Numerical Solution of bilateral obstacle optimal control Problem

HAL Id: hal-01247162

https://hal.archives-ouvertes.fr/hal-01247162

Preprint submitted on 22 Dec 2015

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.

Numerical Solution of bilateral obstacle optimal control Problem

Radouen Ghanem, Billel Zireg

To cite this version:

Radouen Ghanem, Billel Zireg. Numerical Solution of bilateral obstacle optimal control Problem.

2015. �hal-01247162�

Numerical solution of bilateral obstacle optimal control Problem

R. Ghanem

, B. Zireg

†‡ Numerical analysis, optimization and statistical laboratory (LANOS) Badji-Mokhtar, Annaba University

P.O. Box 12, 23000, Annaba Algeria

† radouen.ghanem@univ-annaba.org

Abstract

In this work we consider the numerical resolution of the bilateral obstacle optimal control problem given in Bergounioux et al. Where the main feature of this problem is that the control and the obstacle are the same.

Keywords: Optimal control, obstacle problem, finite differences method.

AMS subject classifications: 65K15 49K10, 35J87 , 49J20 , 65L12 , 49M15.

1 Introduction

Variational inequalities and related optimal control problems have been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as shape optimization theory, image processing and mechanics, (see for example [4], [5], [10], [18], [22]).

Optimal control problem governed by variational inequalities has been studied exten- sively during the last years by many authors, such as [2], [20], [21]. These authors have studied optimal control problems for obstacle problems (or variational inequalities) where the obstacle is a known function, and the control variables appear as variational inequali- ties. In other words, controls do not change the obstacle and, on the other hand , in [1], [7], [11] the authors have studied another class of problems where the obstacle functions are unknowns and are considered as control functions.

In this paper, we investigate optimal control problems governed by variational inequal- ities of obstacle type. This kind of problem is very important and it can lead to the shape optimization problem governed by variational inequality, it may concern the optimal shape of dam [9], for which the obstacle gives the shape to be designed such that the pressure of the fluid inside the dam is close to a desired value. Besides, if we want to design a mem- brane having an expected shape, we need to choose a suitable obstacle. In this case, the obstacle can be considered as a control, and the membrane as the state (see for example [15]).

It should be pointed out that, in the optimal control problem of a variational inequality, the main difficulty comes from the fact that the mappingT between the control and the state (control-to-state operator) is not Gateaux differentiable as pointed it out in [21], [20]

where one can only define a conical derivative forT but only Lipschitz-continuous and so it is not easy to get optimality conditions that can be numerically exploitable.

To overcome this difficulty, different authors (see for example, Kunisch et al.[17] V.

Barbu [2] and the references therein) consider a Moreau-Yosida approximation technique to reformulate the governing variational inequality problem into a problem governed by a variational equation. Our approach is based on the penalty method and Barbu’s treat- ment as a penalty parameter approaching zero. We then obtain a system of optimality for suitable approximations of the original problem which can be easily used from the numerical point of view.

Nevertheless, the optimal control of variational inequalities of obstacle type is still a very active field of research especially for their numerical treatment which are given in the recent publication [13].

The problem that we are going to study can be set in a wider class of problems, which can be formally described as follows

min{J(y, χ), y=T (χ), χ∈ Uad ⊂ U}

where T is an operator which associates y to χ, when y is a solution to

∀y∈ K(y, χ),hA(y, χ), y−vi ≥0, (obs) where K is a multiplication from χ× U to 2^χ when χ is a Banach space and A is a differential operator from Y to the dual Y^′. Let h be an application from R×R to R, then the variational inequality that relates the controlχ to the statey can be written as

hA(y, χ), y−viY,Y^′+h(χ, v)−h(χ, y)≥(χ, v−y),∀y ∈ Y,

where this formulation gives the obstacle problems where the obstacle is the control.

Following the previous ideas, we may apply a smoothed penalization approach to our problem. More precisely, the idea is to approximates the obstacle problem by introducing an approximating parameter δ, where the approximating method is based on the penal- ization method and it consists in replacing the obstacle problem ((obs) by a family of semilinear equations. In [7], Bergounioux et al. considered the following bilateral optimal control obstacle problem

min{J(ϕ, ψ) = 1 2

Z

Ω

(T (ϕ, ψ)−z)²dx+ν 2

Z

Ω

(∆ϕ)²+ (∆ψ)² dx,

(ϕ, ψ)∈ Uad× Uad} (1) where ν is a given positive constant and zbelongs to L²(Ω) as a target profile, such that y=T (ϕ) is a solution of the bilateral obstacle problem given by

hAy, v−yi ≥(f, v−y), for allv inK(ϕ, ψ), where K(ϕ, ψ) is given by

K(ϕ, ψ) =

y∈H₀¹(Ω), ψ≥y≥ϕ , and the set of admissible controls U^ad is defined as follows

U^ad={(ϕ, ψ)∈ U × U |ϕ≤ψ},

where U =H²(Ω)×H₀¹(Ω). As we needH²−priori estimate, we could assume that U^adisH²bounded. For example, we can suppose thatU^ad isBH²(0, R) i.e. a ball of center 0 and radius R, where R is a large enough positive real number, but according to [13], this choice can lead to technical difficulties to get a numerical solution of the optimality system.

