UTMS 2005–34

August 17, 2005

Applications of convexity in some identification
problems
byDan Tiba, Gengsheng Wang
and Masahiro Yamamoto

Ì

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANApplications of convexity in some identification
problems
Dan Tiba1 , Gengsheng Wang2 and Masahiro Yamamoto3

1

Institute of mathematics, Romanian Academy

P.O.Box 1-764, RO-014700 Bucharest, Romania
2

Department of Mathematics, Normal University of Central China
Wuhan, China

3

Graduate School of Mathematical Sciences, The University of Tokyo
3-8-1 Komaba Meguro Tokyo 153, Japan

1

Introduction

We study a new reconstruction technique in some elliptic identification problems. Our method applies to distributed observation on some subset of the
problem domain or to boundary observation via Dirichlet conditions. Moreover our method is applicable to parabolic cases although we mainly discuss theelliptic case.
The basic idea is to divide the classical least squares approach into two steps:
the first one searches for the identification parameters (coefficients, sources) that
ensure one inequality with respect to the observed state, while the second one
1deals with the opposite inequality. In many situations, which are partiallydiscussed in Section 2, the first step has convexity properties, even in the caseof nonlinear partial differential equations (including variational inequalities).
Such properties have been studied by Lemaire [4], Kawohl and Lang [3], Liu
and Tiba [5] in a different context. This is an important advantage for thenumerical solution, although uniqueness is not valid, in general.
Moreover the convexity gives also the equivalence of the first minimization
problem with the corresponding set of optimality conditions. Then, the second
step of the method (which is more difficult) admits again a formulation as an
optimal control problem which may be solved by appropriate techniques. The
argument is indicated in Section 3.

2

The convexity property

Throughout this paper, we set f+ (x) = max{f (x), 0} and f− (x) = − min{f (x), 0}
for a real-valued function f . We note |f (x)| = f+ (x) + f− (x).
We fix our attention on the Dirichlet problem for the nonlinear elliptic equation(2.1)−4y(x) = ϕ(x, y(x), u(x))

(2.2)

y=0

on

in Ω,

∂Ω.

Other boundary conditions such as Neumann, mixed or of Robin types may bealso considered. Parabolic equations can be treated by our method.
We assume that Ω ⊂ Rd , d ∈ N, is a bounded smooth domain and ϕ:Ω × R × Rk → R, k ∈ N, satisfies (2.3) - (2.5):(2.3) ϕ(x, ·, ·) is convex for a.e. x ∈ Ω and there exist C > 0 and a positive2ª©
function M ∈ Ls (Ω), s > max 2, d2 such that |ϕ(x, 0, u)| ≤ M (x) + C|u|,∀(x, u) ∈ Ω × Rk ;©
ª(2.4) there exist C > 0 and a positive function M ∈ Ls (Ω), s > max 2, d2 ,a nondecreasing function η : R+ −→ R+ such that 0 ≥ ϕ0y (x, y, u) ≥− (M (x) + C|u|) η(y), ∀(x, y, u) ∈ Ω × R × Rk ;(2.5) ϕ(x, y, u) and ϕ0y (x, y, u) are Caratheodory mappings, i.e., measurable inx and continuous in y, u.
The vector-valued function u = u(x) describes coefficients or source terms inequation (2.1). This will be made clear in the examples. If u ∈ L∞ (Ω)k , then
(2.1) and (2.2) possess a unique solution y = y(u) in W 2,s (Ω) ∩ H01 (Ω). Thisis a consequence of (2.3)–(2.5) and the well known theory of nonlinear elliptic
equations (e.g., Barbu [1], Neittaanmäki, Sprekels and Tiba [6]). We denote thesolution to (2.1) and (2.2) by y(u) = T (u) for u ∈ L∞ (Ω)k .
Proposition 2.1. Under the above assumptions, the nonlinear operator T :L∞ (Ω)k → L2 (Ω), is convex.
Proof.
Let u1 , u2 ∈ L∞ (Ω)k and λ ∈]0, 1[ be given. Let y1 , y2 , yλ denote the solutions of (2.1), (2.2) corresponding to u1 , u2 and to λu1 + (1 − λ)u2 respectively.
By multiplying the equations corresponding to y1 , y2 respectively by λ, (1 − λ)and by adding them, hypothesis (2.3) yields the inequality:(2.6) −4 (λy1 + (1 − λ)y2 ) ≥ ϕ (x, λy1 + (1 − λ)y2 , λu1 + (1 − λ)u2 ) in Ω.
We subtract (2.6) from the equation corresponding to yλ and we obtain:(2.7)−4 (yλ − λy1 − (1 − λ)y2 ) ≤ ϕ (x, yλ , λu1 + (1 − λ)u2 )−ϕ (x, λy1 + (1 − λ)y2 , λu1 + (1 − λ)u2 )
3

