Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

DOI: 10.1007/s00245-003-0761-y

Appl Math Optim 47:213–230 (2003)

© 2003 Springer-Verlag New York Inc.

Convergence of Distributed Optimal Controls on theInternal Energy in Mixed Elliptic Problems when theHeat Transfer Coefficient Goes to Infinity∗

Claudia M. Gariboldi1 and Domingo A. Tarzia2

1Departamento Matematica, FCEFQyN, Univ. Nac. de Rıo Cuarto,Ruta 36 Km 601, 5800 Rıo Cuarto, [email protected]

2Departamento Matematica-CONICET, FCE, Univ. Austral,Paraguay 1950, S2000FZF Rosario, [email protected]

Communicated by A. Bensoussan

Abstract. We consider a steady-state heat conduction problem Pα with mixedboundary conditions for the Poisson equation depending on a positive parameter α,which represents the heat transfer coefficient on a portion �1 of the boundary of agiven bounded domain in Rn . We formulate distributed optimal control problemsover the internal energy g for each α. We prove that the optimal control gopα andits corresponding system ugopα α

and adjoint pgopα αstates for each α are strongly

convergent to gop, ugop and pgop , respectively, in adequate functional spaces. We alsoprove that these limit functions are respectively the optimal control, and the systemand adjoint states corresponding to another distributed optimal control problem forthe same Poisson equation with a different boundary condition on the portion �1.We use the fixed point and elliptic variational inequality theories.

Key Words. Variational inequality, Distributed optimal control, Mixed ellipticproblem, Adjoint state, Steady-state Stefan problem, Optimality condition, Fixedpoint.

AMS Classification. 49J20, 35J85, 35R35.

∗ This paper has been partially sponsored by the Project “Free Boundary Problems for the Heat-DiffusionEquation” from CONICET - UA, Rosario (Argentina).

214 C. M. Gariboldi and D. A. Tarzia

1. Introduction

We consider a bounded domain� inRn whose regular boundary � consists of the unionof two disjoint portions �1 and �2 with meas(�1) > 0 and meas(�2) > 0. We denotewith meas(�) the (n − 1)-dimensional Lebesgue measure of �.

We consider the following two steady-state heat conduction problems P and Pα (foreach parameter α > 0), respectively, with mixed boundary conditions:

−u = g in �, u|�1 = b, −∂u

∂n

∣∣∣∣�2

= q, (1)

and

−u = g in �, −∂u

∂n

∣∣∣∣�1

= α(u − b), −∂u

∂n

∣∣∣∣�2

= q, (2)

where g is the internal energy in�, b is the temperature on�1 for (1) and the temperatureof the external neighborhood of �1 for (2), q is the heat flux on �2 and α > 0 is the heattransfer coefficient of �1 (Newton’s law on �1), that satisfy the following assumptions:

g ∈ H = L2(�), q ∈ L2(�2), b ∈ H 1/2(�1). (3)

Problems (1) and (2) can be considered as the steady-state Stefan problem for suitabledata q , g and b [5], [8], [11], [17], [18], [20].

Let ug and ugα be the unique solutions of the mixed elliptic problems (1) and (2),respectively, whose variational equalities are given by [14]

a(ug, v) = Lg(v), ∀v ∈ V0, ug ∈ K , (4)

and

aα(ugα, v) = Lgα(v), ∀v ∈ V, ugα ∈ V, (5)

where

V = H 1(�), V0 = {v ∈ V/ v|�1 = 0},

K = v0 + V0, (g, h) = (g, h)H =∫�

gh dx, (6)

a(u, v) =∫�

∇u · ∇v dx, aα(u, v) = a(u, v)+ α∫�1

bv dγ,

Lg(v) = (g, v)H −∫�2

qv dγ, Lgα(v) = Lg(v)+ α∫�1

bv dγ

for a given v0 ∈ V, v0|�1 = b.We consider g as a control variable for the cost functionals J : H→R

+0 and

Jα: H→R+0 respectively given by

J (g) = 12‖ug − zd‖2

H +M

2‖g‖2

H (7)

Convergence of Distributed Optimal Controls Problems 215

and

Jα(g) = 12‖ugα − zd‖2

H +M

2‖g‖2

H , (8)

where zd ∈ H is given and M = const. > 0.Then we can formulate the following distributed optimal control problems [7], [9],

[10], [15]:

Find gop ∈ H such that J (gop) = ming∈H

J (g) (9)

and

Find gopα ∈ H such that Jα(gopα ) = ming∈H

Jα(g), (10)

respectively.The use of variational inequality theory in connection with optimal control problems

was done, for example, in [1]–[4], [6], [13] and [16]. In [12] an optimization problemcorresponding to (1) is studied in order to avoid a change phase process.