In [13], Ghanem et al., have solved numerically the unilateral optimal control of ob- stacle problem given by

min

J(ϕ) = 1 2

Z

Ω

(T (ϕ)−z)²dx+ν 2

Z

Ω

(∆ϕ)²dx, ϕ∈ U

(2) instead of the one defined by

min

J(ϕ) = 1 2

Z

Ω

(T (ϕ)−z)²dx+ν 2

Z

Ω

(∇ϕ)²dx, ϕ∈ U^ad

where

U^ad =

ϕ∈H²(Ω), ϕ∈ BH²(0, R) (3)

such that y=T (ϕ) is a solution of the unilateral obstacle problem given by hAy, v−yi ≥(f, v−y), for allv in K(ϕ)

where K(ϕ) is defined by

K(ϕ) =

y∈H₀¹(Ω), y≥ϕ .

According to the result given in [12] the authors point out that, in spite of the elimi- nation of the inequality constraint given by (3), we still get a local convergence property implied by the constraintkϕ_nkH²(Ω)≤R. Hence, we are again confronted to the inequality constraint (3).

So we note that it is not necessary to suppress the constraint (3), because it is going to appear again to get the local convergence of the algorithm used for the numerical solution of the problem give by (2).

For the numerical solution of optimal control problem, it is usual to use two kinds of numerical approaches: direct and indirect methods. Direct methods consist in discretizing the cost function, the state and the control and thus reduce the problem to a nonlinear optimization problem with constraints. Indirect methods consist of solving numerically the optimality system given by the state, the adjoint and the projection equations.

The aim of this paper is the numerical solution of the optimal control problem given in [7] by using the indirect approach (after optimisation) based on the same idea and techniques given in [13], where the optimality system is characterized by



















Ay^δ+ (β_δ(y^δ−ϕ^δ)−β_δ(ψ^δ−y^δ)) =f in Ω andy^δ= 0 on ∂Ω A^∗p^δ+µ^δ₁+µ₂^δ=y^δ−z in Ω andp^δ= 0 on ∂Ω

µ₁+ϕ^δ−ϕ^∗, ϕ−ϕ^δ

+ µ₂+ψ^δ−ψ^∗, ψ−ψ^δ + +ν ∆ϕ^δ,∆ ϕ−ϕ^δ

+ν ∆ψ^δ,∆ ψ−ψ^δ

= 0, for all ϕinU^ad

For the numerical solution, we first begin by discretizing the optimality system by using finite differences schemes and then by proposing an iterative algorithm based on Gauss- Seidel method that is a combination of damped-Newton-Raphson and a direct method.

The main difficulties of this work compared to the one considered in [13], is to get an optimality system numerically exploitable by the proposed algorithm.

In the sequel, we denote byBV (0, r) theV-ball aroundoof radiusr and byC generic positive constants.

The rest of paper is organized as follows: in section 2 we give precise assumptions and some well-known results. In section 3, we introduce the iterative algorithm and give convergence results to solve the optimality system. Section 4 is devoted to numerical examples that illustrate the theoretical findings and in section 5 we present some remarks and a conclusion.

2 Preliminaries and known results

We consider the bilinear formσ(·,·) defined inH¹(Ω)×H¹(Ω), where we assume that the following conditions are fulfilled

H₁. Continuity

∃C >0,∀u, v ∈H¹(Ω),|σ(u, v)| ≤CkukH¹(Ω)kvkH¹(Ω)

H₂. Coercivity

∃c >0,∀u∈H¹(Ω), σ(u, u)≥ckuk²H¹(Ω)

We call AinL(H¹(Ω), H⁻¹(Ω)) the linear self-adjoint elliptic operator (see [19]) asso- ciated toσ such thathAu, vi=σ(u, v), and assume that the adjoint formσ^∗(·,·) satisfies the conditions H₁ and H₂.