in Ω.

Multiply (2.7) by [yλ − λy1 − (1 − λ)y2 ]+ ∈ H01 (Ω) and integrate in Ω, we have
Z
(2.8)
Ω

¡
¢2
∇ [yλ − λy1 − (1 − λ)y2 ]+ dx ≤ 0.

Here we usedZ−∆(yλ − λy1 − (1 − λ)y2 )[yλ − λy1 − (1 − λ)y2 ]+ dxZΩ∇(yλ − λy1 − (1 − λ)y2 ) · ∇[yλ − λy1 − (1 − λ)y2 ]+ dx=ZΩ(∇[yλ − λy1 − (1 − λ)y2 ]+ )2 dx

=
Ω

and
(ϕ(x, yλ , λu1 + (1 − λ)u2 ) − ϕ(x, λy1 + (1 − λ)y2 , λu1 + (1 − λ)u2 )) × [yλ − λy1 − (1 − λ)y2 ]+≤0

if yλ (x) > (λy1 + (1 − λ)y2 )(x), i.e., [yλ − λy1 − (1 − λ)y2 ]+ (x) > 0,=0

if yλ (x) ≤ (λy1 + (1 − λ)y2 )(x),by (2.4). Clearly (2.8) signifies that [yλ − λy1 − (1 − λ)y2 ]+ = 0 a.e. in Ω, that
is
(2.9)

yλ (x) ≤ λy1 (x) + (1 − λ)y2 (x)a. e. inΩ.

Relation (2.9) is the desired infinite dimensional convexity property of T andends the proof.
Let us now comment briefly on some classes of examples that satisfy assumptions (2.3)-(2.5). A very simple situation is obtained when ϕ arises as asum
(2.10)

ϕ(x, y, u) = a(x)y + b(x)uwith a, b ∈ L∞ (Ω). If a ≤ 0 a. e. in Ω, then (2.3)-(2.5) are clearly fulfilled.
More generally, ϕ may be assumed to be the sum of convex mappings:(2.11)

ϕ(x, y, u) = ψ1 (x, y) + ψ2 (x, u).
4

If ψ1 , ψ2 are convex functions and ψ1 (x, ·) is nonincreasing for a.e. x ∈ Ω, then
(2.3)-(2.5) are valid. As a limit case, for instance, if ψ1 (x, y) = −β(y) with βbeing the maximal monotone graph∅, y < 0,β(y) =(2.12)] − ∞, 0], y = 0,







0, y > 0,then equations (2.1) and (2.2) become a variational inequality and Proposition
2.1 remains valid. Such situations have been studied by Lemaine [4] and theextension of Proposition 2.1 to (2.12) is obtained by first considering the caseof the Yosida approximation βε (y) and then by passing to the limit ε → 0 as it
is standard in the theory of variational inequalities, Barbu [1].
Example 2.2.
We indicate now a situation when u appears as a coefficient in the equation:(2.13)(2.14)¡¢
4 u3 (x)4y(x) = f (x) in Ω,
y = 0,

4y = 0on∂Ω.

