USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 109–122. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

A semilinear control problem involving homogenization ^{∗}

Carlos Conca, Axel Osses, & Jeannine Saint Jean Paulin

Abstract

We consider a control problem involving a semilinear elliptic equation with a uniformly Lipschitz non-linearity and rapidly oscillating coefficients in a bounded domain of ℝ^{N}. The control is distributed on a compact subset interior to the domain. Given an N − 1 dimensional hypersurface at the interior of the domain not intersecting the control zone, the trace of the solution on the curve has to be controlled. We prove that there exists a limit control as the homogenization parameter converges to zero, which results as the limit of fixed points for controllability problems. We link this limit control with the corresponding homogenized problem.

1 Introduction

Let Ω be a connected and open subset of ℝ^{N} with smooth boundary Γ. Let ω ⊂⊂ Ω be a non-empty open subset with indicatrix set 1_{ω} and let S be a N −1 dimensional manifold strictly included in Ω and not intersecting ω. Consider the following control problem. Given ε > 0, α > 0 and y_{1} ∈ L^{2}(S)^{N} find a control function v^{ε} with support in ω such that

− div(A^{ε}∇y^{ε}) + f(y^{ε}) = 1_{ω}v^{ε} in Ω

y^{ε} = 0 on Γ (1.1)

and

∥y^{ε}_{|_{S}} − y_{1}∥_{0,S} ≤ α,  (1.2)

where y^{ε}_{|_{S}} is the trace of y^{ε} on S and ∥ ∥_{0,S} denotes the standard L^{2}-norm on S. The nonlinear function f is such that

f ∈ C^{0}, f(0) = 0,  (1.3)

∗Mathematics Subject Classifications: 35B37, 35B27, 35J60.

Key words: control, homogenization, semilinear elliptic equation.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001 Partially supported by the Mathematical-Mechanics program of FONDAP.

and uniformly Lipschitz, that is

∃γ > 0 such that ∀s ∈ ℝ \ \{0\}, 0 ≤ \frac{f(s)}{s} ≤ γ.  (1.4)

The coefficients of the symmetric matrix A^{ε} are real and piecewise C^{1} in \overline{Ω}. We assume the condition

∃α_{m}, α_{M} > 0 such that ∀ξ ∈ ℝ^{N}, |ξ| = 1, α_{m} ≤ ∑_{i,j=1}^{N} A_{ij}^{ε}(x)ξ_{i}ξ_{j} ≤ α_{M}, (1.5)

for a.e. x ∈ Ω. The following result can be established as in [7, 8] using a fixed point technique introduced in [2].

Theorem 1.1 Assume that each point x_{0} on S can be connected by an arc included in Ω to some point in ω without intersecting S \ {x_{0}}. Then, under the hypotheses (1.3), (1.4) and (1.5), there exists a control v^{ε} ∈ L^{2}(ω)^{N} satisfying (1.1) and (1.2).

Moreover a control v_{∗}^{ε} of minimal norm and solution of (1.1)-(1.2) can be constructed as follows. Using a density argument, we can assume f ∈ C^{1}. Define the real function

g(s) = \{ \begin{matrix}f(s)/s & if s ≠ 0 \\ f′(0) & if s = 0. \end{matrix}

For each z ∈ L^{2}(Ω)^{N} consider the following auxiliary control problem. Given ε > 0, α > 0 and y_{1} ∈ L^{2}(S)^{N} find a control function v^{ε} supported in ω such that

− div(A^{ε}∇y^{ε}) + g(z)y^{ε} = 1_{ω}v^{ε} in Ω

y^{ε} = 0 on Γ (1.6)

and

∥y^{ε}(z)_{|_{S}} − y_{1}∥_{0,S} ≤ α.  (1.7)

For the existence of these controls see [7]. Among the controls satisfying (1.6) and (1.7) we choose as an optimal the minimizer of the functional (see [4, 5])

I_{z}^{ε}(v) = \{ \begin{matrix}\frac{1}{2} ∥v∥_{0,ω}^{2} & if (1.7) is satisfied \\ +∞ & otherwise. \end{matrix}  (1.8)

We denote by v_{∗}^{ε}(z) the point of minimum value, which depends on z and ε of course. Associated to this control we have the solution of (1.6) that we denote by y_{∗}^{ε}(z). Now we define the mapping

F^{ε} : z ∈ L^{2}(Ω)^{N} → y_{∗}^{ε}(z) ∈ L^{2}(Ω)^{N}.  (1.9)

We will show that it has a fixed point \overline{z}^{ε}, that is to say

F^{ε}(\overline{z}^{ε}) = \overline{z}^{ε}.  (1.10)

An admissible control for the semilinear control problem (1.1) and (1.2) is simply

v_{∗}^{ε} = v_{∗}^{ε}(\overline{z}^{ε}).  (1.11)

Our main goal is to study the behavior of v_{∗}^{ε} as ε → 0.

