Identification problem for damping parameters in linear damped second order systems

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IDENTIFICATION PROBLEM FOR DAMPING PARAMETERS IN LINEAR DAMPED SECOND ORDER

SYSTEMS

JONG-YEOUL PARK*, JUN-HONG HA AND HYO-KEUN HAN

ABSTRACT. We state the necessary conditions on optimizing the parameters which the damping differential operators containinab- stract linear damped second order evolution equation on the Gelfand five fold.

1. Introduction

A widely used approach to the identification problem for any system

is

to estimate the unknown parameters appear

in

the system by

minimizing

a quadratic function of the difference between observed value and desired value, so-called output least-square identification problem (OLSIP). We consider the system given by linear damped second order evolution equations, which we refer to"Dautray and Lions[5] as a model, of the forms

~y

dy .

(1.1) dt

₂

+ A

2

(q, t) dt + AI(q, t)y

=

f

^ID

(0, T)

with the initial values, and the cost functional given by the quadratic form

(1.2)

where Al (t, q), A

2

(t, q) are differential operators containing unknown pa- rameter

q E

Q which are given by some bilinear forms on Hilbert spaces,

C is

an observation operator defined on an observation space M,

^Zd^is

a desired value. The OLSIP suject to (1.1) with (1.2)

is

to find an element

Received April 8, 1997.

1991 Mathematics Subject Classification: 35LlO, 49K20, 49N50.

Key words and phrases: identification, optimal control.

• This work was partial supported

by

the Basic Science Research Institute Program, Ministry Education, 1996, Project No. BSRl-96-141O.

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q

E

Q such that inf

qEQ

J(q) = ^J(q) and to give a characterization of such q.

In

this paper we will study the OLSIP to the system (1.1) with (1.2). It is not easy to find the elements q belonging to an admissible set Q of parameters subject to (1.1) with (1.2). That is, we have no general answer for it. Hence we will show the existence of such q when Q is a compact subset of a topological space, and then we are going to concen- trate on giving a characterization of such q. Recently, inspired by the optimal control theoretical studies of Euler-Bernoulli Beam Equations with Kelvin-Voigt Damping, and Love-Kirchoffplate equations with var- ious damping terms, there appeared numerous papers studying optimal control theory and identification problem for the autonomous case of (1.1) on the Gelfand triple spaces.

In

Banks, Ito and Wang [3], Banks and Kunisch[4], they treated the ex- istence of the optimal controls (or minimizing parameters) by using the method of approximations, but the)' didn't deal with the necessary con- ditions (or characterizations) on them. When At (t, q) =

^'Y

^A

^{2 (}

^t, ^q),

^'Y

^> ⁰

in (1.1), the identification problem estimating q via OLSIP

is

studied by Ahmed[l, 2] based on the transposition method.

The aim of this paper is to study the identification problems through the method of OLSIP to (1.1) on the Gelfand five fold which will be stated in preliminary.

As

using the Gelfand five fold structure we may have some advantages that the operators At(t,q) and A

2

(t,q) can be defined with free differential orders in spatial sense. This paper is com- posed of the parts of preliminaries as section 2, necessary conditions as section 3 and applications to partial differential equations as section 4.

2. Preliminaries

First we explain the notations used in this paper. Let X be a Hilbert space. (·,·)x and 11 . lIx denote the inner product and the induced norm on X. X' denotes the dual space of X and (', ')x',x denotes the dual pairing between X' and X. Let us introduce underlying Hilbert spaces to describe damped second order evolution equations. Let H be a real pivot Hilbert space, its norm II . IIH is denoted simply by I . IH. For

i =

1, 2, let Vi be a real separable Hilbert space. Assume that each pair

(Vi, ^H) is a Gelfand triple space with the notation, Vi

^C-...+

^H = ^H'

^C-...+

_l'i'.

