Boundary control problem with an infinite number of variables

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PII. S0161171201007128 http://ijmms.hindawi.com © Hindawi Publishing Corp.

BOUNDARY CONTROL PROBLEM WITH AN INFINITE

NUMBER OF VARIABLES

HUSSAIN A. EL-SAIFY

(Received 15 March 2001)

Abstract.Using a previous result by Gali and El-Saify (1983) and the theory of Kotarski (1989), and Lions (1971), we formulate the boundary control problem for a system gov-erned by Neumann problem involving selfadjoint elliptic operator of 2th order with an inﬁnite number of variables. The inequalities which characterize the optimal control in terms of the adjoint system are obtained, it is studied in order to construct algorithms attainable to numerical computations for the approximation of the control.

2000 Mathematics Subject Classiﬁcation. 49J20, 93C20.

1. Functions spaces. This section covers the basic notations, deﬁnitions, and prop-erties which are necessary to present this work.

Let(pk(t))∞k=1be a sequence of weights, ﬁxed in all that follows, such that

0< pk(t)∈C∞

R1_,

R1pk(t)dt=1, (1.1)

with respect to it we introduce on the regionR∞₌_R1_×_R1_×···_,_{the measure}_dρ(x) by setting [1,2,3],

dρ(x)=p1x1dx1⊗p2x2dx2⊗···, R∞x=xk∞k=1, xk∈R1. (1.2)

OnR∞_{we construct the space}_L2₍_R∞_{, dρ(x))}_{with respect to this measure [}₃_{], that is,} L2₍_R∞_{, dρ(x))}_{is the space of quadratic integrable functions on}_R∞_{. We will often set} L2₍_R∞_{, dρ(x))}₌_L2₍_R∞_).

We next consider a Sobolev space in the case of an unbounded region. For functions which are=1,2, . . . times continuously diﬀerentiable up to the boundaryΓ ofR∞ and which vanish in a neighborhood of∞, we introduce the scalar product

(φ, ψ)_W₍_R∞)=

|α|≤

Dαφ, DαψL2(R∞), (1.3)

whereDα_{is deﬁned by}

Dα= ∂|α|

∂x1α1_∂x2α2_···, |α| = ∞

i=1

αi (1.4)

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As in the case of a bounded region, the spacesW₍_R∞₎_{form a sequence of positive}

spaces. We can construct the negative spacesW−₍_R∞₎_{with respect to the zero space} W0₍_R∞₎₌_L2₍_R∞₎_{and then we have the following equipped [}₁_,₃_],

W_R∞_⊆_L2_R∞₌_W0_R∞_⊆_W−_R∞_,

φ_W₍_R∞)≥ φL2(R∞)≥ φW−₍_R∞).

(1.5)

As in [7] by using the Laplace-Beltrami operator (−∆Γ)on Γ, we choose the scalar product on the spaceH−β−1/2₍_Γ₎_{, fractional order Sobolev spaces, as}

(φ, ψ)H−β−1/2(Γ)=

Γ

−∆Γ−β−1/2

φ·ψ dΓ. (1.6)

2. Facts and results. Letπ (φ, ψ)be a continuous bilinear form which has the rep-resentation

π (φ, ψ)=(Aφ, ψ)L2₍_R∞), A∈L

W_R∞_{, W}−_R∞_, _{φ, ψ}_∈_W_R∞_, _(2.1)

whereAis a bounded selfadjoint elliptic operator of 2th order with an inﬁnite num-ber of variables which takes the form, [2,4,8,9],

(Aφ)(x)= |α|≤

∞

k=1

(−1)|α| 1 pk

xk

∂

2α ∂xk2α

pkxkφ(x)

+q(x)φ(x)

=

|α|≤

∞

k=1

(−1)|α|D2_kαφ(x)+q(x)φ(x),

(2.2)

where

Dαkφ

(x)= 1 pk

xk

∂

α

∂x_kα

pkxkφ(x)

, (2.3)

and the potentialq(x)is a real-valued function fromL2₍_R∞₎_{such that}_q(x)_≥_c0_>_0, c0is a constant [5].

The above bilinear form is coercive inW₍_R∞₎_{that means}

π (φ, φ)≥cφ2

W₍_R∞), φ∈W

_R∞_{, c}_constant_. _(2.4)

Since

π (φ, φ)=

|α|≤

∞

k=1

Dα kφ, Dαkφ

L2₍₍_R∞))+(qφ, φ)L2((R∞))

≥ φW₍_R∞)+c0φW₍_R∞) ≥cφW(R∞), 1≥c >0.

(2.5)

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Lemma_2.1. _{If (}_2.4_{) is satisﬁed then there exists a unique element}_y∈_W₍R∞₎

satis-fying Neumann problem

Ay=f , inR∞,

∂β_y

∂ν_Aβ =h onΓ,

(2.6)

where∂β_y/∂νβ A=

|β|≤−1

∞

k=1(D β

ky)cos(n, xk)onΓ,cos(n, xk)=kth direction co-sine ofn, n being the normal atΓ.