Notation. We will denote by y^{ε} (or y^{ε}(v^{ε})) the solution of the original problem (1.1) and by y^{ε}(z) (or y^{ε}(z, v^{ε})) the solution of the auxiliary problem (1.6).

2 Dual context

For each z ∈ L^{2}(Ω)^{N}, ε > 0, α > 0 and y_{1} ∈ L^{2}(S)^{N} the optimal control v_{∗}^{ε}(z) minimizing (1.8) and satisfying simultaneously (1.6) and (1.7) can be expressed in a dual context. Indeed, we have the relationship [7]

v_{∗}^{ε}(z) = φ_{∗}^{ε}(z)_{|_{ω}},  (2.1)

where φ_{∗}^{ε}(z) is the solution of the following dual problem associated to (1.6) (δ_{S} is a Dirac mass concentrated on S)

− div(^{t}A^{ε}∇φ^{ε}) + g(z)φ^{ε} = δ_{S}φ_{1} in Ω

φ^{ε} = 0 on Γ (2.2)

for

φ_{1} = φ_{1∗}^{ε}(z),  (2.3)

where φ_{1∗}^{ε}(z) is the point of minimum in L^{2}(S)^{N} of the following dual functional of (1.8)

J_{z}^{ε}(φ_{1}) = \frac{1}{2} ∫_{ω} |φ^{ε}|^{2} dx + α ∥φ_{1}∥_{0,S} − ∫_{S} y_{1}φ_{1} ds  (2.4)

in the sense of Fenchel-Rockafellar [1, 5]. Note that in order to evaluate this dual functional we have to solve the dual problem (2.2) for each φ_{1} ∈ L^{2}(S)^{N}.

Notation. We will denote by φ^{ε}(z) (or φ^{ε}(z, φ_{1})) the solution of the auxiliary dual problem (2.2).

3 Main result

Our main result can be summarized as follows (the definition of H-convergence can be found in [6]).

Theorem 3.1 Assume that A^{ε} H-converges to A^{0} and that the hypotheses of Theorem 1.1 are satisfied, then up to a subsequence

v_{∗}^{ε} ⇀ v_{∗}^{0} in L^{2}(ω)^{N} − weakly and y(v_{∗}^{ε}) ⇀ y_{∗}^{0} in H_{0}^{1}(Ω) − weakly as ε → 0,

where v_{∗}^{0} has minimal norm among all controls v satisfying

∥y_{∗}^{0}(v)_{|_{S}} − y^{1}∥_{0,S} ≤ α.

Moreover y_{∗}^{0} is solution of the system

− div(A^{0}∇y_{∗}^{0}) + f(y_{∗}^{0}) = 1_{ω}φ_{∗}^{0} in Ω
y_{∗}^{0} = 0 on ∂Ω
− div(^{t}A^{0}∇φ^{0}) + g(y_{∗}^{0})φ^{0} = δ_{S}φ_{1} in Ω
φ^{0} = 0 on ∂Ω

φ_{1∗} = argmin (\frac{1}{2} ∫_{ω} |φ^{0}|^{2} dx + α ∥φ_{1}∥_{0,S} − ∫_{S} y_{1}φ_{1} ds) ,

(3.1)

where φ_{∗}^{0} is the solution of (3.1c,d) associated to φ_{1∗}. In terms of this dual variable,

v_{∗}^{0} = φ_{∗}^{0}_{|_{ω}}.  (3.2)

The proof of this theorem is developed in the rest of the paper and uses the following Lemma. The proof of this Lemma is similar to the one in [2] (see also [7]) taking care of the ε dependence in bounds and the regularity of A^{ε}.

Lemma 3.1 Assume that the coefficients of A^{ε} are piecewise C^{1} in \overline{Ω}. Then, under the hypotheses of Theorem 1.1, we have

lim inf_{∥φ_{1}∥_{0,S}→∞} \frac{J_{z}^{ε}(φ_{1})}{∥φ_{1}∥_{0,S}} ≥ α > 0.  (3.3)

Proof. We have

\frac{J_{z}^{ε}(φ_{1})}{∥φ_{1}∥_{0,S}} = \frac{1}{2} ∫_{ω} \frac{1}{∥φ_{1}∥_{0,S}} |φ^{ε}|^{2} dx + α − ∫_{S} y_{1} \frac{φ_{1}}{∥φ_{1}∥_{0,S}} ds.