From now on, we write Vi

⁼

V for notational convenience. We shall give

an exact description of damped second order evolution equations. We

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suppose that Q is algebraically contained in a linear topological vector space with topology T and QT

=

(Q, T) is closed. Let T> ° ^{be fixed.}

In

order to define two main differential operators, diffusion and damp- ing, we introduce two bilinear forms ai(t, q; 4>, <p),

i =

1,2, q E Qn

t

E [0, T] on Vi x Vi satisfying

(2.1) ai(t, q;

<P?

<p) = ai(t, q; <p, <p) for

all

<p, <p

^E

Vi and

t E

[0, Tj, there exists

Ci1

> ° ^{such that}

(2.2)

lai(t,q;<p,<p)I~Ci111<p1lv;11<plhf;

forall <p,'ljJEVi and tE[O,T]

and there exist

ai

> ° ^and ^Ai

^E

R such that

(2.3) ai(t, q; <p, <p) + Ail<pl1- 2

aill<plI~

for

all

<p

E

Vi and t

E

[0, Tj, the function t

^--t

ai(t, q; <p, <p) is continuously differentiable in [0, Tjand there exists

Ci2

> ° ^{such that}

(2.4)

la~(t,q;<p,<p)1 ~

Cz"2I1<pIlv;II<plh'i for all <p,'IjJ

E

Vi and t

E

[O,Tj, where

I

= ft. Then we can define the operator A

1

(t, q)

E

£(Vi, Vn for t

E

[0, Tj deduced by the

r~lation

(2.5) ai(t, q; <p, <p) = (Ai(t, q)<p, <p)V;',v; for

all

<p, <p

E

Vi,

We suppose that Vi = ^V is continuously embedded in "2. Then we see that V

~

"2

~

H = ^H'

~

V;

~

V' and the equalities (<p, <p)v',v =

(<p, <P)V;,V2 for <p E V;, <p E V and (<p, <p)v',v = (<p, <P)H for <p E H, <p E V hold.

We often use the notations to express the time derivatives as g', g"

instead of !fi, ~, respectively. We define a Hilbert space, which will be a solution space, as

W(O, T) = {gig E £2(0, T; V), g' E £2(0, T; "2), g" E £2(0, T; V')}

with an inner product

(gl,g2)W(O,T)

=

I ^T {(gl(t),g2(t))V + (g~(t),g~(t))V2 ⁺ (g~(t),g~(t))v,}dt

and the induced norm

1

IIgllw(o,T)

=

(lIglli2(o,T;V) + Ilg'lli2(O,T;V2) + ^IIg"

¹¹

i2(O,T;V'))

2' •

In

OLSIP, many adjoint state equations have a variety of perturbed

terms as depending on observations. Hence we will treat existence,

uniqueness and regularity for the more generalized equation than (1.1),

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(2.6)

Le., we consider the Cauchy problem for the perturbed linear damped second order evolution equation of the form

{

y" + A2(t, q)y' + Al(t, q)y

=

L(t)y + 1

in

(0, T), y(O)

=

Yo E V;

y'(O)

=

Yl

E

H,

where L(·)

E

LOO (0, T; £(V2, V;» and 1

E

L2 (0, T; V;). Then we can show that there

is

a unique weak solution yE W(O, T) n C([O, T]; V) n

Cl ([0, T]; H) to (2.6)

in

the sense that

{

(y"('), 4J)VI,V + a2(-,

q;

y'(.), 4J) + al(-,

q;

y(.), 4J)

= (L(·)y(·) + ^1('),

4J)v~,V2for

all 4J

E

V in the sense of V'(O, T) with the initial conditions -

y(q; 0) = Yo

E

V; y'(q; 0) = Yl EH.

Moreover, the solution

y

to (2.6) satisfies the energy equality for each

t E

[0, T]

al(t, q; y(t), y(t» + ly'(t)lir + 2i

t

a2(0-, q; y'(o-), y'(o-»dn

=

al(O,q;yo,Yo) + !Yllir + it a~(o-,q;y(o-),y(o-»dn

(2.7) +2i

t

(L(o-)y(o-) + 1(0-), y'(o-»V~,V2dn.