Proof. From the coerciveness condition and using the Lax-Milgram lemma [6]

there exists a uniquey∈W₍_R∞₎_{such that}

π (y, ψ)=L(ψ) ∀ψ∈WR∞ (2.7)

which is known as the variational Neumann problem whereL(ψ)is a continuous linear form onW₍_R∞₎_{and take the form}

L(ψ)=

R∞f ψ dρ(x)+

Γhψ dΓ, f∈L

2_R∞_{, h}_∈_H−β−1/2₍_Γ_). _(2.8)

Equation (2.7) is equivalent to

Ay=f onR∞, (2.9)

multiplying both sides byψand applying Green’s formula, we have

R∞(Ay)dρ(x)=

R∞f ψ dρ(x),

R∞

|α|≤

∞

k=1

(−1)|α|_D2α k y

ψ dρ(x)+

R∞q·yψ dρ(x)=

R∞f ψ dρ(x),

R∞

|α|≤

∞

k=1

Dkαy

Dαkψ

dρ(x)−

Γ

|β|≤−1 ∞

k=1 ψDαky

cosn, xk

dΓ

+

R∞q·yψ dρ(x)=

R∞φψ dρ(x).

(2.10)

Then

π (y, ψ)−

Γ

|β|≤−1 ∞

k=1 ψDβky

cosn, xkdΓ=

R∞f ψ dρ(x), (2.11)

sinceπ (y, ψ)=L(ψ)we have

Γ h−

∂β_y ∂νAβ

ψ dΓ=0, (2.12)

so

h=∂ β_y

∂νAβ

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3. Boundary control problem. The spaceU=H−β−1/2₍_Γ₎_{is the space of controls.} For every controlu∈U,Bu∈W−₍_R∞₎_{is given by}

(Bu, ψ)=

Γuψ dΓ, ψ∈W

_R∞_. _(3.1)

The statey(u)∈W₍_R∞₎_{of the system is given by the solution of}

πy(u), ψ=L(ψ)+(Bu, ψ) ∀ψ∈WR∞ _(3.2)

which may be interpreted asLemma 2.1to

Ay(u)=f inR∞_,

∂β_y(u) ∂νAβ

=h+u onΓ. (3.3)

The observationz(u)=y(u)and the cost functionJ(v)is given by

J(v)=y(v)−zd 2

L2₍_R∞)+(Nv, v)U

=

R∞

y(v)−zd

2

dρ(x)+(Nv, v)U,

(3.4)

wherezd∈L2(R∞)andN∈L(U , U ),Nis Hermitian positive deﬁnite operator. We wish to ﬁnd

inf v∈Uad

J(v), (3.5)

whereUad(set of admissible controls) is a closed convex subset ofU. Under the given consideration we have the following theorem.

Theorem_3.1. _{Assume that (}_2.4_{) holds and the cost function being given by (}_3.4_).

The optimal controluis characterized by (3.3) and

−∆Γ−+β+1/2P (u)|Γ+Nu, v−u

U≥0 ∀v∈Uad, (3.6)

whereP (u)is the adjoint state ofy(u).

Outline of the proof. _{As the proof of theorems in [}₄_,₆_{], the control}_u∈_Uad_is

optimal if and only if

J(u)(v−u)≥0 ∀v∈Uad (3.7)

which may be written as

y(u)−zd, y(v)−y(u)

L2₍_R∞)+(Nu, v−u)U≥0. (3.8)

In order to transform (3.8), we deﬁne the adjoint stateP (u)as the solution of the adjoint Neumann problem

Ap(u)=y(u)−zd inR∞,

∂β_{P (u)} ∂νAβ

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Now, multiplying the ﬁrst equation in (3.9) by(y(v)−y(u))and applying Green’s formula, ﬁnally taking into account the conditions in (3.3) and (3.9), we obtain

R∞

y(u)−zd

y(v)−y(u)dρ(x)

=

R∞Ap(u)

y(v)−y(u)dρ(x)

=

ΓP (u)

∂β_y(v) ∂νAβ

−∂βy(u) ∂νAβ

dΓ

−

Γ ∂β ∂νAβ

P (u)y(v)−y(u)dΓ+

R∞P (u)A

y(v)−y(u)dρ(x)

=

ΓP (u)(v−u)dΓ.

(3.10)

Hence (3.8) becomes

ΓP (u)(v−u)dΓ+(Nu, v−u)U≥0. (3.11)

By using the deﬁnition of the scalar product inH−β−1/2₍_Γ₎_{, then (}_3.9_{) is equivalent to}

−∆Γ−+β+1/2P (u)|Γ+Nu, v−u

U≥0 ∀v∈Uad (3.12)

which is equivalent to

Γ

P (u)+−∆Γ−β−1/2Nu(v−u)dΓ≥0 ∀v∈Uad (3.13)

which completes the proof.

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Hussain A. El-Saify: Mathematics Department, Faculty of Science, Branch of Cairo

University, Beni Suef, Egypt