Study of the nodal feedback stabilization of a string-beams network

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JAMC

DOI 10.1007/s12190-010-0412-9

Study of the nodal feedback stabilization of a string-beams network

Kaïs Ammari·Michel Mehrenberger

Received: 29 October 2009

Abstract We consider a stabilization problem for a string-beams network. We prove an exponential decay result. The method used is based on a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. Moreover, we give a numerical illustration based on the methodology introduced in Ammari and Tucsnak (ESAIM Control Optim. Calc.

Var. 6, 361–386,2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system.

Keywords Numerical stabilization·Feedback stabilization·String-beams network Mathematics Subject Classification (2000) 35B40·93D15·93D20·93B07· 65N25

1 Introduction and main results

We study a network ofNbeams and one string, whereN≥1, see [17] (pages: 80–81, for example) concerning the model. More precisely we consider the following initial

K. Ammari (

⁾

Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia e-mail:[email protected]

M. Mehrenberger

Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France

e-mail:[email protected]

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and boundary value problem:

(S)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

∂_t²u−∂_x²u=0, x∈(0, ), ∂_t²ui+∂_x⁴ui=0, x∈(0, i),

∂_xu(t,0)=∂_tu(t,0), ∂_xu_i(t, _i)=0, ∂_x³u_i(t, _i)=∂_tu_i(t, _i),

∂_x²ui(t,0)=0, u(t, )=u_i(t,0),

N j=1

∂_x³u_j(t,0)+∂_xu(t, )=0, u(0, x)=u⁰(x), ∂tu(0, x)=u¹(x), x∈(0, ),

u_i(0, x)=u⁰_i(x), ∂_tu_i(0, x)=u¹_i(x), x∈(0, i), t∈(0,∞), for alli=1, . . . , N,where_i denote the length of the beam numberi andis the length of the string.

In the last years an important literature was devoted to the controllability and sta- bilizability of string network or beam network, see [1–3,5,6,8–11,13,14] and [16].

In this paper, we study a stabilization problem for a string-beams network. By a resolvent method, we prove an exponential stability result of the energy of the system(S).

Moreover, we give some numerical study for the stabilization problem, the method used is based an the methodology introduced in Ammari and Tucsnak [7] where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system.

The paper is organized as follows. In this section we give precise statements of the main results. Section2contains the proof of the main results. Numerical illustration is given in the last section.

We define the energy ofu=(u, u1, . . . , uN)solution of(S)at instanttby

ES(t )=1 2

0

|∂tu(t, x)|²+ |∂xu(t, x)|² dx

+1 2

N i=1

_i

0

|∂_tu_i(t, x)|²+∂_x²u_i(t, x)²

dx. (1.1)

LetH= [V×(L²(0, )×N

i=1L²(0, i))]^t and let the topological supplementH˜of Span(0)inH, i.e.,

Span(0)⊕ ˜H=H, (1.2)

where₀=(1, . . . ,1,0, . . . ,0)^tand

V =

φ∈H¹(0, )× N i=1

H²(0, i), ∂_xφ_i(_i)=0, φ ()=φ_j(0),∀j=1, . . . , N

.

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Then, the wellposedness space for(S)isH, equipped with the inner product˜ (y, v) ;(z,w)) _H_˜ =

0

dy dx

dz dx+vw

dx+

N j=1

_j

0

d²yj

dx² d²zj

dx² +v_jw_j

dx.

(1.3) Denote

D(AS)

=

(u, v)^t∈

V ∩

H²(0, )× N i=1

H⁴(0, i)

×V t

∩ ˜H, ∂_xu(0)=v(0),

∂_x³u_i(_i)=v_i(_i), ∂_x²u_i(0)=0, i=1, . . . , N, N j=1

∂_x³u_j(0)+∂_xu()=0

. (1.4) The corresponding operatorASis defined by

AS(u, v)^t=

v, ∂_x²u,−∂_x⁴u₁, . . . ,−∂_x⁴u_N ^t, ∀(u, v)^t∈D(AS). (1.5) For the well-posedness of the system (S), It can be seen easily that H˜ endowed with this inner product, given by (1.3), is a Hilbert space. We show that the operator(AS,D(AS))defined by (1.4) and (1.5) generates a contraction semigroup on the Hilbert spaceH˜.

We have the following fundamental result.