If Ω ⊂ R or Ω ⊂ R2 , then problem (2.13) and (2.14) is a simplified model forthe deflection of a beam, respectively a plate, under the load f . The coefficientu ∈ L∞ (Ω) with u > 0 represents the thickness and boundary conditions (2.13)
correspond to the simply supported case.
We assume that f ∈ L2 (Ω) satisfies f ≥ 0 a. e. in Ω and by h ∈ H 2 (Ω) ∩H01 (Ω) we denote the unique solution of(2.15)

4h = f(2.16)

h=0

in Ω,
on

5

∂Ω.

By the maximum principle, we clearly have h(x) ≤ 0 in Ω. It is also easy to seethat y given by (2.13), (2.14) can be equivalently defined by(2.17)

4y(x) =h(x)u3 (x)

in(2.18)

y=0

on

∂Ω.

Ω,

Since the mapping u 7→ u−3 is convex for u > 0 and h is negative, for any u1 ,u2 ∈ L∞ (Ω) with u1 , u2 > 0, we have the inequality(2.19)

h(x)

h(x)

h(x)

3 ≥ λ u (x)3 + (1 − λ) u (x)31
2[λu1 (x) + (1 − λ)u2 (x)]in Ω.

Let y1 , y2 ∈ H 2 (Ω) ∩ H01 (Ω) denote the solutions of (2.17), (2.18) corresponding
to u1 , respectively u2 and let yλ denote the solution of (2.17), (2.18) corresponding to λu1 + (1 − λ)u2 . By (2.17)–(2.19), we obtain(2.20)

4yλ ≥ λ4y1 + (1 − λ)4y2

in Ω.

The maximum principle shows
yλ (x) ≤ λy1 (x) + (1 − λ)y2 (x)a. e. inΩ.

Consequently, we see that (2.9), i.e., the conclusion of Proposition 2.1 remains
valid in the case of equations (2.13) and (2.14) (with f positive) as well.
Let us now formulate the identification problem for equations (2.1) and (2.2),
which includes all the examples mentioned above. We assume for simplicity
that some distributed observation yb can be measured on a given measurable
subset ω ⊂ Ω. However the discussion below extends to the case of Dirichlet
observation, when boundary condition (2.2) is replaced by some other type ofboundary condition, as it has been already mentioned.

6

As a first step in the reconstruction method that we are proposing, we modify
the classical least square approach and we introduce the following functional:(2.21)

min
u

1
2

Z
2

(y(u) − yb)+ dx

ωsubject to (2.1), (2.2) and to the constraintu ∈ K ⊂ L∞ (Ω)k ,(2.22)

K being a bounded closed convex subset. We set
1
J(u) =2Z2ω

(y(u) − yb)+ dx.

Here y(u) is the solution to (2.1) and (2.2) for u ∈ L∞ (Ω)k . One example isgiven by
(2.23)

K = {u ∈ L∞ (Ω); a ≤ u(x) ≤ b a. e. in Ω}with some positive constants a, b and k = 1.
If yb is exact data without errors, that is, yb comes from y(bu) with some ub ∈ K,
then J(bu) = 0. This means that some optimal control ub exists. However theuniqueness of minimizers is in general not valid.
Corollary 2.3. The problem (2.1), (2.2), (2.21), (2.22) is a convex control
problem.
Proof.
This is an immediate consequence of Proposition 2.1 and of the fact that the
integrand in (2.21) is convex and nondecreasing in y.
Remark. The convexity property makes the application of descent algorithms very efficient. In case (2.21) is replaced by the usual least square approach, the convexity is lost.

7

Let J(u∗ ) = 0. Then
(2.24)y(u∗ )(x) ≤ yb(x) a. e. inω.

Thus in our optimization problem we are looking for inputs which cause lower
outputs than a given target function yb. For example, if yb corresponds to somecritical value, then it is practically significant to steer the state keeping a lower
level of the critical value.
However, as yb may involve measurement errors, it is possible that (2.24)
is violated on some subset of ω. This also shows the necessity to provide an
existence argument independently of the definition of yb. In the next section, westudy such questions together with the opposite inequality to (2.24).