For any ϕand ψinH₀¹(Ω), we define K(ϕ, ψ) =

y ∈H₀¹(Ω)|ψ≥y≥ϕ in Ω , (4) and consider the following variational inequality

σ(y, v−y)≥(f, v−y), for allv inK(ϕ, ψ), (5) where f belongs to L²(Ω) is a source term. From now on, we define the operator T (control-to-state operator) fromU × U toU, such thaty =T (ϕ, ψ) is the unique solution

to the obstacle problem given by (4) and (5) (see [18]), where U =H²(Ω)∩H₀¹(Ω).

Let U^ad be the set of admissible controls which is assumed to be H²(Ω)-bounded subset of H²(Ω)∩H₀¹(Ω), convex and closed in H²(Ω). We may choose, for example,

U^ad =BH²(0, R) ={v inH²(Ω)∩H₀¹(Ω)|kvkH² ≤R} (6) where R is a large enough positive real number. This boundedness assumption forU^ad is crucial: it gives a prioriH² - estimates on the control functions and leads to the existence of a solution. Now, we consider the optimal control problem (P) defined as follows

min{J(ϕ, ψ) = ¹₂ Z

Ω

(T (ϕ, ψ)−z)²dx+^ν₂ Z

Ω

(∇ϕ)²+ (∇ψ)² dx

,

for all ϕ, ψ∈ U^ad}, (P) where ν is a strictly given positive constant, z in L²(Ω). We seek the obstacles (optimals controls) ϕ,¯ ψ¯

in Uad² , such that the corresponding state is close to a target profilez.

To derive necessary conditions for an optimal control, we would like to differentiate the map (ϕ, ψ)7→ T (ϕ, ψ). Since the map (ϕ, ψ)7→ T (ϕ, ψ) is not directly differentiable (see [21]), the idea here consists in approximating the map T (ϕ, ψ) by a family of maps T^δ(ϕ, ψ) and replacing the obstacle problem (5) and (4) by the following smooth semilinear equation (see [20], [8]):

Ay+ (β_δ(y−ϕ)−β_δ(ψ−y)) =f in Ω, andy = 0 on ∂Ω.

Then, the approximation map (ϕ, ψ) 7→ T^δ(ϕ, ψ) will then be differentiable and ap- proximate necessary conditions will be derived, such that

β_δ(r) = ¹_δ







0 if r ≥0

−r² if r ∈

−¹₂,0 r+ ¹₄ if r ≤ −¹₂

where β(·) is negative and belongs toC¹(R), such that δ is strictly positive and goes to 0. Then β_δ^′(·) is given by

β_δ^′(r) = ¹_δ







0 if r≥0

−2r if r∈

−¹₂,0 1 if r≤ −¹₂

As β_δ(· −ϕ)−β_δ(ψ− ·) is nondecreasing, it is well known (see [14]), that boundary value problem (2) admits a unique solution y^δ in H²(Ω)∩H₀¹(Ω) for a fixed ϕ and ψ in H²(Ω)∩H₀¹(Ω) andf inL²(Ω). In the sequel, we set y^δ =T^δ(ϕ, ψ) and in addition,cor C denotes a general positive constant independent of any approximation parameter. So for any δ >0, we define

J_δ(ϕ, ψ) = ¹₂[ Z

Ω

T^δ(ϕ, ψ)−z2

dx+ν Z

Ω

(∇ϕ)²+ (∇ψ)²

dx]. (P^δ)

Then, the approximate optimal control problem is given by

min{J_δ(ϕ, ψ), ϕ, ψ inUad× Uad}. (7) and by using the same techniques given in [2] and [7], the problem (7) has, at least, one solution denoted by (y^δ, p^δ, ϕ^δ, ψ^δ) and characterized by the following Theorem Theorem 2.1. Since ϕ^δ, ψ^δ

is an optimal solution to P^δ

,andy^δ=T^δ ϕ^δ, ψ^δ .Then there exist p^δ in U, µ^δ₁ = β_δ^′(y^δ−ϕ^δ)p^δ and µ^δ₂ = β_δ^′(ψ^δ−y^δ)p^δ in L²(Ω) such that the following optimality system S^δ

is satisfied























Ay^δ+ (β_δ(y^δ−ϕ^δ)−β_δ(ψ^δ−y^δ)) =f in Ω A^∗p^δ+µ^δ₁+µ^δ₂=y^δ−z in Ω

ν∆ϕ^δ+ν∆ψ^δ+β_δ^′(y^δ−ϕ^δ)p^δ+β_δ^′(ψ^δ−y^δ)p^δ = 0 y^δ=p^δ=ϕ^δ =ψ^δ= 0 on ∂Ω

Now, we give some important results relevant for the sequel of this paper.