Theorem 1.1 The operator(AS,D(AS))generates a strongly continuous contrac- tion semigroup(TS(t ))t≥0onH.˜

Proof To show the dissipativity ofAS, let(u, v)^t∈D(AS). By the definition ofAS

we have

AS(u, v) ^t, (u, v)^t

H

=

v, ∂_x²u,−∂_x⁴u1, . . . ,−∂_x⁴u_N ^t, (u, v) ^t

H

=

0

∂_xv∂_xu dx+

0

∂_x²uv dx+ N j=1

_j

0

∂_x²u_j∂_x²v_jdx− _j

0

∂_x⁴u_jv_jdx

= −

0

v∂_x²u dx+

0

∂_x²uv dx+ N j=1

_j

0

∂_x⁴u_jv_jdx− _j

0

∂_x⁴u_jv_jdx

+v()∂_xu()−v(0)∂xu(0)

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+ N j=1

∂_x²u_j(_j)∂_xv_j(_j)−∂_x²u_j(0)∂xv_j(0)−∂_x³u_j(_j)v_j(_j)

+∂_x³u_j(0)vj(0) .

Since ∂_x²uj(0)=0, ∂xvj(j)=0, vj(0)=v(), ∂_x³uj(j)=v(j), j =1, . . . , N,

∂_xu(0)=v(0)and∂_xu()= −_N

j=1∂_x³u_j(0), we get AS(u, v) ^t, (u, v)^t _H= −|v(0)|²−

N j=1

|v_j(_j)|²,

which shows thatASis dissipative.

Next, we show that(λI−AS)is surjective for someλ >0.

Given a vector(f , g) ^t∈H, we look for(u, v) ^t∈D(AS)such that (λI−AS) (u, v)^t =

f , g ^t. (1.6)

Hence by the definition ofAS, we obtain

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

λu−v=f, λuj−vj=fj, λv−∂_x²u=g,

λvj+∂_x⁴uj=gj, j=1, . . . , N.

This is clearly equivalent to

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

v=λu−f, vj=λuj−fj, λ²u−∂_x²u=g+λf,

λ²uj+∂_x⁴uj=gj+λfj, j=1, . . . , N.

Multiplying the third and the fourth identities byw∈V, and summing we get

0

(λ²u−∂_x²u)w dx+ N j=1

_j

0

(λ²uj+∂_x⁴uj)wjdx

=

0

(g+λf )w dx+ N j=1

_j

0

(gj+λfj)wjdx, ∀ w∈V . By formal integrations by parts, the left hand side is equal to

0

(λ²uw+∂_xu∂_xw)dx+ N j=1

_j

0

(λ²u_jw_j+∂_x²u_j∂_x²w_j) dx

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−∂_xu()w()+∂_xu(0)w(0) +

N j=1

∂_x³u_j(_j)w_j(_j)−∂_x³u_j(0)wj(0)−∂_x²u_j(_j)∂_xw_j(_j)

+∂_x²u_j(0)∂xw_j(0) . We thus find that

(u, w) =F (w), ∀ w∈V , (1.7) where

(u, w) =

0

∂xu∂xw+λ²uw (x) dx+ N j=1

_j

0

(∂_x²uj∂_x²wj+λ²ujwj) dx

+λu(0)w(0)+λ N j=1

uj(j)wj(j)

and F (w) =

0

(g+λf )w dx+ N j=1

_j

0

(g_j+λf_j)w_jdx+f (0)w(0)+ N j=1

f (_j)w_j(_j).

Sinceis a continuous bilinear coercive form on V andF is a continuous linear formV, by the Lax-Milgram lemma, problem (1.7) has a unique solutionu¯ ∈V. Using some integrations by parts, we easily check thatusatisfies

u−∂_x²u=g+λf,

λ²uj+∂_x⁴uj=gj+λfj, j=1, . . . , N,

as well as we have setv=λu−f,andv_j=λu_j−f_j, j=1, . . . , N. This means that (1.6) holds and consequently,(λI−AS)is surjective. The density ofD(AS)inH˜ is clear. Finally, the Lumer-Phillips theorem (see [21]) leads to the claim.

The above theorem provides the well-posedness of the evolution equation (S).

