ON OPTIMIZATION METHOD IN THE NEUMANN PROBLEM FOR WAVE EQUATION

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ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v30i6.5}

ON OPTIMIZATION METHOD IN

THE NEUMANN PROBLEM FOR WAVE EQUATION

Hamlet F. Guliyev1, Vusala N. Nasibzadeh2§ 1_{Baku State University}

23, Z. Khalilov str. AZ 1148, Baku, AZERBAIJAN

1_{Institute of Mathematics and Mechanics}

AZ 1141, B. Vahabzadeh str. 9, Baku, AZERBAIJAN

1,2_{Sumgait State University}

1, Baku str, 43 ward. AZ 5008, Sumgait, AZERBAIJAN

Abstract: In this paper the Neumann problem is considered for the wave equation in two-dimensional case. A theorem on uniqueness of the solution of the appropriate inverse problem is proved. In the optimal control problem compared to the inverse problem, a theorem on the existence of an optimal control is proved, sufficient and necessary condition of optimality is derived in the form of variational inequality.

AMS Subject Classification: 31A25, 49J20, 49N45

Key Words: Neumann problem, inverse problem, optimality condition

1. Introduction

As is known, the Dirichlet and Neumann problems for hyperbolic equations are ill-posed [1]. Such problems arise in different applied problems (see [1], [2] and references therein). Therefore, study of Dirichlet and Neumann problems for hyperbolic equations is of interest both from applied point of view and in con-nection with theoretical issues. In the paper [2] the Dirichlet problem is studied for the wave equation in two-dimensional case. The inverse problem is consid-ered, ill-posedness of this problem is shown, it is reduced to the sequence of

one-Received: November 4, 2017 c 2017 Academic Publications

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dimensional inverse problems and in the obtained problem, iterational solution methods are suggested. In the present paper we consider the Neumann prob-lem for the wave equation also in two-dimensional case. The inverse probprob-lem is formulated and this problem is reduced to the sequence of one-dimensional inverse problems. The uniqueness of the solution of one-dimensional inverse problems is proved. Then an optimal control problem associated to the for-mulated inverse problem is studied, differentiability of the functional is shown, necessary and sufficient condition of optimality is derived in the form of integral inequality.

2. Problem Statement

Let us consider the Neumann problem for the wave equation

∂2u ∂t2 =

∂2u ∂x2 +

∂2u

∂y2,(x, y, t)∈Q= Ω×(0, T), (2.1)

∂u

∂t |t=0 =f1(x, y), ∂u

∂t |t=T =f2(x, y),(x, y)∈Ω, (2.2) ∂u

∂x|x=0 = ∂u

∂x|x=ℓ = 0, (y, t)∈(0, ℓ)×(0, T), (2.3) ∂u

∂y|y=0 = ∂u

∂y|y=ℓ = 0, (x, t)∈(0, ℓ)×(0, T), (2.4)

where Ω = (0, ℓ)_×(0, ℓ), ℓ >0, T >0 are the given numbers,f1(x, y), f2(x, y)

are the given functions, moreover f₁, f₂ _∈ W₂1(Ω). It is known that the Neu-mann problem for hyperbolic equation is ill-posed [1]. We formulate problem (2.1)-(2.4) as an inverse problem to the following direct well-posed problem

∂2u ∂t2 =

∂2u ∂x2 +

∂2u

∂y2, (x, y, t)∈Q, (2.5)

u_|t=0 =υ(x, y),

∂u

∂t |t=0 =f1(x, y), (x, y)∈Ω, (2.6) ∂u

∂x|x=0 = ∂u

∂x|x=ℓ = 0, (y, t)∈(0, ℓ)×(0, T), (2.7) ∂u

∂y|y=0 = ∂u

∂y|y=ℓ = 0, (x, t)∈(0, ℓ)×(0, T). (2.8)

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Let the function υ(x, y) be unknown and about the solution of direct problem (2.5)-(2.8) the following additional information is known

∂u

∂t |t=T =f2(x, y), (x, y)∈Ω. (2.9)

The inverse problem is in determining the pair of function u = u(x, y, t),

υ=υ(x, y) from relations (2.5)-(2.9) for the given functionsf1(x, y),f2(x, y).