Lemma 2.1. From the definition of β(·) and since p^δ_n belongs to BH0¹(Ω)(0,ρ˜₃), where ρ˜₃ is a positive constant, and for y^δ_i, ϕ^δ_i, p^δ_i

in U˜ where U˜=H₀¹(Ω)×H₀²(Ω)×H₀¹(Ω) and i= 1,2, we get

kβ_δ^′

y^δ₂−ϕ^δ₂

p^δ₂−β_δ^′

y^δ₁−ϕ^δ₁

p^δ₁ kL²(Ω)≤ C

δ kp^δ₂−p^δ₁ kH¹(Ω)+ Cρ˜₃

δ ky₂^δ−y^δ₁kL²(Ω) +Cρ˜₃

δ kϕ^δ₂−ϕ^δ₁ kL²(Ω)

Proof. By the definition ofβ^′(·) we get (β_δ^′

y^δ₂−ϕ^δ₂

p^δ₂−β_δ^′

y^δ₁−ϕ^δ₁

p^δ₁) =β_δ^′

y₂^δ−ϕ^δ₂

(p^δ₂−p^δ₁)+

(β_δ^′

y₂^δ−ϕ^δ₂

−β_δ^′

y^δ₁−ϕ^δ₁ )p^δ₁. Then, by Cauchy-Schwarz inequality and since p^δ₁ belongs to BH¹(Ω)(0, ρ₃) and by the Mean-Value Theorem applied in the interval of sides{ y^δ₂−ϕ^δ₂

, y₁^δ−ϕ^δ₁

}, we can deduce

kβ_δ^′

y^δ₂−ϕ^δ₂

p^δ₂−β_δ^′

y^δ₁−ϕ^δ₁

p^δ₁ kL²(Ω)≤ C

δ kp^δ₂−p^δ₁ kH¹(Ω)+ Cρ˜₃

δ k

y^δ₂−y₁^δ

−

ϕ^δ₂−ϕ^δ₁ kL²(Ω)

Lemma 2.2. Let y_i^δ, ϕ^δ_i, p^δ_i

belong toU˜ where i= 1,2 and by the properties ofβ_δ(·), we get

k(βδ(y₂^δ−ϕ^δ₂)−β_δ(y₁^δ−ϕ^δ₁))−(βδ(ψ₂^δ−y₂^δ)−β_δ(ψ^δ₁−y₁^δ))kL²(Ω)≤ C

δ ky₂^δ−y^δ₁kL²(Ω) +C

δ kϕ^δ₂−ϕ^δ₁kL²(Ω) +C

δ kψ^δ₂−ψ^δ₁ kL²(Ω). Proof. It is easy to see that

k(β_δ(y₂^δ−ϕ^δ₂)−β_δ(y₁^δ−ϕ^δ₁))−(β_δ(ψ₂^δ−y₂^δ)−β_δ(ψ^δ₁−y₁^δ))kL²(Ω)≤ kβ_δ(y₂^δ−ϕ^δ₂)−β_δ(y₁^δ−ϕ^δ₁)kL²(Ω)

+kβ_δ(ψ^δ₂−y₂^δ)−β_δ(ψ₁^δ−y^δ₁)kL²(Ω). By the Mean-Value Theorem applied in the interval of sides{(y₂^δ−ϕ^δ₂), (y^δ₁−ϕ^δ₁)}and {(ψ₂^δ−y^δ₂), (ψ₁^δ−y₁^δ)}, we get

k(β_δ(y₂^δ−ϕ^δ₂)−β_δ(y₁^δ−ϕ^δ₁)−(β_δ(ψ₂^δ−y^δ₂)−β_δ(ψ₁^δ−y^δ₁))kL²(Ω)≤ C

δ ky₂^δ−y^δ₁kL²(Ω) +C

δ kϕ^δ₂−ϕ^δ₁kL²(Ω) +C

δ kψ^δ₂−ψ^δ₁ kL²(Ω). Theorem 2.2. For any triplet y^δ_i, ϕ^δ_i, ψ^δ_i

in U˜ that satisfies the optimality system S^δ where i= 1,2, and since δ≤C, we get

ky^δ₂−y₁^δkH¹(Ω)≤l₁(kϕ^δ₂−ϕ^δ₁ kL²(Ω)+kψ₂^δ−ψ₁^δkL²(Ω)).

where l₁ := C

δ. This means that the mapping y^δ:=T^δ ϕ^δ, ψ^δ

, is Lipschitizian, with a Lipschitz constant l₁.

Proof. From the state equation of the optimality system S^δ

and by subtraction of the two previous equations and, if we takev=y^δ₂−y₁^δ, we get

σ(y₂^δ−y₁^δ, y^δ₂−y₁^δ) + (β_δ(y^δ₂−ϕ^δ₂)−β_δ(y₁^δ−ϕ^δ₁)−(β_δ(ψ₂^δ−y₂^δ)−β_δ(ψ^δ₁−y^δ₁)), y^δ₂−y^δ₁) = 0.