For the proofs of these we refer to Ha [6], and Dautray and Lions [5]

whose treatment of equations was done on the Gelfand triple structure by the energy methods.

3. Necessary

conditions

In this paper we consider the case where all the parameters

q

related to the diffusion operator Al(t, q) has already known, Le., the damp- ing operator A

2

(t, q) contains unknown parameters only. Hence letting Al(t, q) = Al(t) the system (2.6)

is

written by

{

y" + A2(t, q)y' + Al(t)y

=

1

in

(0, T), (3.1) - y(q; 0) = Yo

^E

V;

y'(q; 0)

⁼

Yl EH.

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Note that since there is a unique solution

^y

to (3.1) for given

q E

QTl we can have a well-defined mapping y

=

y(q) of QT into W(O, T). We often call (3.1) the state equation and y(q) the state with respect to (3.1).

Let us consider a quadratic cost functional attached to (2.6) as

(3.2) J(q) = IICy(q) -

zdll~w

q

E

QTl

where M is a Hilbert space of observations, C

E

£(W(O, T), M) is an observer and

^Zd

is a desired value belonging to M. Our aim is to find

q

E

QT satisfying

(3.3) J(q)

=

minJ(q)

qEQT

and to give a characterization of such q. We call q the optimal control (or the minimizing parameter) to the system (3.1) and (3.2). Furthermore, we give an assumption to a2(t, q;

</>,

cp):

(3.4)

q

~

a2(t, q;

^</>,

cp) : QT

_~

R is continuous for all t

^E

[0, T],

^</>,

cp

E

\12.

Note that for each

qE

QT,

</>,

cp

E

\12 the following equalities hold:

sup la2(t, q;

</>,

cp)1 = sup I (A2(t, q)</>, cp}v£,V21 = II A2(t, q)</>IIv _2,

IIrp 11V2=1 11 rpllV2=1

whence the assumption (3.4) and the above equality imply that IIA 2(t, q)</>lIv 2 is continuous on q.

LEMMA

3.1. Let us assume that (2.1)-(2.4) and (3.4) hold. Then y(q) is strongly continuous on q, i.e., y(q)

E

C(QTl W(O, T».

Proof. Let us suppose that qn

~

q in QT and let Yn

=

y(qn),Y

=

y(q) be the solutions corresponding to qn, q, respectively. Then by letting Zn = Yn - Y we obtain the equation

(3.5)

_z~

+ _A2(t,

_qn)z~

+ Al (t)zn

=

[A

2

(t, q) - A

2

(t, qn)]Y'.

Since [A

₂

(t, q) - A2(t, qn)]Y'

^E

L2(0, T; Vn, we can apply (3.5) to (2.7).

Hence we have from zn(O) =

z~(O)

= 0

(3.6) al(t; Zn(t), zn(t» + Iz~(t)11 ⁺ 21

^t

a2(0", qn; z~(O"), z~(O"»do"

l

^t

a~ (0"; zn(O"), Zn(O"»do" + 21

^t

[a2(0", q; Y'(O") , z~(O"»

-a2(0", qn; Y'(O"),

z~(CT»]dO".

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Denote Ai = IAil,

i

= 1,2 for simplicity.

If

we estimate the above equality by using (2.3) and (2.4), then we have

(3.7) alll.zn(t)lI~ ⁺ Iz~(t)IJt ⁺ ^2a2it IIz~(a)II~/.lo-

::; AII.zn(t)IJt

+ 2A2it Iz~(a)IJtda ⁺ ^Cl2i ^t IIzn(a)lI~da

+2i

t

11

[A2(a, q) - A2(a, qn)]y'(a)lIv;lIz~(a)llV:!da.