3

The complete problem

As problem (2.21) - (2.22) is convex, the first order optimality conditions givea characterization of the minimizers.
¤£
Theorem 3.1. The pair [y ∗ , u∗ ] ∈ W 2,s (Ω) ∩ H01 (Ω) × K is optimal for
problem (2.21) - (2.22) if and only if there exist p∗ ∈ W 2,s (Ω) ∩ H01 (Ω) satisfying
with y ∗ , u∗ the first order necessary conditions(3.1)−4y ∗ = ϕ (x, y ∗ , u∗ )(3.2)

−4p∗ = ϕ0y (x, y ∗ , u∗ ) p∗ + (y ∗ − yb)+ χω

(3.3)
(3.4)

y ∗ = 0,p∗ = 0

on

in

Ω,
in

Ω,

∂Ω,

0 ∈ p∗ (x)∂ϕu (x, y ∗ (x), u∗ (x)) + ∂IK (u∗ (x))

a.e. Ω.

Here χω denotes the characteristic function of ω, we set IK (u) = 1 if u ∈ K£¤
and = 0 if u 6∈ K, ϕ0y , ∂ϕu denotes the components of the subdifferential ∂ϕ
8with respect to [y, u]. We recall that ϕ is assumed to be differentiable in y andª©
by (2.4), ϕ0y ∈ Ls (Ω), ϕ0y ≤ 0, s > max 2, d2 . Then the Dirichlet problem
for equation (3.2) has a unique weak solution p∗ ∈ H01 (Ω). As (y ∗ − yb)+ χω ispositive and bounded, a comparison technique shows that p∗ ∈ W 2,s (Ω)∩H01 (Ω).
Proof of the sufficiency.
The necessity of conditions (3.1)–(3.4) is well known (see for instance Neittaanmäki, Sprekels and Tiba [6]). Hence we confine ourselves to the proof ofthe sufficiency of (3.1)–(3.4), based on the convexity property from Proposition
2.1.
We multiply (3.2) by (y ∗ − y) and (3.4) by (u∗ − w) where [y, w] is any
admissible pair for problem (2.21) - (2.22). After addition, we obtain:
Z
−
Z

Z

Ω

ϕ0y (x, y ∗ , u∗ ) (y ∗ − y) p∗ dx −Zp∗ ∂ϕu (x, y ∗ , u∗ ) (u∗ − w)dxΩ4p∗ (y ∗ − y) dx +(y ∗ − yb)+ (y ∗ − y) χω dx
Ω
Ω
Z
Z
∗
∗
+
∂IK (u ) (u − w) dx ≥p∗ (4y ∗ − 4y)dx
Ω
Ω
(3.5)
Z
Z
1
1
2
2
(y ∗ − yb)+ dx −(y − yb)+ dx
+
2 ω
2 ω
Z
Z
∗
∗
∗
=−
p ϕ (x, y , u ) dx +p∗ ϕ(x, y, w)dx
Ω
Ω
Z
Z
1
1
2
2
+
(y ∗ − yb)+ dx −(y − yb)+ dx,
2 ω
2 ω
=

where we have used (3.1) and the definition of the subdifferential repeatedly.
By combining the first and the last parts in (3.5), we conclude that
1
2
Z

Z
2

(y ∗ − yb)+ dx −ω∗(3.6) ≤

∗

1
2

Z
2

ω

(y − yb)+ dx

Z∗p [ϕ (x, y , u ) − ϕ(x, y, w)] dx −ΩZΩ

ϕ0y (x, y ∗ , u∗ ) (y ∗ − y) p∗ dx∂ϕu (x, y ∗ , u∗ ) p∗ (u∗ − w) dx ≤ 0.