Instead of this problem let us consider the following problem: to find the func-tion υ(x, y) from the class

V =

υ(x, y) : υ_∈W₂2(Ω), ∂υ ∂x

x=0

= ∂υ

∂x|x=ℓ= 0,

∂υ ∂y

y=0

= ∂υ

∂y

y=ℓ = 0,kυkW₂2(Ω)≤M

, (2.10)

such that together with the solutionu(x, y, t;υ) of problem (2.5)-(2.8) it affords minimum to the functional

J(υ) = 1 2

Z

Ω

∂u(x, y, T;υ)

∂t −f2(x, y)

2

dxdy, (2.11)

where u = u(x, y, t;υ) is the solution of problem (2.5)-(2.8) for υ = υ(x, y),

M > 0 is a given number. We call the function υ(x, y) a control, while the set V a class of admissible controls. We call this problem (2.5)-(2.8), (2.10), (2.11). Note that there is a close relation between inverse problem (2.5)-(2.9) and problem (2.5)-(2.8), (2.10), (2.11) if in problem (2.5)-(2.8), (2.10), (2.11) there exists a control υ∗ from V, such that min

υ∈V J(υ) = J(υ∗) = 0, then in

problem (2.5)-(2.9) the additional condition (2.9) is fulfilled.

From the results of the papers [3,4] it follows that for each fixed control

υ(x, y), problem (2.5)-(2.8) has a unique solution fromW₂2(Q) and this solution has the properties

u_∈C [0, T] ;W₂2(Ω)

, ∂u

∂t ∈C [0, T] ;W

1 2 (Ω)

,

∂2u

∂t2 ∈C([0, T] ;L2(Ω)),

and furthermore, the following estimation is valid

∂2u ∂t2

L2(Ω)

+

∂2u ∂t∂x

L2(Ω)

+

∂2u ∂t∂y

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+

∂2u ∂x2

_L

2(Ω)

+

∂2u ∂x∂y

_L

2(Ω)

+

∂2u ∂y2

_L

2(Ω)

≤ch_kυ_k_W2

2(Ω)+kf1kW21(Ω)

i

,_∀t_∈[0, T]. (2.12)

Here and in the sequel, different constants independent of the estimated quantities and admissible controls will be denoted by c.

3. On Uniqueness of the Solution of Inverse Problem

Let for simplicity ℓ = π. In problem (2.5)-(2.8) we continue evenly all the functions with respect to variableyto the interval (₋π,0). Expand the function

u(x, y, t) and all other functions in Fourier series in the system of functions {cosky_}∞_k=1:

u(x, y, t) =

∞

X

k=1

uk(x, t) cosky, υ(x, y) = ∞

X

k=1

υk(x) cosky, f1(x, y)

=

∞

X

k=1

f₁(k)(x) cosky, f₂(x, y) =

∞

X

k=1

f₂(k)(x) cosky.

Substituting these expressions in conditions (2.5)-(2.9), we get a sequence of one-dimensional inverse problems

∂2uk

∂t2 =

∂2uk

∂x2 −k 2_u

k, x∈(0, π), t∈(0, T), (3.1)

uk(x,0) =υk(x),

∂uk

∂t (x,0) =f

(k)

1 (x), x∈(0, π), (3.2)

∂uk

∂x |x=0 = ∂uk

∂x |x=π = 0, t∈(0, T), (3.3)

and

∂uk(x, T)

∂t =f

(k)

2 (x), k= 1,2, ... . (3.4)

Now we evenly continue the functionsuk(x, t),f₁(k)(x),f₂(k)(x),υk(x) with

respect to x to the interval (₋π,0) and expand them in Fourier series in the system of functions_{cosnx_}∞_n=1:

uk(x, t) = ∞

X

n=1

uk,n(t) cosnx, υk(x) = ∞

X

n=1

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f₁(k)(x) =

∞

X

n=1

f₁(k,n)cosnx, f₂(k)(x) =

∞

X

n=1

f₂(k,n)cosnx.