By the proprieties of β_δ(·), we deduce

σ(y₂^δ−y₁^δ, y₂^δ−y^δ₁) ≤ −(β_δ(y₂^δ−ϕ^δ₂)−β_δ(y^δ₁−ϕ^δ₁),

ϕ^δ₂−ϕ^δ₁ )

−(βδ(ψ₂^δ−y^δ₂)−β_δ(ψ₁^δ−y^δ₁),

ψ₁^δ−ψ₂^δ ).

Then, by using the Mean-Value Theorem and the coercivity condition H2 of σ(·,·), we get

cky₂^δ−y^δ₁k²H¹(Ω)≤ c

δ(cky^δ₂−y₁^δkH¹(Ω)+kϕ^δ₂−ϕ^δ₁ kL²(Ω))kϕ^δ₂−ϕ^δ₁ kL²(Ω)

+c

δ(cky₂^δ−y₁^δkH¹(Ω)+kψ₁^δ−ψ₂^δkL²(Ω))kψ₁^δ−ψ₂^δkL²(Ω) . From above, and since

1. Ifcky^δ₂−y₁^δkH¹(Ω)≤kϕ^δ₂−ϕ^δ₁kL²(Ω) andcky^δ₂−y₁^δkH¹(Ω)≤kψ^δ₁−ψ^δ₂ kL²(Ω)

then

ky₂^δ−y^δ₁kL²(Ω)≤ 1

c(kϕ^δ₂−ϕ^δ₁ kL²(Ω)+kψ^δ₁−ψ^δ₂ kL²(Ω)).

2. Ifcky^δ₂−y₁^δkH¹(Ω)≥kϕ^δ₂−ϕ^δ₁kL²(Ω) andcky^δ₂−y₁^δ)kH¹(Ω)≥kψ₁^δ−ψ₂^δkL²(Ω)

we get

ky₂^δ−y₁^δkH¹(Ω)≤ c

δ(kϕ^δ₂−ϕ^δ₁ kL²(Ω)+kψ^δ₁−ψ^δ₂ kL²(Ω)).

3. Ifcky^δ₂−y₁^δkH¹(Ω)≥kϕ^δ₂−ϕ^δ₁kL²(Ω) andcky^δ₂−y₁^δkH¹(Ω)≤kψ^δ₁−ψ^δ₂ kL²(Ω), we get

ky^δ₂−y₁^δkH¹(Ω)≤ c δ

kϕ^δ₂−ϕ^δ₁ kL²(Ω)+kψ^δ₁−ψ^δ₂ kL²(Ω)

. 4. Ifcky^δ₂−y₁^δkH¹(Ω)≤kϕ^δ₂−ϕ^δ₁kL²(Ω) andcky^δ₂−y₁^δkH¹(Ω)≥kψ^δ₁−ψ^δ₂ kL²(Ω),

we get

cky^δ₂−y₁^δk²H¹(Ω)≤ c

δ kϕ^δ₂−ϕ^δ₁ k²L²(Ω) +c

δ ky₂^δ−y^δ₁kH¹(Ω)kψ₁^δ−ψ₂^δkL²(Ω), and since the hypothesis of the Theorem are satisfied, we deduce that in all the previous cases we have

ky^δ₂−y₁^δkH¹(Ω)≤ c δ

kϕ^δ₂−ϕ^δ₁ kL²(Ω)+kψ^δ₁−ψ^δ₂ kL²(Ω)

.

Lemma 2.3. For any triplet y^δ, ϕ^δ, ψ^δ

in U˜, satisfying the optimality system (S^δ), we have

ky^δkH¹(Ω)≤max

Ckϕ^δkH¹(Ω), Ckψ^δ kH¹(Ω)

,

and moreover when ϕ^δ and ψ^δ belong to B_H²_(Ω)(0, ρ1)∩ W,we deduce that ky^δ kH¹(Ω)≤ρ₂.

This means that y^δ belongs to B_H¹_(Ω)(0, ρ₂)∩ U, where ρ₂ :=Cρ₁. Proof. Letv inK(ϕ^δ, ψ^δ), where K(ϕ^δ, ψ^δ) is given by

K(ϕ^δ, ψ^δ) =n

y ∈H₀¹(Ω)|ψ^δ ≥y≥ϕ^δ in Ωo .

• Ify^δ−ϕ^δ ≥0 and ψ^δ−y^δ ≥0,we getβ(y^δ−ϕ^δ) =β(ψ^δ−y^δ) = 0.