Let

φ^{ε} = \frac{φ^{ε}}{∥φ_{1}∥_{0,S}} and φ̂_{1} = \frac{φ_{1}^{ε}}{∥φ_{1}∥_{0,S}} .

Then

\frac{J_{z}^{ε}(φ_{1})}{∥φ_{1}∥_{0,S}} = \frac{∥φ_{1}∥_{0,S}}{2} ∫_{ω} |φ^{ε}|^{2} dx + α − ∫_{S} y_{1} φ̂_{1} ds .  (3.4)

We write that for a sequence φ_{1,n} such that ∥φ_{1,n}∥_{0,S} → ∞ as n → ∞. Since \left∥{ \atop φ_{1,n}}\right∥_{0,S} = 1 it is easy to see using (1.4) and (1.5) that the associated solutions of (2.2) satisfy

∥φ_{n}^{ε}∥_{1,Ω} ≤ C

where the constant C does not depend on n nor ε and only depends on α_{m}, γ and the norm of the trace operator from H^{1}(Ω) into L^{2}(S). For a fixed ε up to a sequence (in n), we have

φ̂_{1,n} ⇀ φ̃_{1} in L^{2}(S) − weakly
φ_{n}^{ε} ⇀ φ^{ε} in H^{1}(Ω) − weakly.

Then

lim inf_{∥φ_{1}∥_{0,S}→∞} \frac{J_{z}^{ε}(φ_{1})}{∥φ_{1}∥_{0,S}} = lim inf_{n→∞} \frac{J_{z}^{ε}(φ_{1,n})}{∥φ_{1,n}∥_{0,S}} .

We consider two cases. Firstly, if

lim_{n} ∫_{ω} |φ_{n}^{ε}|^{2} dx = ∫_{ω} |φ^{ε}|^{2} dx > 0 ,

then

∥φ_{1,n}∥_{0,S} ∫_{ω} |φ_{n}^{ε}|^{2} dx → +∞

and since ∫_{S} y_{1}φ̂_{1,n} → ∫_{S} y_{1}φ̃_{1}, from (3.4) we obtain (3.3). Secondly, if

lim_{n} ∫_{ω} |φ_{n}^{ε}|^{2} dx = ∫_{ω} |φ^{ε}|^{2} dx = 0

then φ^{ε} = 0 in ω. Next, our aim is to prove that φ^{ε} = 0 in the whole of Ω. The fact that we have supposed the coefficients of A^{ε} piecewise C^{1}, implies that φ^{ε} = 0 till S. Indeed, the classical Holmgren’s unique continuation property [3] shows that φ^{ε} is zero in the regions intersecting ω where A^{ε} is regular and the transmission conditions allow to extend φ^{ε} by zero to the contiguous regions till S. This gives the desired result if S is an open curve. Conversely, if S is closed, the geometrical hypothesis on S and ω introduced in Theorem 1.1 implies that φ^{ε} is zero in the whole Ω. This implies that φ̃_{1} = 0 on S, therefore

lim inf_{n} \frac{J_{z}^{ε}(φ_{1,n})}{∥φ_{1,n}∥_{0,S}} ≥ α + lim inf_{n} (∥φ_{1,n}∥_{0,S} ∫_{ω} |φ_{n}^{ε}|^{2} dx) − 0 ≥ α,

which completes the proof of the lemma.

4 Step 1. Fixed point

We will establish that the operator F^{ε} defined in (1.9) has a fixed point using Schauder’s theorem. We follow the ideas in [2] and [7], taking care of the ε dependence.

Let us prove that F^{ε} is continuous and maps L^{2}(Ω)^{N} into a relatively compact subset of L^{2}(Ω)^{N}. Take

z_{n} → z_{0} in L^{2}(Ω)^{N}

and in order to simplify notations let us set

φ_{n}^{ε} = φ^{ε}(z_{n})

the solution of (2.2) associated to z_{n} and to a fixed φ_{1} ∈ L^{2}(S)^{N}. Now, taking φ_{n}^{ε} as a function test in (2.2) the following estimate is easily obtained

∥φ_{n}^{ε}∥_{1,Ω} ≤ C ∥φ_{1}∥_{0,S} ,  (4.1)

where the constant C depends only on the A^{ε}-ellipticity constant α_{m}, and on trace and Poincaré constants, but is independent on ε (we also use hypothesis (1.4) about f). Thanks to (4.1) we have up to a subsequence

φ_{n}^{ε} ⇀ φ_{0}^{ε} in H_{0}^{1}(Ω) − weakly.