As a result, we get a sequence of inverse problems:

d2uk,n

dt2 + k

2₊_n2

uk,n= 0, (3.5)

uk,n(0) =υk,n,

duk,n(0)

dt =f

(k,n)

1 , (3.6)

duk,n(T)

dt =f

(k,n)

2 . (3.7)

It is clear that the solution of direct problem (3.5), (3.6) is obtained in the form

uk,n =υk,ncospk,nt+

f₁(k,n) pk,n

sinpk,nt, (3.8)

wherepk,n=

√

k2₊_n2 _.

From (3.8) we take the derivative with respect totand instead oftsubstitute

T and provided (3.7). If T ₆= πm

pk,n, m∈Z, then we get

υk,n=

f₁(k,n)cospk,nT −f2(k,n)

pk,nsinpk,nT

. (3.9)

Thus, we can formulate the following theorem.

Theorem 3.1. Suppose that for all k, n_∈N andm_∈Z the parameterT satisfies the conditionT ₆= _pπm

k,n. If for anyk∈N the inverse problem(3.1)-(3.4)

has the solutionυk(x) inW22[0, π],then this solution is unique and its Fourier

coefficients are given by formula (3.9).

4. On Solvability of Problem (2.5)-(2.8), (2.10), (2.11)

Theorem 4.1. If the conditions posed in formulation of problem (2.5)

-(2.8), (2.10), (2.11) are fulfilled, then the set of optimal controls of problem

(2.5)-(2.8),(2.10),(2.11)

V∗=

υ∗∈V : J(υ∗) =J∗ = inf υ∈V J(υ)

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is not empty, is weakly compact inW₂2(Ω)and any minimizing sequence_{υm(x)}

weakly inW₂2(Ω)converges to the setV∗.

Proof. If is easy to see that the set V determined by relation (2.10) is weakly compact in W₂2(Ω). Show that functional (2.11) weakly in W₂2(Ω) is continuous on the set V. Let υ be some element and _{υm} ⊂ V be arbitrary

sequence such thatυm →υ weakly inW₂2(Ω). Hence and from the imbedding

theoryW₂2(Ω)_→C Ω¯

[4] it follows that

υm→υ strongly in C Ω¯

as m_{→ ∞}. (4.1)

In virtue of unique solvability of boundary value problem (2.5)-(2.8) to each controlυm∈V there corresponds a unique solution um=u(x, y, t;υm) of

problem (2.5)-(2.8) and the estimation _kumkW2

2(Q) ≤c, ∀m= 1,2, ..., is valid,

i.e. the sequence_{um} is uniformly bounded in the norm of the spaceW₂2(Q).

Then from imbedding theorem [5] it follows that

um →u strongly in C Q¯, (4.2)

∂2um

∂t2 →

∂2u ∂t2,

∂2um

∂t∂x →

∂2u ∂t∂x,

∂2um

∂t∂y →

∂2u

∂t∂y, (4.3)

∂2um

∂x2 →

∂2u ∂x2,

∂2um

∂x∂y →

∂2u ∂x∂y,

∂2um

∂y2 →

∂2u

∂y2 strongly in L2(Q)

and in particular,

∂um(x, y,0)

∂t →

∂u(x, y,0)

∂t ,

∂um(x, y, T)

∂t →

∂u(x, y, T)

∂t strongly in L2(Ω), (4.4)

whereu=u(x, y, t)_∈W₂2(Q) is some element.

Show thatu(x, y, t) =u(x, y, t;υ), i.e. the functionu(x, y, t) is the solution of problem (2.5)-(2.8) corresponding to the control υ_∈V. It is clear that the following relations are valid:

Z

Q

∂2um

∂t2 −

∂2um

∂x2 −

∂2um

∂y2

ϕ(x, y, t)dxdydt= 0 _∀ϕ_∈L2(Q),

um|t=0 =υm(x, y),

∂um

∂t |t=0 =f1(x, y), ∂um

∂x |x=0 = ∂um

∂x |x=π = 0, ∂um

∂y |y=0 = ∂um

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Then passing to limit in these relations asm _{→ ∞}, considering (4.1)-(4.4) and uniqueness of the solution of problem (2.5)-(2.8), corresponding to the control υ _∈ V, we get u(x, y, t) = u(x, y, t;υ). Now, using second one of relations (4.4) we get J(υm) → J(υ) as m → ∞, i.e. J(υ) W₂2(Ω) is weakly

in continuous on the setV. Then by Theorem 4.1 from [6, p.49] we get that all statements of Theorem 4.1 are valid.