In Section 2 we prove that the functional J is coercive and Gateaux differentiableon H , and J ′ is a Lipschitzian and strictly monotone application on H . We also provethe existence and uniqueness of the distributed optimal control problem (9) and wecharacterize this optimal energy gop as a fixed point on H of a suitable operator W overits adjoint state pg for a large parameter M .

Similary, in Section 3 we prove that the functional Jα is coercive and Gateauxdifferentiable on H , and J ′α is a Lipschitzian and strictly monotone application on H forall α > 0. We also prove the existence and uniqueness of the distributed optimal controlproblem (10) and we characterize this optimal energy gopα as a fixed point on H of asuitable operator Wα over its adjoint state pgα for a large parameter M .

In Section 4 we study the convergence when α→∞ of the optimal control problem(10) corresponding to the state system (2). We prove that the optimal state system ugopα α

and the optimal adjoint system pgopα αof problem (10) are strongly convergent in V to

the corresponding ugop and pgop for problem (9), respectively, when α→∞. Finally,the strong convergence in H of the optimal control gopα of problem (10) to the optimalcontrol gop of problem (9) is also proved when α→∞.

2. Problem P and Its Corresponding Optimal Control Problem

Let C : H→ V0 be the application such that

C(g) = ug − u0, (11)

where u0 is the solution of problem (4) for g = 0 whose variational equality is given by

a(u0, v) = L0(v), ∀v ∈ V0, u0 ∈ K , (12)

with

L0(v) = −∫�2

qv dγ.


Let �: H × H→R and L: H→R be defined by the following expressions:

�(g, h) = (C(g),C(h))+ M(g, h), ∀g, h ∈ H, (13)

L(g) = (C(g), zd − u0), ∀g ∈ H.

We have that a is a bilinear, continuous and symmetric form on V and coercive onV0, that is [14],

∃λ > 0 such that a(v, v) ≥ λ‖v‖2V , ∀ v ∈ V0. (14)

Lemma 2.1.

(i) C is a linear and continuous application.(ii) � is linear, continuous, symmetric and coercive form on H, that is,

�(g, g) ≥ M‖g‖2H , ∀g ∈ H. (15)

(iii) L is linear and continuous on H .(iv) J can be also written as

J (g) = 12�(g, h)− L(g)+ 1

2‖u0 − zd‖2H , ∀g ∈ H. (16)

(v) There exists a unique optimal control gop ∈ H such that

J (gop) = ming∈H

J (g). (17)

(vi) The application g ∈ H→ ug ∈ V is Lipschitzian, that is,

‖ug2 − ug1‖V ≤ 1

λ‖g2 − g1‖H , ∀g1, g2 ∈ H. (18)

Proof. (i)–(iii) This follows as in [12] and [15].(iv) From the definitions of J , � and L , we have

J (g) = 12‖ug + u0 − u0 − zd‖2

H +M

2‖g‖2

H

= 12‖ug − u0‖2

H − (ug − u0, zd − u0)+ 12‖u0 − zd‖2

H +M

2‖g‖2

H

= 12�(g, h)− L(g) + 1

2‖u0 − zd‖2H .

(v) This is a result of (ii)–(iv) [14], [15].(vi) If we take v = ug1 − ug2 ∈ V0 in the variational equality (4) for ug2 , that is,

a(ug2 , ug1 − ug2) = (g2, ug1 − ug2)−∫�2

q(ug1 − ug2) dγ,

and if we take v = ug2 − ug1 ∈ V0 in the variational equality (4) for ug1 , that is,

a(ug1 , ug2 − ug1) = (g1, ug2 − ug1)−∫�2

q(ug2 − ug1) dγ,


then we obtain

a(ug2 − ug1 , ug2 − ug1) = (g2 − g1, ug2 − ug1)

and taking into account that a is a coercive form we get

λ‖ug2 − ug1‖2V ≤ a(ug2 − ug1 , ug2 − ug1) ≤ ‖g2 − g1‖H‖ug2 − ug1‖H ,

and therefore (18).