More precisely, for every (u⁰,u¹)^t ∈ H, the function (u(t ), ∂ _tu(t )) ^t given by TS(t )(u⁰,u¹)^t is the mild solution of(S). In particular, for (u⁰,u¹)^t ∈D(AS), the problem(S)admits a unique classical solution

(u, ∂ _tu) ^t∈C([0,∞),D(AS))∩C¹([0,∞),H).

The well-posedness part follows from Theorem1.1. In order to prove estimate (1.9) it suffices to remark that, by simple integrations by parts, it holds for regular solutions (i.e.(u, ∂ _tu) ^t ∈C([0,+∞);D(AS)). For mild solutions, we simply use the density ofD(AS)inH.

Thus, we have the following proposition.

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Proposition 1.2 The following assertions hold true

1. If(u⁰,u¹)^t∈Hthen the problem(S)admits a unique solution

u∈C([0,+∞);V )∩C¹

[0,+∞);L²(0, )× N i=1

L²(0, i)

such thatu(·,0), ui(·, _i)∈H¹(0, T ), T >0, i=1, . . . , N,and u(·,0)²_H1(0,T )+

N i=1

ui(·, i)²_H1(0,T )≤C(u⁰,u¹)²_H, (1.8)

for a constantC >0.

Moreoverusatisfies the energy identity:

ES(0)−ES(t )= _t

0

|∂tu(s,0)|²ds+ N

i=1

_t

0

|∂tui(s, i)|²ds. (1.9)

2. The estimate limt→∞E_S(t )=0 holds true for any finite energy solution of(S).

Our main result can now be stated as follows.

Theorem 1.3 There exist constantsC, w >0 such that for all(u⁰,u¹)^t∈Hthe so- lutionuof the system(S)satisfies:

E_S(t )≤Ce⁻^wtE_S(0), ∀t >0. (1.10)

2 Proof of the main result

In order to prove the strong stability we need to verify the following property.

Lemma 2.1 The spectrum ofAScontains no point on the imaginary axis.

Proof SinceAS has compact resolvent, its spectrumσ (AS), only consists of eigenvalues ofAS. We will show that the equation

ASz=iβz (2.1)

withz=(y, v)^t∈D(AS)andβ=0 has only the trivial solution.

By taking the inner product of (2.1), withz∈ ˜Hand using ASz, z _H= − |v(0)|²−

N j=1

v_j(_j)², (2.2)

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we obtain thatv(0)=0, vj(_j)=0, j=1, . . . , N. From (2.1), we get iβ(y, v)^t=(v, ∂ _x²y,−∂_x⁴y1, . . . ,−∂_x⁴y_N)^t. Next, we eliminatev=iβyto get:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−β²y−d²y

dx² =0, (0, ),

−β²y_j+d⁴yj

dx⁴ =0, (0, j), y()=yj(0),

N j=1

d³y_j

dx³(0)+dy

dx()=0, d²y_j

dx²(0)=0, y(0)=0, yj(j)=0, dy

dx(0)=0, dy_j

dx (j)=0, d³y_j

dx³(j)=0, j=1, . . . , N.

(2.3)

By a simple calculation we show that the above system only has trivial solution.

In fact, we have y(x)=Asin(βx), since y(0)=0 and −β²y − ^d_dx²^y2 =0. From

dy

dx(0)=0, we getA=0 and thus y(x)=0. This implies that y_j(0)=0. We thus have

yj(0)=d²y_j

dx²(0)=dy_j

dx (j)=d³y_j

dx³ (j)=0, (2.4) together with−β²yj+^d_dx⁴^y4^j =0. We can write

y_j(x)=A₁exp(i

βx)+A₂exp(−i

βx)+A₃exp(

βx)+A₄exp(− βx).

By using the conditions (2.4), we getA₁=A₂=A₃=A₄=0, and thusz=0 Proof of Proposition1.2 As the imbedding ofD(AS)inHis obviously compact, in order to prove the energy identity it suffices to remark that they hold true for regular solutions (i.e.(u, ∂ _tu)^t ∈C([0,+∞);D(AS))and to use the density ofD(AS)in H). Since AS is a maximal-dissipative operator inH, AS has no purely imaginary eigenvalues (see Lemma2.1), andAS has compact resolvent. Then, the strong stability estimate at the end of Proposition1.2can be obtained by applying the result in

Sect. 5 of [18] (see also [19]).