Thus Theorem 4.1 is proved.

5. Differentialbility of Functional (2.11) and Optimality Condition

Now let us study Frechet differentiability of functional (2.11) and get optimality condition in problem (2.5)-(2.8), (2.10), (2.11).

Letψ=ψ(x, y, t;υ) be the generalized solution fromW₂1(Q) of the adjoint problem

∂2ψ ∂t2 =

∂2ψ ∂x2 +

∂2ψ

∂y2, (x, y, t) ∈Q, (5.1)

ψ_|t=T =

∂u(x, y, T;υ)

∂t −f2(x, y), ∂ψ

∂t |t=T = 0,(x, y)∈Ω, (5.2) ∂ψ

∂x |x=0 = ∂ψ

∂x|x=π = 0,(y, t)∈(0, π)×(0, T), (5.3) ∂ψ

∂y |y=0 = ∂ψ

∂y |y=π = 0,(x, t)∈(0, π)×(0, T). (5.4)

Under the generalized solution of boundary value problem (5.1)-(5.4) for the givenυ_∈V we will understand the function ψ=ψ(x, y, t;υ) fromW₂1(Q) which equals to ∂u(x,y,T_∂t ;υ)₋f2(x, y) fort=T and satisfies the integral identity

Z

Q

∂ψ ∂t

∂η

∂t −

∂ψ ∂x

∂η

∂x−

∂ψ ∂y

∂η ∂y

dxdydt

+ Z

Ω

∂ψ(x, y,0;υ)

∂t η(x, y,0)dx= 0 (5.5)

for all η=η(x, y, t)_∈W₂1(Q).

As

∂u(x, y, T;υ)

∂t −f2(x, y)

∈W₂1(Ω), from the results of the work [7] it follows that for each given υ_∈V problem (5.1)-(5.4) has a unique generalized solution from W₂1(Q) and this solution has the properties

ψ_∈C [0, T] ;W₂1(Ω)

, ∂ψ

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moreover, the following estimation is valid:

kψ_k_W1 2(Ω)+

∂ψ ∂t

L2(Ω)

≤c

∂u(x, y, T;υ)

∂t −f2(x, y)

W1 2(Ω)

, _∀t_∈[0, T].

Considering here the estimation (2.12) and boundedness of the imbedding

W₂2(Q)_→W₂1(Ω) [4], we get _∀t_∈[0, T]: kψ_k_W1

2(Ω)+

∂ψ ∂t

_L

2(Ω)

≤ch_kυ_k_W2

2(Ω)+kf1kW21(Ω)+kf2kW21(Ω)

i

. (5.6)

Theorem 5.1. Let the conditions in the statement of problem(2.5)-(2.8),

(2.10), (2.11) be fulfilled. Then the functional (2.11) is continuously Fr´echet differentiable on V and its differential at the point υ _∈ V at the increment δυ_∈W₂2(Ω)is determined by the expression

J′(υ), δυ =₋

Z

Ω

∂ψ(x, y,0;υ)

∂t δυ(x, y)dxdy. (5.7)

Proof. Let us consider the increment of functional (2.11):

∆J(υ) = Z

Ω

∂u(x, y, T;υ)

∂t −f2(x, y)

∂δu(x, y, T)

∂t dxdy

+1 2

Z

Ω

∂δu(x, y, T)

∂t

2

dxdy, (5.8)

where δυ_∈W₂2(Ω) is the increment of the control on the elementυ _∈V such thatυ+δυ_∈V,and byδu(x, y, t) we denote the differenceu(x, y, t;υ+δυ)₋

u(x, y, t;υ).