−Ω

Relation (3.6) is a consequence of the definition of the subdifferential applied toϕ(x, ·, ·) and of the positivity of p∗ which may be inferred from (3.2). This ends
9

the proof since [y, w] is an arbitrary admissible pair.
The solution triple [y ∗ , u∗ , p∗ ] of (3.1)–(3.4) is not unique, in general. The
set of [y ∗ , u∗ ] generated in this way coincides with the set of optimal pairs for(2.21) and (2.22). In many applications it may be enough to solve (3.1)–(3.4)
and to by some ad-hoc means select a solution which is appropriate for solving
the identification problem associated to (2.1) - (2.2) and the observation yb.
Otherwise, one should study the second step in the proposed reconstruction
procedure, formulated as follows
(3.7)

1
min
u∈K 2

Z
2

ω

(y − yb)− dx

subject to (3.1)–(3.4). Then we can derive the corresponding relations to (3.1)–(3.4).
Assuming that the observation yb is without errors, that is, yb = y(bu)|ω withub ∈ K, then both problems (2.21) and (3.7) have ub as solution with minimalvalue zero (and not necessarily unique). This shows that both (2.21) and (3.7)
solve the original identification problem.
In problem (3.7), the optimality conditions (3.1)–(3.3) play the role of the
new state system with state variables y and p and control parameter u ∈ K.
Relation (3.4) is a new constraint of mixed type, which is a state-control constraint involving k equalities. As its form is rather unusual, we examplify by
the simpler situation when ϕ(x, y, ·) is differentiable on Rk and the indicatorfunction IK is regularized in the Yosida–Moreau sense by Iε , ε > 0. We obtain(3.8)

p(x)ϕ0u (x, y(x), u(x)) + Iε0 (u(x)) = 0

a. e. in Ω.

Relation (3.8) has now a usual form of a nonlinear mixed constraint involving thestate y, p and the control u with k components. The approximation properties
10

for ε → 0 are well known in the literature on optimal control, Barbu [1], Barbu
and Precupanu [2].
In the sequel we assume that ϕ0u is a Caratheodory mapping and satisfies
the growth condition(3.9)|ϕ0u (x, y, u) | ≤ (M (x) + C|u|) η(y).

Here C > 0 is a constant and M , η satisfy the same conditions as in (2.3). We
consider constraint (3.4) in the form:(3.10)

0 ∈ p(x)ϕ0u (x, y(x), u(x)) + ∂IK (u(x)) .

Proposition 3.2. Assume that K is compact in L∞ (Ω)k . Then problem(3.1)–(3.3), (3.10) has at least one solution [y, u, p]. Therefore minimization
problem (3.7) has at least one solution.
Proof.
We notice that the admissibility property is automatically fulfilled for system(3.1)–(3.3) and (3.10) as it comes from problem (2.1), (2.2), (2.21), (2.22). Let{un } be a minimizing sequence. Since K is compact in L∞ (Ω)k , we can extract
a subsequence denoted again by un such that un → ue ∈ K strongly in L∞ (Ω)k .
Multiplying (3.1) by yn and integrating by parts, we obtainZ

Z
2(3.11)|∇yn | dx =Ωϕ (x, yn , un ) yn dx.
Ω

Relation (3.11), (2.3) and (2.4) show that {yn } is bounded in H01 (Ω). Then,standard techniques in the theory of semilinear elliptic equations and Tiba [7],¡ ¢yield that {yn } is bounded in W 2,s (Ω) and relatively compact in C Ω by the
Sobolev embedding theorem.
A similar argument applied to (3.2) proves that {pn } is bounded in W 2,s (Ω)¡ ¢and relatively compact in C Ω . We may assume that yn → ye, pn → pe weakly in11

W 2,s (Ω) and uniformly on Ω. The Caratheodory assumptions on ϕ and ϕ0y (see(2.3)–(2.5)) show that ϕ (·, yn , un ) → ϕ (·, ye, ue) and ϕ0y (·, yn , un ) → ϕ0y (·, ye, ue) inLs (Ω), by the Lebesgue theorem. Then one can pass to the limit in (3.1)–(3.3).
The Caratheodory assumption and (3.9) give that ϕ0u (x, yn , un ) → ϕ0u (x, ye, ue)strongly in Ls (Ω)k . As pn → pe uniformly, we obtain that pn ϕ0u (x, yn , un ) →peϕ0u (x, ye, pe) strongly in Ls (Ω)k . By (3.10), we see that ∂IK (un ) → α strongly
in Ls (Ω)k . As un → u
e uniformly, the closedness of the maximal monotone
operator ∂IK gives that α ∈ ∂IK (eu) a. e. in Ω.
One can conclude that [ey, ue, pe] is admissible for problem (3.1)–(3.3) and(3.10). As the passage to the limit in the cost functional (3.7) is obvious, theproof is finished.
We continue with the definition of a penalization and regularization of problem (3.1)–(3.3), (3.7), (3.10). The penalized and regularized cost functionalis(3.12)