In order to pass to the limit in a variational formulation of (2.2), note that

∫_{Ω} g(z_{n})φ_{n}^{ε}φ dx − ∫_{Ω} g(z_{0})φ_{0}^{ε}φ dx
= ∫_{Ω} g(z_{n})(φ_{n}^{ε} − φ_{0}^{ε})φ dx + ∫_{Ω}(g(z_{n}) − g(z_{0}))φ_{0}^{ε}φ dx,

but g(z_{n}) is bounded in L^{∞}(Ω) and since z_{n} converges to z_{0} a.e. then

g(z_{n}) ⇀ g(z_{0}) in L^{∞}(Ω) − weakly*.  (4.2)

Therefore

∫_{Ω} g(z_{n})φ_{n}^{ε}φ dx → ∫_{Ω} g(z_{0})φ_{0}^{ε}φ dx ∀φ ∈ H_{0}^{1}(Ω).

Remark 4.1 Convergence (4.2) implies weak but not strong convergence in H^{−1}(Ω).

Nevertheless, a technical argument allows to obtain the strong convergence in H^{−1}(Ω). Indeed, for all φ ∈ H_{0}^{1}(Ω), we have

| ∫_{Ω} g(z_{n})φ_{n}^{ε}φ dx − g(z_{0})φ_{0}^{ε}φ dx| ≤
≤ | ∫_{Ω} g(z_{n})(φ_{n}^{ε} − φ_{0}^{ε})φ dx| + | ∫_{Ω}(g(z_{n}) − g(z_{0}))φ_{0}^{ε}φ dx| .

On the one hand

|∫_{Ω} g(z_{n})(φ_{n}^{ε} − φ_{0}^{ε})φ dx| ≤ ∥g(z_{n})∥_{L^{p_{1}}} ∥φ_{n}^{ε} − φ_{0}^{ε}∥_{L^{p_{2}}} ∥φ∥_{L^{p_{3}}} .

Choosing p_{1} = N, p_{2} = p_{3} = \frac{2N}{N−1} if N ≥ 2 otherwise p_{1} = p_{3} = 4 and p_{2} = 2, thanks to this choice of p_{2}, the injection from H^{1}(Ω) to L^{p_{2}}(Ω) is compact and then

∥φ_{n}^{ε} − φ_{0}^{ε}∥_{L^{p_{2}}} → 0 as n → ∞.

Note that g is bounded and

∥g(z_{n})∥_{L^{p_{1}}} ≤ γ meas(Ω)^{1/p_{1}}.

Finally, the injection from H_{0}^{1}(Ω) to L^{p_{3}}(Ω) is continuous so

∥φ∥_{L^{p_{3}}} ≤ ∥i∥_{L(H_{0}^{1}(Ω);L^{p_{3}} (Ω))} ∥φ∥_{1,Ω} .  (4.3)

On the other hand

∥ ∫_{Ω}(g(z_{n}) − g(z_{0}))φ_{0}^{ε}φ dx∥ ≤ ∥g(z_{n}) − g(z_{0})∥_{L^{q_{1}}} ∥φ_{0}^{ε}∥_{L^{q_{2}}} ∥φ∥_{L^{q_{3}}}

with q_{1} = \frac{N}{2} , q_{2} = q_{3} = \frac{2N}{N−2} for N ≥ 3, otherwise q_{1} = 2, q_{2} = q_{3} = 4. Thanks to this choice the injection from H^{1}(Ω) into L^{q_{3}}(Ω) is continuous and a bound can be obtained as in (4.3). In virtue of dominated convergence theorem and bounds on g(z_{n}) we have

∥g(z_{n}) − g(z_{0})∥_{L^{q_{1}}} → 0 as n → ∞.

From the above convergences we see that

g(z_{n})φ_{n}^{ε} → g(z_{0})φ_{0}^{ε} in H_{0}^{1}(Ω) − strongly as n → ∞.