• Ify^δ−ϕ^δ ≥0 and ψ^δ−y^δ <0,then

β(ψ^δ−y^δ)(ψ^δ−y^δ)≥0 and β(y^δ−ϕ^δ) = 0.

For all v inK(ϕ^δ, ψ^δ),we have v−y^δ< v−ψ^δ<0, then

−β(ψ^δ−y^δ)(v−y^δ)≤0.

Then, by the following equation

σ(y^δ, v−y^δ) + (βδ(y^δ−ϕ^δ)−βδ(ψ^δ−y^δ), v−y^δ) =

=

f, v−y^δ

, for all v inK(ϕ^δ, ψ^δ), and, if we take v=ψ^δ, we get

σ(y^δ, y^δ)≤σ(y^δ, ψ^δ) +

f, y^δ−ψ^δ .

By the coercivity and continuity conditions ofσ(·,·) given respectively by H1 and H2, we get

cky^δk²H¹(Ω)≤Cky^δkH¹(Ω)kψ^δkH¹(Ω)+kfkL²(Ω)

Cky^δkH¹(Ω)+kψ^δkL²(Ω)

. From the above inequality, we can deduce that we have two cases

a. If kψ^δkL²(Ω)≤Cky^δkH¹(Ω)

Then, we get

ky^δkH¹(Ω) ≤Ckψ^δkH¹(Ω). b. If Cky^δkH¹(Ω)≤ kψ^δkL²(Ω), then we obtain

ky^δkH¹(Ω) ≤ Ckψ^δkH¹(Ω).

• Ify^δ−ϕ^δ<0 andψ^δ−y^δ≥0, we deduce

β(y^δ−ϕ^δ)(y^δ−ϕ^δ)≥0 and β(ψ^δ−y^δ) = 0.

For all v inK(ϕ^δ, ψ^δ),we have v−y^δ > v−ϕ^δ ≥0, then β(y^δ−ϕ^δ)(v−y^δ)≤0.

The following equation

σ(y^δ, v−y^δ) + (βδ(y^δ−ϕ^δ)−βδ(ψ^δ−y^δ), v−y^δ) =

f, v−y^δ

, for all vin K(ϕ^δ, ψ^δ), gives

σ(y^δ, y^δ−v)≤

f, y^δ−v

, for allv inK(ϕ^δ, ψ^δ).

If we takev =ϕ^δ, and by the coercivity and continuity conditions ofσ(·,·) given respec- tively by H₁ and H₂, we get

cky^δk²H¹(Ω) ≤Cky^δkH¹(Ω)kϕ^δkH¹(Ω)+kfkL²(Ω)

Cky^δkH¹(Ω)+kϕ^δkL²(Ω)

. From the above inequality, we can deduce the following two cases

a. If kϕ^δkL²(Ω) ≤Cky^δkH¹(Ω), we get

ky^δkH¹(Ω) ≤Ckϕ^δkH¹(Ω). b. If Cky^δkH¹(Ω)≤ kϕ^δkL²(Ω),we have

ky^δkH¹(Ω) ≤Ckϕ^δkH¹(Ω). Therefore, in all cases we get

ky^δkH¹(Ω) ≤ Ckϕ^δkH¹(Ω).

• The case wheny^δ−ϕ^δ≤0 and ψ^δ−y^δ≤0, is similar to case 3.

Lemma 2.4. For any pair p^δ, y^δ

in U ×(U ∩B_H¹(0, ρ₂)), satisfying the optimality system(S^δ), we have

kp^δkH¹≤Cky^δkH¹(Ω), and when y^δ belongs to B_H¹(0, ρ2)∩ U, we deduce that

kp^δ kH¹(Ω)≤ρ₃.

This means that p^δ belongs to B_H¹(0, ρ₃)∩ U, where ρ₃:=Cρ₂. Proof. From the adjoint equation of optimality system (S^δ),we have

σ^∗ p^δ, v

+

µ^δ₁+µ^δ₂, v

=

y^δ−z, v

, for allv in H₀¹(Ω) if we take v=p^δ, and by the coercivity condition H₂ of σ^∗(·,·), we obtain

kp^δkH¹(Ω)≤C ky^δkH¹(Ω).

3 Convergence study of an iterative algorithm

In this section, we give an algorithm to solve problem (P^δ). Roughly speaking, we pro- pose an implicit algorithm to solve the necessary optimality system (S^δ). The proposed algorithm is based on the Gauss-Seidel method and is given below.