Noting that IZn(t)\1- ::; T

I~

lz'n(a)l1-da and using Cauchy-Schwarz in- equality it follows from (3.7) that

(3.8) alll.zn(t)lI~ ⁺ Iz~(t)IJt ⁺ ^a2it Ilz~(a)II~2da

~ ^(>'1T ⁺ ^{2A2) i t} Iz~(a)IJtda ⁺ cl2i

^t

IIZn(a)lI~da

+2-i

t

11

[A2(a, q) - A2(a, qn)]y'(a)II~,da.

~ 0 2

Put a = min{l, ab a2} > ° ^and

^(J

⁼ ^a-I ^max{A1T ⁺ ^2A2, ^{C12, a2} ¹ ^{} and}

CPn(t)

= IIzn(t)lI~

+

Iz~(t)IJt·

It follows from (3.8) and Bellman-Gronwall's lemma that

(3.9) CPn(t) ~ ^(iT

¹¹

^[A ^{2(a, q) -} ^A ^2(a, qn)]y'(a)II~;da

⁾ ^(Je?'.

Using the continuity of a2(t,

q;

<p, 'ljJ) on

q,

the right hand side of (3.9) goes to zero, and so, we have CPn(t)

--t

° ^{for all} ^t

^E

^[0, ^Tj. ^Applying

this fact to (3.8) we conclude that Yn

--t

Y in C([O, T], V),

~ ^--t

y' in C([O, T], H) and

~ ^--t

y' in L2(0, T; V2). In particular, we also have

Yn

^--t

Y in W(O, T)

if

estimating (3.5). 0

LEMMA

3.2. Let us assume that (2.1)-(2.4) and (3.4) hold. There

is

at least one optimal control q

ifQr is

compact.

Proof

It

is clear from Lemma 3.1 and continuity of norm. 0

Now we present the necessary condition (the minimizing condition) for

the optimal controls q

E

Qr to the system (3.1) with the cost functional

J(p) given by (3.2).

If

J(p)

is

Gateaux differentiable at q in the direction

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q -

q, the necessary condition on q

is

characterized by the following inequality

(3.10) 6J(q)(q - q)

~

0 for all q

E

Q,

where 6J(q)(q - q) denotes the Gateaux derivative at q in the direction q - q. Note that since J(q)

is

composed of the term y(q), the Gateaux differentiability of J(q) follows from that of y(q). Hence to obtain that of y(q) we assume that U2(t,

q;

</J, cp) satisfies the following condition:

(3.11)

q -- A

2

(t, q) is weakly Gateaux differentiable for

all

t

and 6A2(t, q)(P) == 6A2(t, q;p) ^{E £2(0,} T; £(V2, VD) for

all

pE Qn where 6A

2

(t, q;p) denotes the Gateaux derivative at

q

in the direction of p.

LEMMA

3.3. Let us assume that (2.1)-(2.4), (3.4) and (3.11) are sat-

isfied.

Then y(q)

is

weakly Gateaux differentiable at q

in

the direction q - q, denote the Gateaux derivative of y(q) by z

=

6y(t, q; q - q), which satisfies the following Cauchy problem:

( 3 12) { z" + A2(t, q)z' + ^A

¹

^(t)z

⁼

-6A2(t, q; q - q)y(q)'

in

(0, T),

. z(O) = z'(O) = O.

Proof. For A

^E

(0,1) let y>.

=

y(q>.) and y

=

y(q) be weak solutions to (3.1) for given parameters q>. = q + A(q - q) and q, respectively. Then z>. = (y>. - y) / A satisfies the following equation .

(3.13)

{

z1 + _A2(t; q>.)z~ + A

₁

(t)z>.

=

A2(t, q) ~ A2(t, q>.)y' in (0, T), z>.(O) = z>.(O) = O.