Let us continue with our problem. Multiplying (2.2) by ϕ ∈ H_{0}^{1}(Ω) and integrating by parts we obtain

∫_{Ω} A^{ε}∇φ_{n}^{ε} · ∇ϕ dx + ∫_{Ω} g(z_{n})φ_{n}^{ε}ϕ dx = ∫_{S} φ_{1}ϕ dσ,

for a fixed ε and we take n → ∞ to obtain

∫_{Ω} A^{ε}∇φ_{0}^{ε} · ∇ϕ dx + ∫_{Ω} g(z_{0})φ_{0}^{ε}ϕ dx = ∫_{S} φ_{1}ϕ dσ,

and this shows that φ_{0}^{ε} = φ^{ε}(z_{0}). Let us now show that

φ_{n}^{ε} → φ_{0}^{ε} in H_{0}^{1}(Ω) − strongly.  (4.4)

Take φ_{n}^{ε} as a test function in the problem

− div(^{t}A^{ε}∇φ_{n}^{ε}) + g(z_{n})φ_{n}^{ε} = δ_{S}φ_{1} in Ω
φ_{n}^{ε} = 0 on Γ .

Passing to the limit, we obtain

lim_{n→∞} ∫_{Ω} ^{t}A^{ε}∇φ_{n}^{ε} · ∇φ_{n}^{ε} dx = ∫_{S} φ_{1}φ_{0}^{ε} dσ − ∫_{Ω} g(z_{0})φ_{0}^{ε}φ_{0}^{ε} dx .

Now, taking φ_{0}^{ε} as a test function in

− div(^{t}A^{ε}∇φ_{0}^{ε}) + g(z_{0})φ_{0}^{ε} = δ_{S}φ_{1} in Ω
φ_{0}^{ε} = 0 on Γ ,

we obtain

∫_{Ω} A^{ε}∇φ_{0}^{ε} · ∇φ_{0}^{ε} dx = ∫_{S} φ_{1}φ_{0}^{ε} dσ − ∫_{Ω} g(z_{0})φ_{0}^{ε}φ_{0}^{ε} dx .

By comparison

lim_{n→∞} ∫_{Ω} ^{t}A^{ε}∇φ_{n}^{ε} · ∇φ_{n}^{ε} dx = ∫_{Ω} ^{t}A^{ε}∇φ_{0}^{ε} · ∇φ_{0}^{ε} dx.

We conclude (4.4) since (∫_{Ω} ^{t}A^{ε}∇v · ∇v dx)^{1/2} is equivalent to the standard norm in H_{0}^{1}(Ω).

By a method analogous to the one that yields (4.4) from (4.1), we show that

∥φ_{1∗}^{ε}∥_{0,S} ≤ C  (4.5)

with C independent of n and of ε, that is

φ_{1∗}^{ε}(z_{n}) ⇀ ξ^{ε} in L^{2}(S) − weakly

and

φ^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n})) → φ^{ε}(z_{0}, ξ^{ε}) in H_{0}^{1}(Ω) − strongly.  (4.6)

Let us show by contradiction that (4.5) holds. Otherwise, there exists a sequence {φ_{1∗}^{ε}(z_{n})}_{n≥0} such that

∥φ_{1∗}^{ε}(z_{n})∥_{0,S} → +∞ as n → ∞,  (4.7)

but for each z_{n}, the function φ_{1∗}^{ε}(z_{n}) minimizes J_{z_{n}}^{ε} and consequently

J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})) ≤ J_{z_{n}}^{ε}(φ_{1}) ∀φ_{1} ∈ L^{2}(S).  (4.8)

At the same time, we see that

J_{z_{n}}^{ε}(φ_{1}) = \frac{1}{2} ∫_{ω} |φ^{ε}(z_{n})|^{2} dx + α ∥φ_{1}∥_{0,S} − ∫_{S} y_{1}φ_{1} dσ

converges as n → ∞ to

J_{z_{0}}^{ε}(φ_{1}) = \frac{1}{2} ∫_{ω} |φ^{ε}(z_{0})|^{2} dx + α ∥φ_{1}∥_{0,S} − ∫_{S} y_{1}φ_{1} dσ.

Therefore from (4.8), for each fixed φ_{1}

J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})) ≤ C

with C independent of n (and of ε). This last upper bound contradicts (4.7) since

lim inf_{∥φ_{1∗}^{ε}(z_{n})∥_{0,S}→∞} \frac{J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n}))}{∥φ_{1∗}^{ε}(z_{n})∥_{0,S}} ≥ α > 0.  (4.9)

Proof of (4.9) is similar to the proof of Lemma 3.1 since

∥φ_{∗}^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n}))∥_{1,Ω} ≤ C ∥φ_{1∗}^{ε}(z_{n})∥_{0,S}

with a constant C independent of n (and of ε). Since (4.7) does not hold, we have up to a subsequence

φ_{1∗}^{ε}(z_{n}) ⇀ ξ^{ε} in L^{2}(S) − weakly as n → ∞.  (4.10)

It remains to identify the limit. Let us show that ξ^{ε} minimizes J_{z_{0}}^{ε}, that is to say