Proof of Theorem1.3 By a classical result (see Huang [15] and Prüss [22]) it suffices to show thatAsatisfies the following two conditions:

ρ(AS)⊃

iββ∈R

≡iR, (2.5)

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and

lim sup

|β|→+∞(iβI−AS)⁻¹<∞, (2.6) whereρ(AS)denotes the resolvent set of the operatorAS.

By Lemma2.1the condition (2.5) is satisfied for all, j>0, j =1, . . . , N. Sup- pose that the condition (2.6) is false. By the Banach-Steinhaus Theorem (see [12]), there exist a sequence of real numbersβ_n→ ∞and a sequence of vectors Z_n= _y_n

vn

∈D(AS)withZnH=1 such that

(iβ_nI−AS)Z_n_H→0 asn→ ∞, (2.7) i.e.,

iβ_ny_n− v_n≡ f_n→0 inV , (2.8)

iβnvn−d²y_n

dx² ≡gn→0 inL²(0, ), (2.9)

iβnvj,n+d⁴y_j,n

dx⁴ ≡kj,n→0 inL²(0, j), j=1, . . . , N. (2.10) Our goal is to derive from (2.7) thatZnHconverges to zero, thus, a contradiction.

The proof is divided in three steps:

First step. We notice that from (2.2) we have (iβ_nI−A)Z_nH≥

(iβ_nI−A)Z_n, Z_n _H= |v_n(0)|²+ N j=1

v_j,n(_j)².

Then, by (2.7)

v_n(0)→0, v_j,n(_j)→0, d³yj,n

dx³ (_j)=v_j,n(_j)→0, dy_n

dx (0)=v_n(0)→0, j=1, . . . , N.

This further leads to

βnyn(0)= −ifn(0)−ivn(0)→0, βnyj,n(j)= −ifj,n(0)−ivj,n(0)→0,

j=1, . . . , N, (2.11)

due to (2.8) and the trace theorem.

Second step. We express now vn, vj,n in terms of yn, yj,n, j =1, . . . , N, from (2.8)–(2.10) and substitute it into (2.8)–(2.10) to get

−β_n²y_n−d²y_n dx²

=g_n+iβ_nf_n, (2.12)

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−β_n²y_j,n+d⁴yj,n

dx⁴

=k_j,n+iβ_nf_j,n, j=1, . . . , N. (2.13)

Next, we take the inner product of (2.12) with q(x)^dy_dxⁿ inL²(0, )whereq(x)∈ C¹([0, ])andq()=0. We obtain that

0

−β_n²y_n−d²yn

dx²

q(x)dy¯n

dx dx

=

0

(g_n+iβ_nf_n)q(x)dy¯n

dx dx

=

0

g_nq(x)dy¯_n dx dx−i

0

qdf_n

dx β_ny¯_ndx

−i

0

fn

dq

dx βny¯ndx−ifn(0)q(0)βny¯n(0). (2.14) It is clear that the right-hand side of (2.14) converges to zero sincef¯_n, g_nconverge to zero inH¹andL², respectively.

By a straight-forward calculation,

0

−β_n²y_nqdy¯_n dx dx

!

=1

2q(0)|β_ny_n(0)|²+1 2

0

dq

dx|β_ny_n|²dx and

0

−d²yn

dx² qdy¯n

dx dx

!

=1 2q(0)

dyn

dx(0) ²+1

2

0

dyn

dx ²dq

dxdx. (2.15) According to (2.11), we simplify (2.14), then take its real parts. This leads to

0

dq

dx|β_ny_n|²dx+

0

dq dx

dyn

dx

²dx→0. (2.16)

By takingq(x)=e^x−−1,which satisfies|q(x)| ≤1, x∈ [0, ], we obtain that βnyn_L²_(0,)→0, ""

""dy_n dx

""

L²(0,)

→0, vn_L²_(0,)→0. (2.17)

Similarly, we take the inner product of (2.12) withqj(x)^dy_dx^j,n inL²(0, j)withqj∈ C²([0, j])andqj(0)=0, j =1, . . . , N, then repeat the above procedure, this will give

_j 0

dq_j

dx |β_ny2,n|²dx+ _j

0

3dq_j dx

d²y_j,n dx²

2

dx+2 _j

0

d²y_j,n dx²

d²q_j dx²

dy¯_j,n dx dx

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−2

d²y_j,n dx² (j)

2

qj(j)→0, j=1, . . . , N. (2.18)

By integrating by parts, we get _j

0

dyj,n

dx

²dx= − _j

0

y_j,nd²yj,n

dx² dx−y_j,n(0)dyj,n

dx (0)

= − _j

0

y_j,nd²y_j,n

dx² dx−y_n()dy_j,n dx (0)

= − 1 iβ_n

_j

0

v_j,nd²yj,n

dx² dx− 1 iβ_n

_j

0

(iβ_ny_j,n−v_j,n)d²yj,n

dx² dx

−dyj,n

dx (0)yn().