We define the adjoint state pg corresponding to (1) or (4), for each g ∈ H, as theunique solution of the following mixed elliptic problem:

−pg = ug − zd in �, pg|�1 = 0,∂pg

∂n

∣∣∣∣�2

= 0, (19)

whose variational formulation is given by

a(pg, v) = (ug − zd , v), ∀v ∈ V0, pg ∈ V0. (20)

Now we will obtain some useful properties of the functional J .

Lemma 2.2.

(i) J is a Gateaux differentiable functional and J ′ is given by

〈J ′(g), h〉 = (ug − zd ,C(h))+ M(g, h) = �(g, h)− L(g),

∀g, h ∈ H. (21)

(ii) The adjoint state pg satisfy the following equalities:

(pg, h) = (ug − zd ,C(h)) = a(pg,C(h)). (22)

(iii) The Gateaux derivative of J can be written as

J ′(g) = pg +Mg, ∀g ∈ H. (23)

(iv) The optimality condition for problem (9) is given by J ′(gop) = 0 in H, that is,

pgop +Mgop = 0 in H. (24)

Proof. (i) For t > 0, we have

1

t[J (g + t ( f − g))− J (g)] = t

2(u f − ug, u f − ug)+ (ug − zd , u f − ug)

+ M(g, f − g)+ Mt

2( f − g, f − g),

and passing to the limit t→ 0, we obtain (21).


(ii) This results from the definition of pg and taking into account that

a(pg,C(h)) = a(pg, uh − u0) = a(pg, uh)− a(pg, u0) = (pg, h).

(iii), (iv) They follow from (21), (22) and [14] and [15].

Let the operator W : H→ V0 ⊂ H be defined by

W (g) = − 1

Mpg, g ∈ H. (25)

We will prove the following property:

Lemma 2.3. W is a Lipschitz operator over H , i.e.

‖W (g2)−W (g1)‖H ≤ 1

λ2 M‖g1 − g2‖H , ∀g1, g2 ∈ H, (26)

and it is a contraction for all M > 1/λ2.

Proof. If we take v = pg2 − pg1 in the variational equality (20) for g2 we obtain

a(pg2 , pg2 − pg1) = (ug2 − zd , pg2 − pg1)

and in a similar way we have

a(pg1 , pg1 − pg2) = (ug1 − zd , pg1 − pg2).

Therefore we obtain

a(pg2 − pg1 , pg2 − pg1) = (ug2 − ug1 , pg2 − pg1)

and by using the coercivieness of the bilinear form a we have

λ‖pg2 − pg1‖2V ≤ a(pg2 − pg1 , pg2 − pg1) ≤ ‖ug2 − ug1‖H‖pg2 − pg1‖H ,

therefore

‖pg2 − pg1‖V ≤ 1

λ‖ug2 − ug1‖H . (27)

Next, taking into account inequalities (18) and (27) we obtain

‖W (g2)−W (g1)‖H ≤ 1

M‖pg2 − pg1‖H

≤ 1

λM‖ug2 − ug1‖H

≤ 1

λ2 M‖g1 − g2‖H ,

that is, (26).


Now we are in the condition for proving other properties of the functional J .

Lemma 2.4.