Since for each A, [(A2(t, q) - A2(t, q>.))/A]Y' ^{E £2(0,} T; V;), we can apply (3.13) to (2.7). Hence from z>.(O) = zHO) = 0 we have

(3.14) al(t; z>.(t), z>.(t)) + Iz~(t)l~ ⁺ 21

^t

a2(0', q>.; z~(O'), ~(O'))do-

l

^t

a~(O'; z>.(O') , z>.(O'))dO' + ~ l

^t

[a2(0" q;y'(O'), z~(O'))

-a2(0', q).; y' (0'),

z~

(0' ))]do-.

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(3.15)

Let us consider the last term in (3.14). From the assumption (3.11) and the homogeneous property oA

2

(t,q;ep) = eoA

2

(t,q;p) there is v E (0,1) such that

~ Lt [a2(U, q;y'(u), z~(u)) - a2(U, q>.;y'(u), z~(u))]du

= Lt ^(-oA

²

^{(u, q} ⁺ >.v(q - q); q - q)y', z~(u)v;,v2du

depending on z>.

E

£2(0, T; V). By the similar calculations to Lemma 3.1 we have the following estimation

IIz>.(t)lI~ ⁺ 1~(t)l~ ⁺ Lt IIz~(u)lI~du ^S ^C Lt [I~(u)l~ ⁺ IIz>.(u)II~]du

+c LT

¹¹

^oA

²

^{(u, q} ⁺ >.v(q -q); q -7j)y'(u)II~;du(= ^K), ^.

where C > ° ^is a proper constant. Using Bellman-Gronwall's lemma we have

IIz>.(t)lI~ ⁺ Iz~(t)l~ ⁺ Lt 1I~(u)II~2du ^{S CTKe}

^CT

⁺ ^{K, v>.}

^E

^(0,1),

which implies that {z>. : >. E (0, 1)}

is

bounded in £2(0, T; V) c £00(0, T;

V) and

_{z~

: >. E (0,1)} is bounded in £2(0, T; V2) n £00(0, T; H). Since

£2(0,

T; V) and £2(0, T; V2) are reflexive, we can extract subsequences {Z>.,.} C {z>.} and find Z E £2(0, T; V) such that

(3.16)

z>.,.

-+Z

weakly in £2(0, T; V), z'>.,.

-+W

weakly in £2(0, T; V2).

Since Z>.,.

E

C([O, T], H) and z>.,.(O)

=

° in H, we have that for each t

E

[O,T]

z>.,.(t) = Lt z~Ju)du

in the H sense. Since by (3.16) J: ^z).,.(u)du

^-+

J: ^w(u)du ^{weakly in} ^H,

we obtain

z(t) = Lt ^w(u)du

in the weak H sense, which implies that z'(t) exists in H sense and

that wet)

=

z'(t) in H with z(O)

=

0. Since w

E

L

²

(0, T; V2), we

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have also w =

z'

in £2(0, T;

_~).

It follows by multiplying (3.13) by

</>

= «t)'Ij;, (

E

1'(0, T) and integrating it by parts that (3.17)l

T

[(

-z~Jt),

^</>'

^(t))H ⁺ a2(t, q>'n; z~Jt), </>(t)) + al (t; z>'n (t), </>(t))]dt

=

iT ^{(A2(t, q)} ^~nA2{t, q>.Jy'(t) , </>(t))v;,~dt.

Since the assumption (3.11) induce that for all v

E

[0,1]

la2(t, q>'n;

z~n

(t), </>{t)) - a2(t, q;

z~

(t), if>(t)) I

::; IAn(q - q)lrIl 8A2(u, q + Av(q - q); q -

q)</>lIv;lIz~J~,

we have from (3.16)

(3.18) l

^T

~(t,q>.n;z~(t),if>(t))dt

^{- t}

l

^T

a2(t,q;z'(t),</>(t))dt as An

^{- t}

O. Hence

if

we take An

^{- t}

0 in (3.17) by using (3.16), (3.11) and (3.18) then we have

(3.19) l

^T^[(

-z'(t), if>'(t))H + a2(t, q; z'(t), if>(t)) + al{t; z(t), </>(t))]dt

= iT (-8A2(t, q; q - q)y'(t), </>(t))v;,~dt,

which implies

(3.20) I

^T

[(z''(t), </>(t))v',V + a2{t, q; z'(t), if>(t)) + al(t; z(t), </>(t))]dt

= LT ^(-c5A

²

^(t, q; q - q)y'(t), </>(t))v;,~dt.