min

½ Z
¾
Z
1
1
2
2
(y − yb)− +[pϕ0u (x, y, u) + Jε0 (u)] dx
2 ω
2ε Ω

subject to u ∈ K and to the state system (3.1)–(3.3). Here Jε , ε > 0, isa regularization of the Moreau-Yosida approximation of IK (via a Friedrichsmollifier, for instance). Consequently, Jε0 exists for any u ∈ Rk .
The formulation (3.12), (3.1)–(3.3) includes the mixed constraint (3.4) inthe cost functional and involves just control constraints u ∈ K. As in Proposition 3.2, we can prove the existence of at least one optimal triple, denoted by[yε , u² , p² ], ε > 0, if K is compact in L∞ (Ω)k .
Proposition 3.3. When ε → 0, a sequence [yε , u² , p² ] converges to [y ∗ , u∗ , p∗ ]strongly in W 2,s (Ω)×L∞ (Ω)×W 2,s (Ω) by taking a subsequence, and [y ∗ , u∗ , p∗ ]is some optimal triple for problem (3.1)–(3.3), (3.7), (3.10).

12

Proof.
Any admissible triple [ey, ue, pe] for the problem (3.1)–(3.3), (3.7), (3.10) is
admissible for the problem (3.1)–(3.3), (3.12) and we have inequality:Z
12[pε ϕ0u (x, yε , uε ) + Jε0 (uε )] dx
2ε Ω
ω
Z
1
1
2
2
≤
(e
y − yb)− dx +[e
pϕ0u (x, ye, ue) + Jε0 (eu)] dx
2
2ε
ωZ
ZΩ
1
1
2
2
=
(e
y − yb)− dx +[J 0 (eu) − ∂IK (eu)] dx.
2 ω
2ε Ω ε12Z(3.13)

Z2(yε − yb)− dx +Here we choose the element ze ∈ ∂IK (eu) which occurs in (3.10), that is, ze =−epϕ0u (x, ye, ue) a. e. in Ω and Jε0 may be assumed to be a regularization ofthis precise mapping defined on K. In the case of (2.23), such a construction
is simple. By taking a Friedrichs mollification with a very small parameter εm ,m ∈ N, (although we preserve the notation Jε ), we may assume thatZ
1
2
[J 0 (eu) − ∂IK (eu)] dx2ε Ω εZ12e)] dx → 0u) + peϕ0u (x, ye, u=[J 0 (e2ε Ω ε

(3.14)as ε → 0. By combining (3.13) and (3.14), we obtain(3.15)ZZ1
122(yε − yb)− dx +[pε ϕ0u (x, yε , uε ) + Jε0 (uε )] dx
2 ω
2ε Ω
Z
1
2
≤
(e
y − yb)− dx + C (ε, u
e) ,
2 ω

where limε→0 C (ε, ue) = 0 for any ue ∈ K, admissible for the problem (3.1)–(3.3),(3.7), (3.10).
As K ⊂ L∞ (Ω)k is compact, we may assume that uε → u ∈ K strongly in
L∞ (Ω)k , on a subsequence. The same argument as in the proof of Proposition
3.2 gives that yε → y, pε → p strongly in W 2,s (Ω) and uniformly on Ω. Then,by (3.9), we obtain that
(3.16)

pε ϕ0u (x, yε , uε ) → pϕ0u (x, y, u)13strongly in Ls (Ω). The construction of Jε0 ensures that Jε0 (uε ) → zb ∈ ∂IK (u)and that (3.10) is satisfied by [y, u, p]. This is a consequence of (3.15), (3.16).
The triple [y, u, p] is admissible for problem (3.1)–(3.3), (3.7), (3.10). By passing
to the limit in (3.15), we obtain
1
2