(i) The application g ∈ H→ pg ∈ V0 is strictly monotone. Moreover, we have

(pg2 − pg1 , g2 − g1) =∥∥ug2 − ug1

∥∥2H ≥ 0, ∀g1, g2 ∈ H. (28)

(ii) J is coercive or H-elliptic, that is,

(1− t)J (g2)+ t J (g1)− J ((1− t)g2 + tg1)

= t (1− t)

2[‖ug2 − ug1‖2

H + M‖g2 − g1‖2H ]

≥ Mt(1− t)

2‖g2 − g1‖2

H , ∀g1, g2 ∈ H, ∀t ∈ [0, 1]. (29)

(iii) J ′ is a Lipschitzian and strictly monotone application, that is,

‖J ′(g2)− J ′(g1)‖H ≤(

M + 1

λ2

)‖g1 − g2‖H (30)

and

〈J ′(g2)− J ′(g1), g2 − g1〉 = ‖ug2 − ug1‖2H + M‖g2 − g1‖2

H

≥ M‖g2 − g1‖2H , ∀g1, g2 ∈ H. (31)

Proof. (i) We have

(pg2 − pg1 , g2 − g1) = (pg2 , g2 − g1)− (pg1 , g2 − g1)

= (ug2 − zd ,C(g2 − g1))− (ug1 − zd ,C(g2 − g1))

= (ug2 − ug1 ,C(g2 − g1))

= ‖ug2 − ug1‖2H ≥ 0, ∀g1, g2 ∈ H.

(ii) For all g1, g2 ∈ H, t ∈ [0, 1] we get

(1− t)J (g2)+ tJ(g1)− J ((1− t)g2 + tg1)

= (1− t)

[12‖ug2 − zd‖2

H +M

2‖g2‖2

H

]

+ t

[12‖ug1 − zd‖2

H +M

2‖g1‖2

H

]

−[

12‖u(1−t)g2+tg1

− zd‖2H +

M

2‖(1− t)g2 + tg1‖2

H

]

= 12 [(1− t)‖ug2 − zd‖2

H + t ‖ug1 − zd‖2H − ‖(1− t)ug1 + tug2 − zd‖2

H

+ (1− t)M‖g2‖2H + tM‖g1‖2

H − M‖(1− t)g2 + tg1‖2H ]


= t (1− t)

2[ ‖ug2 − ug1‖2

H + M‖g2 − g1‖2H ]

≥ Mt (1− t)

2‖g2 − g1‖2

H .

(iii) By using (18), (23) and (27) we have

‖J ′(g2)− J ′(g1)‖H ≤ ‖pg2 − pg1‖H + M‖g2 − g1‖H

≤(

M + 1

λ2

)‖g2 − g1‖H

and

〈J ′(g2)− J ′(g1), g2 − g1〉 = (pg2 + Mg2 − (pg1 +Mg1), g2 − g1)

= (pg2 − pg1 , g2 − g1)+ M(g2 − g1, g2 − g1)

= ‖ug2 − ug1‖2H + M‖g2 − g1‖2

H ≥ M‖g2 − g1‖2H ,

that is, (30) and (31), respectively.

We present an iterative algorithm in order to obtain gop. For each ρ we define thefollowing sequence (gn) given by [7], [10]

g0 ∈ H (given, arbitrarily), gn+1 = (1− ρM)gn − ρpgn , ∀n ≥ 0, (32)

which will converge to gop for a suitable ρ.

Lemma 2.5. If ρ is chosen satisfying the inequalities

0 < ρ <2M

(M + 1/λ2)2, (33)

then the algorithm (32) is strongly convergent in H to the optimal control gop of (9)independently of g0, that is,

limn→∞‖gn − gop‖H = 0, for any g0 ∈ H. (34)

Proof. The operator T : H→ H defined by

T (g) = (1− ρM)g − ρpg (35)

is a Lipschitz operator, that is,

‖T (g2)− T (g1)‖H ≤√γ (ρ)‖g2 − g1‖H , ∀g1, g2 ∈ H, (36)

where γ (ρ) is given by

γ (ρ) = 1− 2Mρ +(

M + 1

λ2

)2

ρ2, (37)


because

‖T (g2)− T (g1)‖2H = ‖g2 − ρ J ′(g2)− g1 + J ′(g1)‖2

H

= ‖g2 − g1‖2H − 2ρ (g2 − g1, J ′(g2)− J ′(g1)

+ ρ2‖J ′(g2)− J ′(g1)‖2H

≤ ‖g2 − g1‖2H − 2ρM‖g2 − g1‖2

H

+ ρ2

(M + 1

λ2

)2

‖g2 − g1‖2H

=[

1− 2Mρ +(

M + 1

λ2

)2

ρ2

]‖g2 − g1‖2

H .