It is clear that the functionδu=δu(x, y, t) is the solution fromW₂2(Q) of the boundary value problem

∂2δu ∂t2 =

∂2δu ∂x2 +

∂2δu

∂y2 ,(x, y, t)∈Q, (5.9)

δu_|t=0 =δυ(x, y),

∂δu

∂t |t=0 = 0,(x, y)∈Ω, (5.10) ∂δu

∂x |x=0 = ∂δu

∂x |x=π = 0, (y, t)∈(0, π)×(0, T), (5.11) ∂δu

∂y |y=0 = ∂δu

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Then, in particular, for t = 0 the condition δu_|_t=0 = δυ(x, y) is fulfilled, and integral identity

Z

Q

−∂δu_∂t ∂µ_∂t +∂δu

∂x ∂µ

∂x+

∂δu ∂y

∂µ ∂y

dxdydt

+ Z

Ω

∂δu(x, y, T)

∂t µ(x, y, T)dxdy= 0 (5.13)

for all µ=µ(x, y, t)_∈W₂1(Q).

If in (5.5) we putη=δu(x, y, t),in (5.13)µ=ψ(x, y, t;υ) and put together the obtained relations, we have

Z

Ω

∂ψ(x, y,0;υ)

∂t δu(x, y,0)dxdy

+ Z

Ω

∂δu(x, y, T)

∂t ψ(x, y, T;υ)dxdy= 0. (5.14)

Then taking into account δu(x, y,0) =δυ(x, y) and

ψ(x, y, T;υ) = ∂u(x, y, T;υ)

∂t −f2(x, y),

from (5.8) and (5.14) we get

∆J(υ) =₋ Z

Ω

∂ψ(x, y,0;υ)

∂t δυ(x, y)dxdy

+1 2

Z

Ω

∂δu(x, y, T)

∂t

2

dxdy. (5.15)

The first addend in the right hand side of (5.15), i.e. expression (5.7) for the given υ _∈ V determines linear bounded functional of δυ on W₂2(Ω). Linearity of functional (5.7) with respect to δυ is obvious. Furthermore, using the Cuchy-Bunyakowskiy inequality, we get

Z

Ω

∂ψ(x, y,0;υ)

∂t δυ(x, y)dxdy

≤

∂ψ(x, y,0;υ)

∂t

L2(Ω)

kδυ_k_L_2(Ω)_≤c

∂ψ ∂t

W1 2(Ω)

kδυ_k_W2 2(Ω).

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Now let us estimate the residual term R _≡ 1₂R Ω ∂δu(x,y,T) ∂t 2 dxdy, con-tained in (5.15). Under the above suppositions, for solving problem (5.9)-(5.12) the following estimation is valid [3,4]

∂2δu ∂t2 _L 2(Ω) +

∂2δu ∂t∂x _L 2(Ω) +

∂2δu ∂t∂y _L 2(Ω) +

∂2δu ∂x2 _L 2(Ω) +

∂2δu ∂x∂y

L2(Ω)

+

∂2δu ∂y2

L2(Ω)

≤c_kδυ_k_W2

2(Ω),∀t∈[0, T]. (5.16)

Taking into account

∂δu(x, y, t)

∂t =

Z t

0

∂2δu(x, y, τ)

∂t2 dτ,

hence and from (5.16) it follows that

R= 1

2 Z

Ω

∂δu(x, y, T)

∂t

2

dxdy _≤c_kδυ_k2_W2 2(Ω).

Then from (5.15) it follows that functional (2.11) is Frechet differentiable on V and formula (5.7) is valid.

At last we show that the mappingυ_→J′₍_υ₎_,_{determined by equality (5.7)}

continuously acts from V to the space W₂2(Ω) associated to W₂2(Ω)∗ . Letδψ(x, y, t) =ψ(x, y, t;υ+δυ)₋ψ(x, y, t;υ).From (5.1)-(5.4) it follows that δψ(x, y, t) is the generalized solution from W₂1(Q) of the boundary value problem

∂2δψ ∂t2 =

∂2δψ

∂x2 +

∂2δψ

∂y2 ,(x, y, t)∈Q,

δψ_|_t=T = ∂δu(x, y, T)

∂t ,

∂δψ

∂t |t=T = 0,(x, y)∈Ω, ∂δψ

∂x |x=0 = ∂δψ

∂x |x=π = 0,(y, t)∈(0, π)×(0, T), ∂δψ

∂y |y=0 = ∂δψ

∂y |y=π = 0,(x, t)∈(0, π)×(0, T).