This algorithm can be seen as a successive approximation method to compute the five points of the functionF that we are going to define. From the different steps of the above algorithm, we define the following functionsFi, fori= 1, 2,3,4 as

Algorithm 1 Gauss-Seidel algorithm (Continuous version)

1: Input :

y₀^δ, p^δ₀, ϕ^δ₀, ψ₀^δ, λ^δ₀, δ, ν, ε choose ϕ^δ₀, ψ^δ₀ inU, ε and δ inR^∗₊;

2: Begin:

3: Solve Ay^δ_n+ ¹_δ β y^δ_n−ϕ^δ_n−1

−β ψ^δ_n−1−y_n^δ

=f ony_n^δ

4: Solve A+ β_δ^′ y_n^δ −ϕ^δ_n−1

+β_δ^′ ψ_n−^δ ₁−y_n^δ

p^δ_n =y_n^δ−z onp^δ_n.

5: Calculate λ^δ_n =ν∆ϕ^δ_n−1+β_δ^′ y_n^δ −ϕ^δ_n−1

p^δ_n

6: Solve ν∆ψ_n^δ +β_δ^′ ψ_n^δ−y_n^δ

p^δ_n =−λ^δ_n onψ^δ_n.

7: Solve −λ^δ_n+ν∆ϕ^δ_n+β_δ^′ y^δ_n−ϕ^δ_n

p^δ_n= 0 on ϕ^δ_n.

8: If the stop criteria is fulfilled Stop.

9: Ensure : s^δ_n := y_n^δ, ϕ^δ_n, ψ^δ_n, p^δ_n

is a solution

10: Else; n←n+ 1, Go to Begin.

11: End if

12: End algorithm.

• From step 1, we defineF₁:U × U → U, such that y^δ_n:=F₁

ϕ^δ_n−1, ψ_n^δ−1

,

we see thatF₁ depends onϕ^δ_n−1, ψ^δ_n−1, and givesy^δ_nas the solution of the following state equation

Ay_n^δ+β_δ

y_n^δ−ϕ^δ_n−1

−β_δ

ψ^δ_n−1−y^δ_n

=f in Ω, andy_n^δ = 0 on ∂Ω. (8)

• From step 2, we defineF₂:U × U × U → U, such that p^δ_n:=F₂

y_n^δ, ϕ^δ_n−1, ψ^δ_n−1

, (9)

we see that F₂ depends on ϕ^δ_n−1, ψ_n^δ₋₁ and y^δ_n, and gives p^δ_n as the solution of the following adjoint state equation

Ap^δ_n+β_δ^′

y^δ_n−ϕ^δ_n−1

p^δ_n+β_δ^′

ψ^δ_n−1−y^δ_n

p^δ_n=y_n^δ−zin Ω, and p^δ_n= 0 on ∂Ω.

(10)

• From step 3, we defineF₃:U × U → U, such that ψ^δ_n:=F₃

y_n^δ, p^δ_n ,

we see thatF₃ depends onp^δ_n and y^δ_n, and gives ψ^δ_n as the solution of the following equation

−λ^δ_n=ν∆ψ^δ_n+β_δ^′

ψ^δ_n−y_n^δ

p^δ_n in Ω, and ψ_n^δ = 0 on ∂Ω. (11)

• From step 4, we defineF₄:U × U → U, such that ϕ^δ_n:=F₄

y_n^δ, p^δ_n ,

we see thatF₄ depends onp^δ_n, andy_n^δ, and givesϕ^δ_nas the solution of the optimality condition equation

−λ^δ_n+ν∆ϕ^δ_n+β_δ^′

y_n^δ−ϕ^δ_n

p^δ_n= 0 in Ω, and ϕ^δ_n= 0 on ∂Ω. (12) Remark 3.1. We note that the equation given by (11) is only used to solve the equation given by (12).

Then according the above definitions ofFi,wherei= 1,2,3,4, let us define the map F :U × U → U × U, as

ϕ^δ_n, ψ^δ_n

:=F

ϕ^δ_n−1, ψ_n^δ−1

, where

ϕ^δ_n:= ˜F₁

ϕ^δ_n−1, ψ_n^δ−1

:=F₄ F₁

ϕ^δ_n−1, ψ^δ_n−1

, F₂

F₁

ϕ^δ_n−1, ψ^δ_n−1

, ϕ^δ_n−1, ψ^δ_n−1

, and

ψ^δ_n:= ˜F₂

ϕ^δ_n−1, ψ_n^δ−1

:=F₃ F₁

ϕ^δ_n−1, ψ_n^δ−1

, F₂

F₁

ϕ^δ_n−1, ψ_n^δ−1

, ϕ^δ_n−1, ψ_n^δ−1

such that

F

ϕ^δ_n−1, ψ_n^δ₋₁

= F˜₁

ϕ^δ_n−1, ψ^δ_n−1 ,F˜₂

ϕ^δ_n−1, ψ^δ_n−1 Proposition 3.1. Let ϕ^δ_n−1 and ψ_n−^δ ₁belong to U and y_n^δ, ϕ^δ_n, ψ^δ_n, p^δ_n