Therefore T will be a contraction if and only if 0 ≤ γ (ρ) < 1, that is, inequality (33).Moreover, the ρ and γ optimals are given by

ρop = M

(M + 1/λ2)2, γop = 1−

(M

M + 1/λ2

)2

. (38)

3. Problem Pα and Its Corresponding Optimal Control Problem

Let �α: H × H → R, Lα: H → R and Cα: H → V be defined by

�α(g, h) = (Cα(g),Cα(h))+ M(g, h), ∀g, h ∈ H,

Lα(g) = (Cα(g), zd − u0α), ∀g ∈ H, (39)

Cα(g) = ugα − u0α, ∀g ∈ H,

where ugα is the unique solution of the variational equality (5), u0α is the unique solutionof (5) for g = 0 whose variational equality is given by

aα(u0α, v) = L0α(v), ∀v ∈ V, u0α ∈ V, (40)

with

L0α(v) = α∫�1

bv dγ −∫�2

qv dγ, (41)

and aα is a bilinear, continuous, symmetric and coercive form on V, that is,

aα(v, v) ≥ λα‖v‖2V , ∀v ∈ V, (42)

where λα = λ1 min(1, α) > 0 for all α > 0 and λ1 is the coerciveness constant for thebilinear form a1 [19].

We can obtain similar properties to Lemma 2.1, following [12], [14], [15] and [18],the proof of which is omitted.


Lemma 3.1.

(i) Cα is a linear and continuous application.(ii) �α is linear, continuous, symmetric and coercive on H, that is,

�α(g, g) ≥ M‖g‖2H , ∀g ∈ H. (43)

(iii) Lα is linear and continuous on H .(iv) Jα can be also written as

Jα(g) = 12�α(g, h)− Lα(g)+ 1

2‖u0α − zd‖2H , ∀g ∈ H. (44)

(v) There exists a unique optimal control gopα ∈ H such that

Jα(gopα ) = ming∈H

Jα(g). (45)

(vi) The application g ∈ H → ugα ∈ V is Lipschitzian, that is,

‖ug2α − ug1α‖V ≤ 1

λα‖g2 − g1‖H , ∀g1, g2 ∈ H. (46)

We define the adjoint state pgα as the unique solution of the following mixed ellipticproblem corresponding to (2) or (5), for each g ∈ H and α > 0:

−pgα = ugα − zd in �, −∂pgα

∂n

∣∣∣∣�1

= αpgα,∂pgα

∂n

∣∣∣∣�2

= 0, (47)

whose variational formulation is given by

aα(pgα, v) = (ugα − zd , v), ∀v ∈ V, pgα ∈ V, (48)

where ugα is the unique solution of (5).Now we obtain some properties of the functional Jα .

Lemma 3.2. Let α > 0.

(i) The Gateaux derivative J ′α is given by

〈J ′α(g), h〉 = (ugα − zd ,Cα(h))+ M(g, h) = �α(g, h)− Lα(g),

∀g, h ∈ H. (49)

(ii) The adjoint state pgα satisfies the following equalities:

(pgα, h) = (ugα − zα, Cα(h)) = aα(pgα, Cα(h)), ∀g, h ∈ H. (50)

(iii) The Gateaux derivative of Jα can be written as

J ′α(g) = pgα +Mg, ∀g ∈ H. (51)

(iv) The optimality condition for problem (10) is given by J ′α(gopα ) = 0 in H, thatis,

pgopα α+Mgopα

= 0 in H. (52)


Proof. (i) We have

1

t[Jα(g + t ( f − g))− Jα(g)]

= t

2(u f α − ugα, u f α − ugα)+ (ugα − zd , u f α − ugα)

+ M(g, f − g)+ Mt

2( f − g, f − g)

and passing to the limit t → 0, we obtain (49).(ii) This results from the definition of pgα and taking into account that

a(pgα,Cα(h)) = a(pgα, uh − u0) = a(pgα, uh)− a(pgα, u0) = (pgα, h).

Remark 1. We note the double dependence on the parameter α for the optimal statesystem ugopα α

and the adjoint state pgopα α.

