) is bounded in L2(Ω) by constant M. Hence, there exists a subsequence still denoted by (yεγ), such that(46)yεγ⇀yγweakly  inL2Ωas ε→0. We deduce from the first equation in (45) that(47)Δηεγ≤εyεγ≤εM⟶0.That is,(48)Δηεγ⇀0weakly  inL2Ω.Using the same arguments and from the last two equations in (45) we have(49)∂ηεγ∂ν⇀0,ηεγ⇀0inL2Γ1when ε→0. We resume by(50)Δηγ=0inΩ,∂ηγ∂ν=0,ηγ=0onΓ1.Using the unique continuation theorem of Mizohata [19], we deduce from (50) that we also have(51)ηγ≡0everywhere  onΩ¯.Then,(52)∂ηγ∂ν=ηγ=0onΓ0.From another side, estimates (36) also give(53)uε0γ,uε1γ⇀u0γ,u1γweakly  inL2Γ0×L2Γ0.We come back to the notation ηγ=Δyγ. System (45) transforms to(54)Δyγ=0,inΩ,yγ=u0γ,∂yγ∂ν=u1γonΓ0.Again, we use the estimates of Lemma 7 and from (36) we deduce the following limits:(55)εγξεγ⇀λ0γweakly  inL2Γ1,εγ∂ξεγ∂ν⇀λ1γweakly  inL2Γ1.Hence, from the optimality system (28) we deduce that both(56)ε2γξεγ,ε2γ∂ξεγ∂νtend  to0whenε⟶0.Now, as above we use the same arguments and we obtain ρε⇀ργ in L2(Ω) solution to (41). Finally we have(57)Δξγ=yγ,inΩ,ξγ=0,∂ξγ∂ν=0onΓ0.From another side, we deduce from (29)(58)pεγ=-N0u0εγ-N1u1εγinL2Γ0and finally, from (36) and (53), that(59)pεγ⇀pγ=-N0u0γ-N1u1γweakly  inL2Γ0.We now easily deduce the singular optimality system of the no-regret control for (5) by the following.Corollary 10.
The no-regret controlu for problem (5) is characterized by the unique solution {y,ξ,ρ,p} of the optimality system:(60)Δy=0,Δξ=0,Δρ=0,Δp=y-zd+ρinΩ,y=u0,ξ=0,ρ=0,p=0,∂y∂ν=u1,∂ξ∂ν=0,∂ρ∂ν=0,∂p∂ν=0onΓ0,with the adjoint equation(61)p+N0u0+N1u1=0inL2Γ0.Proof.
We here pass to the limit inγ→0. From the equalities in (41) we easily deduce the limits:(62)ξγ⇀ξ=0,ργ⇀ρ=0,pγ⇀p=0onΓ0.Then from the adjoint equation (42) we obtain(63)u0γ,u1γ⇀u0,u1weakly  inL2Γ0×L2Γ0.Hence sequences yγγ and ∂yγ/∂νγ are bounded in L2(Γ0) and we obtain the weak convergence:(64)yγ⇀y=u0,∂yγ∂ν⇀∂y∂ν=u1onΓ0.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 information 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.



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*Source: 468918-2015-11-23.xml*