Then, from the boundedness ofv_j,n,iβ_ny_j,n−v_j,n, ^d

2y_j,n

dx² , inL²(0, j), (2.17) and (2.11), we have ^dy_dx^j,n converges to zero inL²(0, j), j=1, . . . , N, which implies

_j

0

dqj

dx |β_ny_2,n|²dx+ _j

0

3dqj

dx

d²yj,n

dx²

2

dx−2

d²yj,n

dx² (_j)

2

q_j(_j)→0,

j =1, . . . , N. (2.19)

Third step. Next, we show that ^d²^y^j,n

dx² (j), j =1, . . . , N,converge to zero. We take the inner product of (2.13) with ¹

φ_n^1/2e⁻^φ^1/2ⁿ ^h^j^(x) in L²(0, j) where φ_n =

|βn|, hj(x)=j−x, j=1, . . . , N.

This leads to _j

0

φn^3/2e⁻^φ

1/2

n hjyj,n− 1 φ^1/2_n

e⁻^φ

1/2

n hjd⁴y_j,n dx⁴

dx→0, j=1, . . . , N. (2.20) Performing integration by parts to the second term on the left-hand side of (2.20), we obtain

_j

0

φ_n^3/2e⁻^φ^1/2ⁿ ^h^jy_j,n− 1 φ^1/2n

e⁻^φ^1/2ⁿ ^h^jd⁴yj,n

dx⁴

dx

=

− 1 φ_n^1/2

e⁻^φⁿ^1/2⁽^j⁻^x)d³y_j,n dx³

_x=_j

x=0

−

−e⁻^φⁿ^1/2⁽^j⁻^x)d²y_j,n dx²

_x=_j

x=0

+

#

−φ^1/2_n e⁻^φ

1/2

n (j−x)dy_j,n dx

$x=j

x=0

−%

−φne⁻^φ

1/2

n (j−x)yj,n

&x=_j x=0 .

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Note thaty_j,n^k (0)is bounded fork≤3, sincey_j,n∈H⁴(0, j), thatφ_n→ ∞, and also that ^d³^y^j,n

dx³ (_j)→0 andβ_n^dy_dx^j,n(_j)→0. We thus get that _j

0

φ_n^3/2e⁻^φ^1/2ⁿ ^h^jyj,n− 1 φ_n^1/2

e⁻^φ^1/2ⁿ ^h^jd⁴y_j,n dx⁴

dx

=d²y_j,n

dx² (j)+φnyj,n(j)+Rj,n, j=1, . . . , N, (2.21) whereR_j,n→0, n→ +∞,∀j=1, . . . , N.Thus, according to (2.11), we simplify (2.21) to

d²y_j,n

dx² (_j)→0, j=1, . . . , N.

Consequently we have:

_j

0

dq_j

dx |βnyj,n|²dx+ _j

0

3dq_j dx

d²y_j,n dx²

2

dx→0, j=1, . . . , N. (2.22)

Finally, we chooseqj(x), j =1, . . . , N,so that ^dq_dx^j, j=1, . . . , N,is strictly nega- tive. This can be done by taking

q_j(x)=e⁻^x−1, j=1, . . . , N.

Therefore, (2.22) implies βnyj,nL²(0,j)→0,

""

"

d²y_j,n dx²

""

"

L²(0,_j)

→0, j=1, . . . , N.

In view of (2.8), we also get

v_j,nL²(0,_j)→0, j=1, . . . , N,

which clearly contradicts the normalization conditionZ_n_H_˜ =1.

3 Numerical illustration

Another way to study the stabililization problem is to adopt the strategy introduced in Ammari and Tucsnak [7] where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system. We give here some numerical illustration of this approach. A theoretical study remains to be done.