On the other hand, it follows from taking </>(t) = «t)'Ij; such that (

E

GI[O, T], «T) = 0 in (3.19), that .

(3.21) l

^T

[(z"(t) ,'Ij;)v',v + a2(t, q; z'(t), 'Ij;) + al (t; z(t), 'Ij;)]«t)dt

=

(z'(O) , </>(O))H + LT (-c5A2(t, q; q - q)y'(t), 'Ij;)v;,~«t)dt

for all

</> E

V. Substituting (3.20) into (3.21) we have z'(O)

=

o. ⁰

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By Lemma 3.3, the cost functional J(q) is weakly Gateaux differen- tiable at 71 in the direction q-71, and so, the condition (3.10) is rewritten by

(3.22)

(CY(71) - Zd, CZ)M

=

(C AM (CY(71) - Zd), Z)W(O,T)',W(O,T)*

_~

0,

"IqE

Qn where

^Z

is a unique weak solution to (3.12), C*

E

£(M', W(O, Ty) is the adjoint operator of C and AM is the canonical isomorphism of M onto M' in the sense that

(i) (AMcb,4»M',M = 114>lIlt,

(ii) IIAM4>IIM' = 114>IIM for

all

4>

E

M. In order to avoid the complexity of setting up observation spaces, we consider the following two types of distributive and terminal value observations in time sense. That is, the following cases:

(i) We take

Cl E

£(L2(0, T; V-;), M) and observer z(q) = Cly(q);

(ii) We take C

2 E

£(H, M) and observer z(q) =

~y(T,q).

3.1.

The case where

Cl E

£(L2(0, T; V-;), M) In this case the cost functional is given by

J(q) = IICly(q) -

zdll~,

q

E

Qn and then the necessary condition (3.22) is equivalent to

(3.23) LT (C; AM(Cly(t, (1) - Zd), z(t»)V2,V2dt ~ ^0,

^"Iq^E

^Qr'

Let us introduce an adjoint state p(71) satisfying (3.24)

{ r1"(71) - A2(t, (1)r1'(71) + ^{(AI(t) -}

A~(t,

(1)}'l(71)

=

Ci AM (ClY(71) - Zd), 1](T, (1)

= 1]'

(T, (1)

=

0. SinceCiAM(ClY(71)-Zd) E L2(0,T;V;)

andA~(·,71)

E LOO(O,T;£(V-;, VD),

the equation (3.24) is well-posed and permits a unique weak solution

1](71)

E

W(O, T) if we consider the change of the time variable as t

^---7

T -to

Multiplying (3.24) by z, which is a weak solution to (3.12), integrating

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it by parts after integrating it on [0, T], we obtain

iT

(''l(t, q), z"

+

A

2

^{(t, q)z}

⁺

Al (t)z)v,v,dt

=

iT

("l(t, q), -8A2(t, q;q - q)y'(t, q))v,v,dt

~ ^0.

Here we used the inequality (3.23). Summarizing these we have the following theorem.

THEOREM

3.1. Let us assume that (2.1)-(2.4), (3.4) and (3.11) hold.

Then the optimal control q is characterized by state and adjoint equa- tions and inequality:

{

y(q)"

+

A₂(t, q)y(q)'

+

Al (t)y(q)

= f in

(0,T), y(O,q)

= Yo E V;

y'(O, q)

=

Yl

EH,

{

"l"(q) - A2(t, q)"l'(q)

+

(A1(t) - A~(t,q))"l(q)

=C~AM(Cly(q)-Zd) in (O,T),

"l(T, q)

=

"l'(T, q)

= 0,

iT

^("l(t,

^q),

^8A

2

^{(t, q;}

^{q -}

q)y'(t, q))v,v,dt

:S

0,

'<Iq

E

QT.