Let the operator Wα: H → V ⊂ H be defined by

Wα(g) = − 1

Mpgα, ∀g ∈ H. (53)

We have the following property:

Lemma 3.3. Wα is a Lipschitz operator over H, that is,

‖Wα(g2)−Wα(g1)‖H ≤ 1

λ2αM‖g1 − g2‖H , ∀g1, g2 ∈ H, (54)

and it is a contraction for all M > 1/λ2α .

Proof. If we take v = pg2α− pg1α in variational equality (48) for g2 and g1, respectively,by substracting them and by using the coerciveness of aα we have

λα‖pg2α − pg1α‖2H ≤ aα(pg2α − pg1α, pg2α − pg1α)

= (ug2α − ug1α, pg2α − pg1α)H

≤ ‖ug2α − ug1α‖H‖pg2α − pg1α‖H ,

therefore

‖pg2α − pg1α‖V ≤ 1

λα‖ug2α − ug1α‖H . (55)

Next, taking into account inequalities (46) and (55) we obtain

‖Wα(g2)−Wα(g1)‖H = 1

M‖pg2α − pg1α‖V

≤ 1

λαM‖ug2α − ug1α‖H

≤ 1

λ2αM‖g1 − g2‖H ,

that is, (54).


Now, we prove other properties of the functional Jα .

Lemma 3.4.

(i) The operator g ∈ H → pgα ∈ V is strictly monotone, that is,

(pg2α − pg1α, g2 − g1) = ‖ug2α − ug1α‖2H ≥ 0, ∀g1, g2 ∈ H. (56)

(ii) Jα is coercive or H-elliptic, that is,

(1− t)Jα(g2)+ t Jα(g1)− Jα((1− t)g2 + tg1)

= t (1− t)

2[‖ug2α − ug1α‖2

H + M‖g2 − g1‖2H ]

≥ Mt(1− t)

2‖g2 − g1‖2

H , ∀g1, g2 ∈ H, ∀t ∈ [0, 1]. (57)

(iii) J ′α is a Lipschitzian and strictly monotone operator, that is,

‖J ′α(g2)− J ′α(g1)‖H ≤(

M + 1

λ2α

)‖g1 − g2‖H , ∀g1, g2 ∈ H, (58)

and

〈J ′α(g2)− J ′α(g1), g2 − g1〉 = ‖ug2α − ug1α‖2H + M‖g2 − g1‖2

H

≥ M‖g2 − g1‖2H , ∀g1, g2 ∈ H. (59)

Proof. (i) We have

(pg2α − pg1α, g2 − g1) = (ug2α − ug1α,Cα(g2 − g1))

= ‖ug2α − ug1α‖2H ≥ 0, ∀g1, g2 ∈ H.

(ii) For all g1, g2 ∈ H, t ∈ [ 0, 1] we obtain

(1− t)Jα(g2)+ t Jα(g1)− Jα((1− t)g2 + tg1)

= (1− t)

[12‖ug2α − zd‖2

H +M

2‖g2‖2

H

]

+ t

[12‖ug1α − zd‖2

H +M

2‖g1‖2

H

]

−[

12‖u(1−t)g2+g1α − zd‖2

H +M

2‖(1− t)g2 + tg1‖2

H

]

= t (1− t)

2[‖ug2α − ug1α‖2

H + M‖g2 − g1‖2H ]

≥ Mt(1− t)

2‖g2 − g1‖2

H .


(iii) By using (46) and (55) we have

‖J ′α(g2)− J ′α(g1)‖H ≤ ‖pg2α − pg1α‖H + M‖g2 − g1‖H

≤(

M + 1

λ2α

)‖g2 − g1‖H ,

then J ′α is a Lipschitzian application. On the other hand we get

〈J ′α(g2)− J ′α(g1), g2 − g1〉 = (pg2α +Mg2 − (pg1α +Mg1), g2 − g1)

= ‖ug2α − ug1α‖2H + M‖g2 − g1‖2

H

≥ M‖g2 − g1‖2H ,

and J ′α is a strictly monotone application.

Now, we prove the following result of convergence when α→∞.