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3.1 Some background on a class of dynamical systems

LetH be a Hilbert space with the norm.H, and let A:D(A)→H be a self- adjoint, positive and boundedly invertible operator. We introduce the Hilbert space H1

2 =D(A¹²), with the normz¹

2 = A¹²zH. The spaceH₋1

2 is defined by duality with respect to the pivot spaceH.

Let the bounded linear operatorB:U→H₋1

2, whereU is another Hilbert space which will be identified with its dual.

The system we consider is described by

¨

w(t )+Aw(t )+By(t )=0, w(0)=w0, w(0)˙ =w1, t∈ [0,∞), (3.1)

y(t )=B^∗w(t ),˙ t∈ [0,∞). (3.2)

The system (3.1)–(3.2) is well-posed:

For(w0, w1)∈H1

2 ×H,the problem (3.1)–(3.2) admits a unique solution w∈C([0,∞);H1

2)∩C¹([0,∞);H )

such thatB^∗w(·)∈H_loc¹ (0,∞;U ). Moreover,wsatisfies the energy identity, for all t≥0

(w₀, w₁)²_H₁

2×H− (w(t ),w(t ))˙ ²_H₁

2×H=2 _t

0

""

""d

dtB^∗w(s)""

""²

U

ds. (3.3) For (3.3) we remark that the mappingt→ (w(t ),w(t ))˙ ²_H₁

2×H is non-increasing.

Consider the initial value problem:

¨

ϕ(t )+Aϕ(t )=0, (3.4)

ϕ(0)=w₀, ϕ(0)˙ =w₁. (3.5)

It is well known that (3.4)–(3.5) is well posed inH1×H1

2 and inH1 2 ×H. Now, we consider the unbounded linear operator

Ad:D(Ad)→ H1

2 ×H

t

, Ad=

0 I

−A −BB^∗

, (3.6)

where

D(Ad)= (u, v)^t∈ H1

2 ×H ^t, Au+BB^∗v∈H, v∈H1 2

! .

The result below, proved in [7], shows that, under a certain regularity assumption, the exponential stability of (3.1)–(3.2) is equivalent to a strong observability inequality for (3.4)–(3.5). More precisely, we have:

(13)

Theorem 3.1 (Ammari-Tucsnak [7]) Assume that for anyγ >0 we have sup

Reλ=γ

""

"λB^∗(λ²I+A)⁻¹B"""

L(U )<∞. (3.7)

Then, there exist constantsC, δ >0 such that for allt >0 and for all(w⁰, w¹)∈ H1

2 ×H, we have

(w(t ),w(t ))˙ H1

2×H≤Ce⁻^δt(w⁰, w¹)H1 2×H,

if and only if there exist constantsT , C >0 such that: for all(w₀, w₁)∈H₁×H1 2, we have

""B^∗ϕ(t )""

L²(0,T;U )≥C(w0, w1)H₁

2×H, (3.8)

whereϕ(t )is the solution of the system (3.4)–(3.5).

We will study the inequalities (3.7) and (3.8) numerically.

3.2 Transfert function LetA:V →V,

A=

⎛

⎝−_dx^d²2 0 0

d⁴

dx⁴ i=1,...,N

⎞

⎠,

D(A)=

⎧⎪

⎪⎨

⎪⎪

⎩y¯^t=(y, y1, . . . , yN)^t∈V^t

dy

dx ∈H¹(0, ),^d_dx²^y2ⁱ ∈H²(0, i),

dy

dx(0)=0,^dy_dxⁱ(_i)=0,^d_dx²^y2ⁱ(0)=0,

d³yi

dx³(_i)=0,_N

i=1 d³yi

dx³(0)+^dy_dx()=0

⎫⎪

⎪⎬

⎪⎪

⎭ andB:R^N⁺¹→V, Bk¯^t =ADk¯^t,∀¯k=(k, k1, . . . , k_N)∈R^N⁺¹, where V is the dual space ofV obtained by means of inner product inL²(0, )×_N

i=1L²(0, i), Dk¯^t= ¯W=(W, W₁, . . . , W_N)^t is the solution of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ d²W

dx² =0, (0, ), d⁴Wi

dx⁴ =0, (0, i), dWi

dx (_i)=0, d²Wi

dx² (0)=0, W ()=W_i(0),

dW

dx (0)=k, d³W

dx³(i)=ki, N

i=1

d³Wi

dx³ (0)+dW

dx ()=0.