3.2. The case where C

2E

£(H, M) In this case the cost functional is given by

J(q) = IIC

2y(T,

q) -

_zdll~,

q

E

Qn and then the necessary condition (3.22) is equivalent to (3.25)

(C2AM(~y(T, q) - Zd), Z(T))H ~0,

'<Iq

E

QT"

Let us introduce an adjoint state

"l(q)

satisfying

{

"l"(q)_- A2(t, q)"l'(q)

+

(A1(t) - A~(t,q))"l(q)

= 0, (3.26)

"l(T,

q) = 0,

"l'(T, q)

=

C2AM(~y(T,q) - Zd).

It

follows by the same reason as the case 3.1 that there is a unique weak

solution

"l(q) E W(O, T),

because

C

2

^{AM (C}²y(T, q) - Zd)

EH.

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THEOREM

3.2. Let us assume that (2.1)-(2.4), (3.4) and (3.11) hold.

Then the optimal control q is characterized by state and adjoint equa- tions and inequality:

{

y(q)"+A2(t,q)y(q)'+Al(t)y(q)=f in (O,T), y(O,q)

=

Yo E V;

y'(0, q)

= Yl E

H,

{

rJ"(q)_- A2(t, q)7]'(q) + (A1(t) -

_A~(t,

q))7](q) = ° ^{in (0,} ^T),

7](T, q) = 0,

7]'(T, q)

=

CiA

_M

(C2y(T, q) -

^Zd),

l

^T

(7](t, q), c5A2(t, q;

^q

-7j)y'(t, q))v,v,dt ;::: 0, V'q

E

Qr'

Proof

We prove the optimal control only. Multiplying (3.26) by

z,

which

is

a weak solution to (3.12), integrating it by parts after integrating it on [0, T], we obtain

° (z(T), 7]'(T, q))H + l

^T

(7J(t, q), z" + A2(t, q)z' + A1(t)z)v,V'dt (z(T), 7]'(T, q))H + l

^T

(7J(t, q), -c5A2(t, q;

^{q -}

q)y'(q))v,v,dt.

Hence from (3.26) and (3.25) we conclude that (z(T), c;A

M

(C

2

y(T, q) -

Zd))H

= l

^T

(7](t, q), c5A2(t, q; ^{q -} q)y'(t, q))v,v,dt 2': 0.

o

4. Applications to partial differential equations

Let n be an open bounded subset of Rn with the smooth boundary

f, and let n

T =

(0, T) x n and :E = (0, T) x f. We will give some

examples for the case where the operator A1(t) has the second (or the

fourth) differential order and A

2

(t) has the second (or o-th) differential

order in the variable x

E

n.

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(4.1)

EXAMPLE 4.1.

One of the simplest example in the linear damped second order equation is the following damped wave equation

! ^fPy

^- - K,A- - Ay

^ay = f

^ill

^.

^QT

at

²

at '

y

= 0 on E,

y(O) =

Yo,

it ⁽⁰⁾ ⁼

^Yl

ⁱⁿ

^Q.

Let us take V = V2 = HJ(Q), H = L

²

(Q) and

Yo E

V,

Yl E

H, and let

f

E

L

²

(Q

_{T ).}

Define a bilinear form u as

(4.2) u(q; </J,

'If;)

= In q(x)V'</J(x) . V''lf;(x)dx, </J,

'If; E

HJ(Q),

and take

al (q;

</J,

'If;)

=

u(l;

</J,

'If;)

and

a2(q;

</J,

'If;)

=

u(K,;

</J,

'If;),

where

K,

> .0 is a positive constant. It is easy to check for

ai

satisfying all conditions to be verified. Therefore there is a unique weak solution

Y

to

(4.1),

whose solution satisfies

ay ay

2

y, V'y, at' V' at EL (Q

T ).