Lemma 3.5. For all α > 0, q ∈ L2(�2), b ∈ H 1/2(�1), we have the following limits:

(i) limα→∞ ‖ugα − ug‖V = 0,∀g ∈ H,

(ii) limα→∞ ‖u0α − u0‖V = 0, (60)

(iii) limα→∞ ‖pgα − pg‖V = 0,∀g ∈ H.

Proof. (i) If we take v = ugα − ug in the variational equality (5), for g, α, with α > 1(because α→∞), following [18] and [19] we obtain

λ1‖ugα − ug‖2V + (α − 1)

∫�1

(ugα − ug)2 dγ ≤ aα(ugα − ug, ugα − ug)

≤ C1‖ugα − ug‖V , (61)

with C1 a constant independent of α. Next for large α we obtain

(a) ‖ugα − ug‖2V ≤

C1

λ1, (b) (α − 1)

∫�1

(ugα − ug)2 dγ ≤ (C1)

2

λ1, (62)

and we deduce that there exists wg ∈ V such that

(a) ugα ⇀ wg weakly in V, (b)∫�1

(ugα − b)2 dγ ≤ (C1)2

λ1

1

(α − 1)→ 0,

as α→∞, (63)

that is, wg ∈ K and taking the limit of the variational equality (5) as α→∞ we have

a(wg, v) = Lg(v), ∀v ∈ V0, wg ∈ K , (64)

and, by uniqueness, we have wg = ug .


Therefore, ugα → ug strongly in V as α→∞ because of the following inequality:

λ1‖ugα − ug‖2V ≤ Lg(ugα − ug)− a(ugα, ugα − ug).

For case (ii) we take g = 0 in (i).(iii) In this case we take v = pgα − pg in the variational equality (48) for g, α and

following a similar method as before we obtain

λ1‖pgα − pg‖2V + (α − 1)

∫�1

(pgα − pg)2 dγ ≤ aα(pgα − pg, pgα − pg)

≤ C2‖pgα − pg‖V ,

with C2 a constant independent of α . Next, for large α, we have

(a) ‖pgα − pg‖2V ≤

C2

λ1, (b) (α − 1)

∫�1

(pgα − pg)2 dγ ≤ (C2)

2

λ1, (65)

and we deduce that there exists ξg ∈ V such that

(a) pgα ⇀ ξg weakly in V,

(b)∫�1

(pgα − pg)2 dγ ≤ (C2)

2

λ1(α − 1)→ 0, as α→∞, (66)

that is, ξg ∈ V0 and taking the limit on the variational equality (48) for pgα we have

a(ξg, v) = (ug − zd , v), ∀v ∈ V0, ξg ∈ V0, (67)

and, by uniqueness, we obtain ξg = pg . Therefore, taking into account the followinginequality,

λ1‖pgα − pg‖2V ≤ (ugα − zd , pgα − pg)− a(pg, pgα − pg),

we have that pgα → pg strongly in V .

4. Convergence of Problem Pα and Its Corresponding Optimal Control asα→∞

In this section we prove that the optimal control gopα of problem (10) and its correspondingadjoint state pgopα α

(48) are convergent to the optimal control gop of problem (9) and itscorresponding adjoint state pgop (20), respectively, when the parameter α (heat transfercoefficient on �1) goes to infinity.

Theorem 4.1. Let M > 1/λ21. Then we have:

(i) If pgop and pgopα αare the corresponding adjoint states of problems (9) and (10),

respectively, then

limα→∞‖pgopα α

− pgop‖V = 0. (68)


(ii) If gop and gopα are the solutions of problems (9) and (10), respectively, then

limα→∞‖gopα − gop‖H = 0. (69)

(iii) If ugop and ugopα αare the corresponding solutions of problems P and Pα ,

respectively, then

limα→∞‖ugopα α

− ugop‖V = 0. (70)

Proof. We prove some preliminary results for the three cases.Since gopα is the solution of problem (10), we have the following inequality:

12‖ugopα α

− zd‖2H +

M

2‖gopα‖2

H ≤ 12‖ugα − zd‖2

H +M

2‖g‖2

H , ∀g ∈ H,

then, taking g = 0, we have

12‖ugopα α

− zd‖2H +

M

2‖gopα‖2

H ≤ 12‖u0α − zd‖2

H ≤ C3, ∀α > 0,

where C3 is a constant independent of parameter α because u0α is convergent whenα→∞. Therefore