J_{z_{0}}^{ε}(ξ^{ε}) ≤ J_{z_{0}}^{ε}(φ_{1}) ∀φ_{1} ∈ L^{2}(S)^{N}.  (4.11)

First, note that φ_{1∗}^{ε}(z_{n}) is optimal for J_{z_{n}}^{ε}, that is

J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})) ≤ J_{z_{n}}^{ε}(φ_{1}) ∀φ_{1} ∈ L^{2}(S)^{N}

hence

lim inf_{n} J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})) ≤ lim inf_{n} J_{z_{n}}^{ε}(φ_{1}) = J_{z_{0}}^{ε}(φ_{1}) ∀φ_{1} ∈ L^{2}(S)^{N}.

In order to get (4.11) it remains to proof that

J_{z_{0}}^{ε}(ξ^{ε}) ≤ lim inf_{n} J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})).  (4.12)

Let us recall that

J_{z_{n}}^{ε}(φ_{1∗}^{ε}(z_{n})) = \frac{1}{2} ∫_{ω} |φ^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n}))|^{2} dx + α ∥φ_{1∗}^{ε}(z_{n})∥_{0,S} − ∫_{S} y_{1}φ_{1∗}^{ε}(z_{n}) dσ,

so from (4.10) we have

lim inf_{n} α ∥φ_{1∗}^{ε}(z_{n})∥_{0,S} − ∫_{S} y_{1}φ_{1∗}^{ε}(z_{n}) dσ ≥ lim inf_{n} α ∥ξ^{ε}∥_{0,S} − ∫_{S} y_{1}ξ^{ε} dσ

and from (4.6)

lim inf_{n} ∫_{ω} |φ^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n}))|^{2} dx ≥ ∫_{ω} |φ^{ε}(z_{0}, ξ^{ε})|^{2} dx.

In this way, we obtain (4.12) and consequently (4.11), in other words

ξ^{ε} = φ_{1∗}^{ε}(z_{0}).

With this relation, convergence in (4.6) becomes

φ^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n})) → φ^{ε}(z_{0}, φ_{1∗}^{ε}(z_{0})) in H_{0}^{1}(Ω) − strongly.  (4.13)

The rest of the proof is straightforward since

v_{∗}^{ε}(z_{n}) = φ^{ε}(z_{n}, φ_{1∗}^{ε}(z_{n}))_{|_{ω}}
v_{∗}^{ε}(z_{0}) = φ^{ε}(z_{0}, φ_{1∗}^{ε}(z_{0}))_{|_{ω}}

and it is clear from (4.13) that v_{∗}^{ε}(z_{n}) → v_{∗}^{ε}(z_{0}) in H^{1}(ω) - strongly. An analogous proof as for the adjoint problem shows that

y^{ε}(z_{n}, v_{∗}^{ε}(z_{n})) → y^{ε}(z_{0}, v_{∗}^{ε}(z_{0})) in H^{1}(Ω) − strongly,

proving the continuity of the map F^{ε} for a fixed ε > 0.

Next we show that F^{ε} is compact (uniformly in ε). Let z ∈ L^{2}(Ω)^{N} since

∥g(z)∥_{∞,Ω} ≤ γ

then

∥φ^{ε}(z, φ_{1})∥_{1,Ω} ≤ C ∥φ_{1}∥_{0,S}

with C independent of z (and of ε). This implies that |J_{z}^{ε}(φ_{1})| ≤ C(φ_{1}), therefore

|J_{z}^{ε}(φ_{1∗}^{ε}(z))| ≤ C(φ_{1}).

Using again the coercitivity of J_{z}^{ε} we see that ∥φ_{1∗}^{ε}∥_{0,S} is bounded independently of z (and ε). Then ∥φ^{ε}(z, φ_{1∗}^{ε})∥_{1,Ω} is bounded independently of z (and ε) and consequently the same is true for v_{∗}^{ε}(z) = φ^{ε}(z, φ_{1∗}^{ε})_{|_{ω}} and y^{ε}(z, v_{∗}^{ε}(z)). ♢

5 Step 2. H-convergence

We first consider the H-convergence in the original problem (1.1) with fixed control v ∈ L^{2}(ω)^{N}, that is the H-convergence in the problem

− div(A^{ε}∇y^{ε}) + f(y^{ε}) = 1_{ω}v in Ω

y^{ε} = 0 on Γ, (5.1)

under the hypotheses (1.3) and (1.4) on f. To have a priori estimates, we multiply (5.1) by y^{ε} and we integrate in Ω to obtain

∫_{Ω} A^{ε}∇y^{ε} · ∇y^{ε} dx + ∫_{Ω} f(y^{ε}) y^{ε} dx = ∫_{ω} v y^{ε} dx,

but from (1.3)

f(y^{ε})y^{ε} = \frac{f(y^{ε})}{y^{ε}} |y^{ε}|^{2} ≥ 0

and it is true also in the case y^{ε}(x) = 0. Hence ∥y^{ε}∥_{1,Ω} ≤ C ∥v∥_{0,ω} with C independent of ε. Up to a subsequence

y^{ε} ⇀ y^{0} in H_{0}^{1}(Ω) − weakly.