Take M = L

²

(Q

T )

and C = I, the identity observation, and consider the cost functional defined as

(4.3) J(K,) = [ (y(t,X,K,) - Zd(t,X))2dx,

^K,

>

^O.

lnT

If

we take Q the set of all positive constants and give a

DXl-

norm topol- ogy on Q, then Q is compact in

DXl(Q).

Hence there is at least one optimal control subject to (4.1) and (4.3), say it k. Denote by y(R) the solution of the state equation (4.1). Then by Theorem 3.1 adjoint state equation and necessary condition on R are given as follows:

(4.4)

and

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(4.5)

(4.8)

EXAMPLE

4.2. As one of the realistic system, we can give the struc- turally damped plate equation of

I ^y=Lly=O ^y(O) ^{f2y ^{{)t2 -}

⁼

^V'. ^Yo,

⁽

^a(x)V' ^Z ^on

⁽⁰⁾

^E,

⁼

^ay) ^{)t

^Yl

⁺

ⁱⁿ

^Ll

²^O.

^y

⁼

^f

^ill^•

^0T,

We take a parameter set Q = {a

^E

Loo(O) : a ::; a(x) ::; b <

⁰⁰

a.e. x

^E

O} for some positive constants a and b. Let us take V =

~(O),

Vi =

HJ(O), H = L

²

(0). Define a bilinear form al as

(4.6)

al(ifJ,7jJ) = 1 LlifJ(x)Ll7jJ(x)dx, ifJ,7jJ

^E

Hg(O),

and take a2(q; ifJ, 7jJ) =

(T(a;

ifJ, 7jJ), which

is

given Example 4.1. For Yo

E

V;

Yl E

H and f

E L2(~),

there exists a unique weak solution

y

to

(4.5)

such that

ay ay

²

y, Lly, {)t' Ll {)t

E

L (OT).

Take M = L

²

(0) and

C

= I, the identity observation, and consider the cost functional defined as

(4.7)

J(a) = 1 ^(y(T, ^x, ^{a) -} ^Zd(X»2dx, ^a

^E

^Q,

where Zd

E

L

²

(0). Let a be the optimal control subject to

(4.5)

and

(4.7)

and denote yea) the solution of the state equation

(4.5).

Then by Theorem

3.2

adjoint state equation and necessary colldition on

a

are given as follows:

{f2~~a) ⁺ ^{V' .} (a(x)V'a~a») ⁺ ^Ll

²

^1](a) ⁼ ⁰

ⁱⁿ

^OT,

1](a) = Ll1](a)

=

0 on E, 1](T, a) = 0 in 0,

{)t1](T, a a) = yeT, a) - Zd

in 0

and iT ^In ^{(a(x) -} ^a(x»V'a~a)

. V'1](a)dxdt

~

^0, ^"la^E

^Q.

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References

[1] N. U. Ahmed, Optimization and identification of systems governed by evolu- tion equations on Banach space, Pitman Research Notes in Mathematics Series, Longman Scientific& Technical,184(1988).

[2] N. U. Ahmed, Necessary conditions of optimality for a class of second-order hyperbolic systems with spatially dependent controls in the coefficients,J.Optim.

Theory and Appl., 38 (1982),423-446.

[3] H. T. Banks, K. Ito and Y. Wang, Well Posedness for damped second order systems with unbounded input opemtors, Center for Research in Scientific Com- putation, North Carolina State Univ., 1993.

[4] H. T. Banks andK.Kunisch, Estimations Techniques for distributed pammeter systems, Birkhiiuser, 1989.

[5] R. Dautray and J. L. Lions,Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, 5, Evolution Problems, 1992.

[6] Junhong Ha, Optimal control problems for hyperbolic distributed pammeter sys- tems with damping terms, Doctoral Thesis, Kobe Univ., Japan, 1996.

Identification problem for damping parameters in linear damped second order systems