satisfies equations (8), (10), (11)and (12) given respectively by F₁, F₂, F₃ and F₄ such that δ≤C, then we get

ky_n^δ kH¹(Ω)≤ c

δ kϕ^δ_n−1 kH²(Ω) +c

δ kψ^δ_n−1 kH²(Ω)+ckf kL²(Ω) (13) kp^δ_nkH¹(Ω)≤C+C ky^δ_nkH¹(Ω) (14) kϕ^δ_nkH²(Ω)≤ _δν^C kp^δ_nkH¹(Ω)+C (15) and

kψ_n^δ kH²(Ω)≤ δν^C kp^δ_nkH¹(Ω)+C (16)

Proof. From the state equation (8), we write σ

y_n^δ, y_n^δ +

β_δ

y^δ_n−ϕ^δ_n−1

−β_δ

ψ^δ_n−1−y_n^δ , y_n^δ

= f, y^δ_n

.

Then, by the definition of βδ(·), and by the coercivity condition H₂ of the bilinear form σ(·,·) and thanks to the Mean-Value Theorem applied in the interval of sides {0, y^δ_n−ϕ^δ_n−1

}and {0, ψ^δ_n−1−y^δ_n

}, we obtain

cky_n^δ k²H¹(Ω)≤ c δ

cky^δ_nkH¹(Ω) +kϕ^δ_n−1kL²(Ω)

kϕ^δ_n−1kL²(Ω) + +c

δ

kψ_n^δ−1 kL²(Ω)+cky_n^δ kH¹(Ω)

kψ^δ_n−1 kL²(Ω)+ckf kL²(Ω)ky_n^δ kH¹(Ω) (17) Then from the above inequality (17), we deduce that we have four cases:

1. Ifkϕ^δ_n−1 kL²(Ω)≤cky^δ_nkH¹(Ω) andkψ^δ_n−1 kL²(Ω)≤cky_n^δ kH¹(Ω), then ky_n^δ kH¹(Ω)≤ c

δ kϕ^δ_n−1 kL²(Ω)+c

δ kψ_n^δ₋₁ kL²(Ω)+ckf kL²(Ω) . 2. Ifkϕ^δ_n−1 kL²(Ω)≤cky^δ_nkH¹(Ω) andkψ^δ_n−1 kL²(Ω)≥cky_n^δ kH¹(Ω), then, we get

ky_n^δ kH¹(Ω)≤c

δ kϕ^δ_n−1 kL²(Ω)+ckf kL²(Ω)

+ c

√δ kψ_n^δ−1kL²(Ω) . 3. Ifkϕ^δ_n−1 kL²(Ω)≥cky^δ_nkH¹(Ω) andkψ^δ_n−1 kL²(Ω)≥cky_n^δ kH¹(Ω), thus

ky^δ_nkH¹(Ω)≤ckf kL²(Ω)+ c

√δ kϕ^δ_n−1 kL²(Ω)+ c

√δ kψ^δ_n−1 kL²(Ω). 4. Ifkϕ^δ_n−1 kL²(Ω)≥cky^δ_nkH¹(Ω) andkψ^δ_n−1 kL²(Ω)≤cky_n^δ kH¹(Ω), therefore

ky^δ_nkH¹(Ω)≤ c

δ kψ^δ_n−1 kL²(Ω)+ckf kL²(Ω)+ c

√δ kϕ^δ_n−1kL²(Ω) . Finally, in any cases we obtain

ky^δ_nkH¹(Ω)≤c

δ kϕ^δ_n−1 kL²(Ω)+c

δ kψ_n^δ₋₁kL²(Ω)+ckf kL²(Ω)

. Now, from the adjoint state equation (10), we get

hAp^δ_n, p^δ_ni+ β_δ^′

y^δ_n−ϕ^δ_n−1

p^δ_n+β_δ^′

ψ_n^δ₋₁−y^δ_n

p^δ_n, p^δ_n

=

y^δ_n−z, p^δ_n in Ω.

By the coercivity condition of σ^∗(·,·) given by H2, we obtain that kp^δ_nkH¹(Ω)≤Cky^δ_nkH¹(Ω),

by using the following equation λ^δ_n=ν∆ϕ^δ_n−1+β_δ^′

y^δ_n−ϕ^δ_n−1

p^δ_n and λ^δ_n= 0 on∂Ω we deduce that