Now let us see which is the limit of f(y^{ε}). Take φ ∈ H_{0}^{1}(Ω), then

|∫_{Ω}(f(y^{ε})φ − f(y^{0})φ) dx| = |∫_{Ω}(g(y^{ε})y^{ε}φ − g(y^{0})y^{0}φ) dx| ≤  (5.2)
≤ |∫_{Ω}(g(y^{ε})(y^{ε} − y^{0})φ) dx| + |∫_{Ω}((g(y^{ε}) − g(y^{0}))y^{0}φ) dx| .

Starting from this and reasoning as in Remark 4.1 we can show that

f(y^{ε}) = g(y^{ε})y^{ε} → g(y^{0})y^{0} = f(y^{0}) in H^{−1}(Ω) − strongly.

Thanks to the H-convergence definition, we immediately deduce that

− div(A^{0}∇y^{0}) + f(y^{0}) = 1_{ω}v in Ω

y^{0} = 0 on Γ, (5.3)

where A^{0} is the H-limit of A^{ε} (and A^{ε}∇y^{ε} ⇀ A^{0}∇y^{0} in L^{2}(Ω)^{N}-weakly).

Consider now the H-convergence with the optimal control v_{∗}^{ε} = v_{∗}^{ε}(\overline{z}^{ε}) satisfying (1.6)-(1.11) where \overline{z}^{ε} is the fixed point of F^{ε}. We have already seen at the end of the previous section that ∥v_{∗}^{ε}(z)∥_{0,ω} is bounded independently of z ∈ L^{2}(Ω) and ε. In particular

∥v_{∗}^{ε}∥_{0,ω} ≤ C

with C independent of ε. Hence there exists v^{0} ∈ L^{2}(ω)^{N} such that

v_{∗}^{ε} ⇀ v^{0} in L^{2}(ω) − weakly
1_{ω}v_{∗}^{ε} → 1_{ω}v^{0} in H^{−1}(Ω) − strongly.

(5.4)

The same proof as in the case of a fixed v shows that the solution y_{∗}^{ε} of

− div(A^{ε}∇y_{∗}^{ε}) + f(y_{∗}^{ε}) = 1_{ω}v_{∗}^{ε} in Ω
y_{∗}^{ε} = 0 on Γ

converges weakly to y_{0}, i.e.,

y_{∗}^{ε} ⇀ y^{0} in H_{0}^{1}(Ω) − weakly  (5.5)

where y^{0} is a solution of

− div(A^{0}∇y^{0}) + f(y^{0}) = 1_{ω}v^{0} in Ω

y^{0} = 0 on Γ, (5.6)

and A^{0} is the H-limit of A^{ε}.

Notation. In the following sections v^{0} stands for the L^{2}-weak limit of the control in (5.4) and y^{0} (or y^{0}(v^{0})) stands for the weak H^{1}-limit of the solution in (5.5), which is solution of the limit problem (5.6).

6 Step 3. Limit of optimal controls

The objective is now to identify v^{0}. Is it an optimal solution? First at all, note that

∥y_{∗}^{ε}(v_{∗}^{ε})_{|_{S}} − y_{1}∥_{0,S} ≤ α .

Since weak convergence in (5.5) implies

y_{∗}^{ε}_{|_{S}} ⇀ y_{|_{S}}^{0} in H^{1/2}(S)^{N} − weakly (L^{2}(S)^{N}-strongly),

we conclude that v^{0} satisfies the approximate controllability inequality

∥y^{0}(v^{0})_{|_{S}} − y_{1}∥_{0,S} ≤ α.

Also from (5.5), since \overline{z}^{ε} is a fixed point (see (1.9), (1.10)), we have

\overline{z}^{ε} = y_{∗}^{ε} ⇀ y^{0} in H_{0}^{1}(Ω) − weakly.