It is clear that the following estimation is valid for solving this problem

kδψ_k_W1 2(Ω)+

∂δψ ∂t L2(Ω) ≤c

∂δu(x, y, T)

∂t W1 2(Ω)

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Hence and from estimation (5.16) it follows that

∂δψ(x, y,0)

∂t

L2(Ω)

→0 as _kδυ_k_W2

2(Ω) →0. (5.17)

(5.7) yields that

J′(υ+δυ)−J′(υ)

(W2 2(Ω))

∗ ≤c

∂δψ(x, y,0)

∂t

L2(Ω)

.

By (5.17) the right hand side of this inequality tends to zero as_kδυ_k_W2 2(Ω)→

0. Hence it follows thatυ_→J′₍_υ_{) is continuous mapping from}_V _in _W2 2 (Ω)

∗ . Theorem 5.1 is proved.

Theorem 5.2. Let the conditions of theorem 5.1 be fulfilled. Then for the optimality of control υ∗=υ∗(x, y)∈V in problem(2.5)-(2.8),(2.10),(2.11) it

is necessary and sufficient the inequality

Z

Ω

∂ψ∗(x, y,0)

∂t (υ(x, y)−υ∗(x, y))dxdy ≤0 (5.18)

be fulfilled for any υ = υ(x, y) _∈ V, where ψ∗(x, y, t) = ψ(x, y, t;υ∗) is the

solution of problem(5.1)-(5.4) forυ=υ∗.

Proof. The set V determined by relation (2.10) is convex in W₂2(Ω). Fur-thermore, according to theorem 5.1, the functional (2.11) is continuously Fr´echet differentiable on V and its differential at the point υ _∈ V is determined by equality (5.7).

Then by Ttheorem 5 from [6, p. 28] on the element υ∗∈V∗ the inequality

hJ′₍_υ

∗), υ−υ∗i ≥0 should be fulfilled for all υ∈V.

Hence and from (5.7) it follows validity of inequality (5.18) for all υ _∈ V. Show that inequality (5.18) is also sufficient for optimality of the con-trol υ∗(x, y) ∈ V. Functional (2.11) under conditions (2.5)-(2.8) is convex on

W₂2(Ω). Indeed, the solution of problem (2.5)-(2.8) has the property

u(x, y, t;λυ+ (1₋λ)w) =λu(x, y, t;υ) + (1₋λ)u(x, y, t;w) (5.19)

for all υ, w_∈W₂2(Ω) and for all real λ. Further, the functional

g(z) = 1 2

Z

Ω

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is convex with respect to the variable z_∈ W₂1(Ω). Therefore, hence and from (5.19) it follows the convexity of the functionalJ(υ) on W₂2(Ω). Then, again by theorem 5 [6, p.28] we get that the proved necessary condition of optimality (5.18) is also a sufficient condition.

Theorem 5.2 is proved.

Remark 5.1. As the gradient of functional (2.11) is determined by formula (5.7), i.e.

J′(υ) =₋∂ψ(x, y,0;υ)

∂t ,

for the numerical solution of inverse problem (2.5)-(2.9) we can use the method of projection of a gradient on the setV [6, p.71-76].

References

[1] S.I. Kabanikhin, Inverse and Ill-posed Problems, Sib. Pub., Novosibirsk, 2009, 457 p.

[2] S.I. Kabanikhin, M.A. Bektemesov, D.B. Nurseitov, O.I. Krivorotko and A.N. Alimova, An optimizition method in the Dirichlet problem for the wave equation,J. Inverse ILL-Posed Probl.20 (2012), 193-211.

[3] O.A. Ladyzhenskaya,Mixed Problem for a Hyperbolic Equation, Gostekhiz-dat, I, 1953.

[4] O.A. Ladyzhenskaya,Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973, 408 p.

[5] S.L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Nauka, Moscow, 1988, 334 p.

[6] F.P. Vasil’ev, Methods for Solving Extremal Problems, Nauka, Moscow, 1981, 400 p.