‖gopα‖H ≤ C4 and ‖ugopα α‖H ≤ C5, (71)

where C4 and C5 are constants independent of α.Now, if we take v = ugopα α

− ugop in the variational equality (5), following [18] weobtain, for α > 1,

λ1‖ugopα α− ugop‖2

V + (α − 1)∫�1

(ugopα α− ugop)

2 dγ

≤ aα(ugopα α− ugop , ugopα α

− ugop)

≤ C6‖ugopα α− ugop‖V ,

where C6 = C6(gop, q, ugop) is independent of α. Next, we have

(a) ‖ugopα α− ugop‖2

V ≤C6

λ1,

(b) (α − 1)∫�1

(ugopα α− ugop)

2 dγ ≤ (C6)2

λ1,

(72)

and therefore we deduce that

∃ η ∈ V such that ugopα α⇀ η weakly in V, (73)


and because of the following inequalities,

0 ≤∫�1

(η − ugop)2 dγ ≤ lim inf

α→∞

∫�1

(ugopα α− ugop)

2 dγ = 0,

we obtain that η ∈ K .Next, if we take v = pgopα α

− pgop in the variational equality (48) we get

λ1‖pgopα α− pgop‖2

V + (α − 1)∫�1

(pgopα α− pgop)

2 dγ

≤ aα(pgopα α− pgop , pgopα α

− pgop)

≤ C7‖pgopα α− pgop‖V ,

with C7 = C7(C5, pgop). Next, we obtain

(a) ‖pgopα α− pgop‖2

V ≤C7

λ1,

(b) (α − 1)∫�1

(pgopα α− pgop)

2 dγ ≤ (C7)2

λ1,

(74)

and therefore we deduce that

∃ξ ∈ V such that pgopα α⇀ ξ weakly in V (75)

and by the following inequality,

0 ≤∫�1

(ξ − pgop)2 dγ ≤ lim inf

α→∞

∫�1

(pgopα α− pgop)

2 dγ = 0,

we obtain ξ ∈ V0.Now we consider v ∈ V0 and, taking into acount (73) and (75), from the variational

equality (48) we have

a(ξ, v) = (η − zd , v), ∀v ∈ V0, ξ ∈ V0. (76)

Next, from (71) we deduce that there exists f ∈ H such that gopα ⇀ f weakly inH . Therefore if we put v ∈ V0 in the variational equality (5) and we pass to the limitα→∞, we obtain

a(η, v) = ( f, v)−∫�2

qv dγ, ∀v ∈ V0, η ∈ K . (77)

Now, taking into account Lemma 3.3 and the facts that gopα ⇀ f weakly in H andpgopα α

⇀ ξ weakly in V , we have

f = − 1

Mξ in H. (78)


Therefore from the uniqueness of fixed point we have

gop = − 1

Mpgop in H, (79)

and then we obtain that f = gop, η = ugop and ξ = pgop .Moreover, from (75) and the following computation,

λ1‖pgopα α− pgop‖2

V ≤ aα(pgopα α− pgop , pgopα α

− pgop)

= aα(pgopα α, pgopα α

− pgop)− a(pgop , pgopα α− pgop)

= (ugopα α− zd , pgopα α

− pgop)− a(pgop , pgopα α− pgop)

we have (68).From Lemmas 2.3 and 3.3 it results that

‖gopα − gop‖H = 1

M‖pgop − pgopα α

‖H ≤ 1

M‖pgop − pgopα α

‖V

and therefore (69) holds.Now we have

λ1‖ugopα α− ugop‖2

V ≤ aα(ugopα α− ugop , ugopα α

− ugop)

= aα(ugopα α, ugopα α

− ugop)− aα(ugop , ugopα α− ugop)

= Lgopα α(ugopα α

− ugop)− a(ugop , ugopα α− ugop)

− α∫�1

b(ugopα α− b) dγ

= a(ugopα, ugopα α

− ugop)− a(ugop , ugopα α− ugop)

and taking into account (69) and the fact that ugopα→ ugop strongly in V when α→∞

because of (18), we get (70).

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Accepted 16 September 2002. Online publication 22 April 2003.

Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

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