Let v_{∗}^{0} be the minimizer in L^{2}(ω)^{N} of the functional

I(v) = \{ \begin{matrix}\frac{1}{2} ∥v∥_{0,ω}^{2} & if ∥y^{0}(v)_{|_{S}} − y^{1}∥_{0,S} ≤ α \\ +∞ & otherwise, \end{matrix}  (6.1)

where for each v ∈ L^{2}(S)^{N}, we denote y^{0}(v) the solution of

− div(A^{0}∇y^{0}) + f(y^{0}) = 1_{ω}v in Ω

y^{0} = 0 on Γ. (6.2)

We will establish that

v^{0} = v_{∗}^{0}.  (6.3)

In virtue of Fenchel-Rockafellar duality, the minimum v_{∗}^{0} can be characterized as follows. Let us consider the dual problem associated to (6.2), that is, for each φ_{1} ∈ L^{2}(S), find φ^{0} ∈ L^{2}(Ω)^{N} such that

− div(^{t}A^{0}∇φ^{0}) + g(y^{0})φ^{0} = δ_{S}φ_{1} in Ω

φ^{0} = 0 on Γ (6.4)

and let us define the respective dual functional of (6.1) as

J^{0}(φ_{1}) = \frac{1}{2} ∫_{ω} |φ^{0}|^{2} dx + α ∥φ_{1}∥_{0,S} − ∫_{S} y_{1}φ_{1} ds.  (6.5)

If φ_{1∗}^{0} is the point of minimum of J^{0} in L^{2}(S)^{N}, and if φ_{∗}^{0} is the solution of (6.4) associated to it, then the duality theory gives the relationship

v_{∗}^{0} = φ_{∗}^{0}_{|_{ω}}.  (6.6)

We will pass to the limit in (2.1), (2.2), (2.3), (2.4) with z = \overline{z}^{ε} as ε → 0. An argument similar to the one used for obtaining (5.3) shows that if we pass to the limit in (2.2) with z = \overline{z}^{ε} as ε → 0 then

φ^{ε}(\overline{z}^{ε}) ⇀ φ^{0} in H_{0}^{1}(Ω) − weakly

where φ^{0} is the solution of

− div(^{t}A^{0}∇φ^{0}) + g(y^{0})φ^{0} = δ_{S}φ_{1} in Ω

φ^{0} = 0 on ∂Ω. (6.7)

Taking the limit in (2.4) with φ_{1} fixed,

J_{\overline{z}^{ε}}^{ε}(φ_{1}) → J^{0}(φ_{1}).

Let us consider now the sequence φ_{1∗}^{ε}(\overline{z}^{ε}). From the uniform coecivity of J_{\overline{z}^{ε}}^{ε} with respect to ε (an analogous to Lemma 3.1 with z = \overline{z}^{ε}), we deduce that φ_{1∗}^{ε}(\overline{z}^{ε}) is bounded in L^{2}(S)^{N} then, up to a subsequence

φ_{1∗}^{ε}(\overline{z}^{ε}) ⇀ φ_{1}^{0} in L^{2}(S)^{N} − weakly.

Then (see the proof of (4.12)) for each φ_{1} ∈ L^{2}(S)

J^{0}(φ_{1}^{0}) ≤ lim inf_{ε} J_{\overline{z}^{ε}}^{ε}(φ_{1∗}^{ε}(\overline{z}^{ε})) ≤ lim inf_{ε} J_{\overline{z}^{ε}}^{ε}(φ_{1}) = J^{0}(φ_{1}).

Therefore φ_{1}^{0} = φ_{1∗}^{0} and consequently φ^{0} = φ_{∗}^{0}. Finally, passing to the limit as ε → 0 in (2.1), we obtain

v_{∗}^{ε} = φ_{∗}^{ε}_{|_{ω}} → φ_{∗}^{0}_{|_{ω}} = v_{∗}^{0} in L^{2}(ω) − strongly.

This, together with (5.4), implies (6.3).

Acknowledgments. This research was carried out when J. Saint Jean Paulin visited the Department of Mathematical Engineering at the University of Chile. The authors thank warmly this institution for supporting the Fondap’s Program on Mathematical-Mechanics. The authors also want to thank the Chilean and French Governments for supporting the Scientific Committee Ecos-Conicyt.

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Carlos Conca & Axel Osses Departamento de Ingeniería Matemática, CMM, UMR 2071 CNRS-Uchile, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile e-mail: cconca@dim.uchile.cl, axosses@dim.uchile.cl Jeannine Saint Jean Paulin Département de Mathématiques, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France e-mail: sjpaulin@poncelet.sciences.univ-metz.fr