Z
2

ω

(y − yb)− dx ≤

1
2

Z
2

ω

(e
y − yb)− dx

for any admissible [ey, ue, pe]. This shows the optimality of [y, u, p] which we denoteagain by [y ∗ , u∗ , p∗ ] and the proof is finished.
Remark. If we assume that ϕ0u , ϕ0y are differentiable, then standard gradientmethods may be applied for numerical solutions approximating problem (3.1)–(3.3), (3.12). As the nonconvexity appears in the formulation of the problem(3.1)–(3.3), (3.7), (3.10), just local optimal solutions are to be expected in thissecond step.
Remark. An alternative formulation of the second step in the solution of
the identification problem associated to (2.1), (2.2) and the observation yb is:
½ Z
¾
1
2
min
(y − yb)− dx
2 ω
subject to (2.1), (2.2), u ∈ K and to the state constraint(3.17)y ≤ yb a. e. inω.

Based on the results from Section 2, it is easy to see that the set of admissible
controls u ∈ K (i. e. such that (3.17) is fulfilled) is convex. This is an important
advantage. In view of this remark an interesting open question is to be studied
under what assumptions quasiconvexity properties are valid in identification
problems. However, by penalizing (3.17) one obtains the classical least squareapproach in identification problems. That is, this is just a reformulation of theusual approach and does not provide a new solution method.
14

References[1] V. Barbu (1984) — “Optimal Control of Variational Inequalities”, RNM
100, Pitman, London.
[2] V. Barbu and Th. Precupanu (1986) — “Convexity and Optimization in
Banach Spaces”, D. Reidel, Dordrecht.
[3] B. Kawohl and J. Lang (1997) — “Are some optimal shape problems convex?”, J. Convex Analysis[4] B. Lemaire (1985) — “Application of a subdifferential of a convex composite
functional to optimal control in variational inequalities”, LNMEMS 255,
Springer-Verlag, Berlin, P. 103–118.
[5] W. B. Liu and D. Tiba (2001) — “Error estimates in the approximationof optimization problem governed by nonlinear operators”, Numer. Funct.
Anal. Optimiz. 22, p. 953–972.
[6] D. Neittaanmäki, J. Sprekels and D. Tiba (2005) — “Optimization of Elliptic Systems, Theory and Applications”, Springer-Verlag, New York.
[7] D. Tiba (1990) — “Optimal Control of Nonsmooth Distributed Parameter
Systems”, Lecture Notes in Mathematics 1459, Springer-Verlag, Berlin.

15

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
UTMS
2005–24 Hirotaka Fushiya: Limit theorem of a one dimensional Marokov process to
sticky reﬂected Brownian motion.
2005–25 Jin Cheng, Li Peng, and Masahiro Yamamoto: The conditional stability in line
unique continuation for a wave equation and an inverse wave source problem.
2005–26 M. Choulli and M. Yamamoto: Some stability estimates in determining sources
and coeﬃcients.
2005–27 Cecilia Cavaterra, Alfredo Lorenzi and Masahiro Yamamoto: A stability result
Via Carleman estimates for an inverse problem related to a hyperbolic integrodiﬀerential equation.
2005–28 Fumio Kikuchi and Hironobu Saito: Remarks on a posteriori error estimation
for ﬁnite element solutions.
2005–29 Yuuki Tadokoro: A nontrivial algebraic cycle in the Jacobian variety of theKlein quartic.
2005–30 Hao Fang, Zhiqin Lu and Ken-ichi Yoshikawa: Analytic Torsion for Calabi-Yau
threefolds.
2005–31 Ken-ichi Yoshikawa: On the singularity of Quillen metrics.
2005–32 Ken-ichi Yoshikawa: Real K3 surfaces without real points, equivariant determinant of the Laplacian, and the Borcherds Φ-function.
2005–33 Taro Asuke: Inﬁnitesimal derivative of the Bott class and the Schwarzian
derivatives.
2005–34 Dan Tiba, Gengsheng Wang and Masahiro Yamamoto: Applications of convexity in some identiﬁcation problems.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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