Controllability of a system of parabolic equations with

IMA Journal of Mathematical Control and Information (2005) 22, 187–199 doi:10.1093/imamci/dni023

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic condition for the approximate controllability of the following system of parabolic equations with Dirichlet boundary condition: z t = D∆z + b1 (x)u 1 + · · · + bm (x)u m , t 0, z ∈ Rn , z = 0, on ∂Ω where Ω is a sufficiently smooth bounded domain in R N , bi ∈ L 2 (Ω ; Rn ), the control functions u i ∈ L 2 (0, t1 ; R); i = 1, 2, . . . , m and D is an n × n non-diagonal matrix whose eigenvalues are semi-simple with positive real part. This algebraic condition is checkable since it is given in terms of the nγ j × m matrices D P j and P j B, i.e.   ·· ·· 2 ·· nγ −1 Rank  P j B ·D P j B ·D P j B · · · · D j P j B  = nγ j , where P j Bu = P j b1 u 1 + · · · + P j bm u m . Finally, this result can be applied to those systems of partial differential equations that can be rewritten as a diffusion system (see de Oliveira, 1998). Keywords: parabolic equation; algebraic condition; approximate controllability.

1. Introduction Most of the results from Fattorini (1966, 1967), Russell (1978) and Triggiani (1976) are collected in Curtain & Pritchard (1978). In these references the following linear control system is considered in a separable Hilbert space Z : z = −Az +

m

bi u i (t),

t > 0,

(1.1)

i=1

where b1 , b2 , . . . , bm ∈ Z , and u i ∈ L 2 [0, t1 ] and A : D(A) ⊂ Z → Z is an unbounded linear operator with the following spectral decomposition: Az =

∞

λj

j=1

γj

z, φ j,k φ j,k =

k=1

∞

λ j E j x,

j=1

where ·, · is the inner product in Z and Ejz =

γj

z, φ j,k φ j,k .

k=1 c Institute of Mathematics and its Applications 2005; all rights reserved. For permissions please email: [email protected]

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H UGO L EIVA Universidad de los Andes, Facultad de Ciencias, Departamento de Matem´atica, M´erida 5101, Venezuela [Received on 20 June 2003]

188

H . LEIVA

The eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞ of A have finite multiplicity γ j equal to the dimension set of eigenvectors of A. So, {E j } of the corresponding eigenspace and {φ j,k } is a complete orthonormal is a complete family of orthogonal projections in Z and z = ∞ z ∈ Z. j=1 E j z, −A generates a strongly continuous semigroup {e−At } given by ∞

e−λ j t E j z.

j=1

In the above references, particularly in Curtain & Pritchard (1978), the following statement is proved: The system (1.1) is approximately controllable on [0, t1 ], t1 > 0 iff each of the following finitedimensional systems are controllable on [0, t1 ]: y = −λ j y + B j u(t),

y ∈ R(E j ),

(1.2)

where u ∈ L 2 (0, t1 ; Rm ) and B j is the following matrix: 

b1 , φ j,1  ·· Bj =   · b1 , φ j,γ j

b2 , φ j,1 · · · · · · ·· ·· · · b2 , φ j,γ j · · · · · ·

 bm , φ j,1  ·· .  · bm , φ j,γ j

From the classical finite-dimensional theory it is easy to see that the controllability of system (1.2) is equivalent to Rank(B j ) = dimR(E j ) = γ j .

(1.3)

Using the above ideas, in this note we give a necessary and sufficient algebraic condition for the approximate controllability of the following system of parabolic equations with Dirichlet boundary conditions: z t = D∆z + b1 (x)u 1 + · · · + bm (x)u m , t 0, z ∈ Rn , (1.4) z = 0, on ∂Ω where Ω is a sufficiently smooth bounded domain in R N , bi ∈ L 2 (Ω ; Rn ), the controls u i ∈ L 2 (0, t1 ; R); i = 1, 2, . . . , m and D is an n × n non-diagonal matrix whose eigenvalues are semi-simple with positive real part. One of the goal in this work is to prove the following statement: System (1.4) is approximately controllable on [0, t1 ], t1 > 0 iff each of the following finite-dimensional systems are controllable on [0, t1 ]: y = −λ j D P j y + P j b1 u 1 + · · · + P j bm u m , y ∈ R(P j ),

(1.5)

where j = 1, 2, . . . , ∞, λ j are the eigenvalues of −∆ with Dirichlet boundary condition and γ j the corresponding multiplicity, P j , s are the projections on the corresponding eigenspace and R(P j ) denotes the range of P j .


e−At z =

CONTROLLABILITY OF PARABOLIC

e

189

Since dimR(P j ) = nγ j < ∞, the controllability of (1.5) is equivalent to the following algebraic condition:   · · ·   · · 2 · nγ j −1 Rank  Pj B (1.6)  P j B ·D P j B ·D P j B · · · · D  = nγ j ,

The thermoelastic plate equation with Dirichlet boundary condition  2  wtt + ∆ w + α∆θ = a1 (x)u 1 + · · · + am (x)u m , θt − β∆θ − α∆wt = d1 (x)u 1 + · · · + dm (x)u m ,   θ = w = ∆w = 0,

t 0, t 0, t 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω .

(1.8)

Some notation for this work can be found in Curtain & Pritchard (1978), Curtain & Zwart (1995) and Leiva (1996, 1999, 2003). 2. Notation and preliminaries In this section we shall choose the space where this problem will be set and give the definition of approximate controllability. Also, we will present some results to be used in the next section. Let X = L 2 (Ω ) = L 2 (Ω , R) and consider the linear unbounded operator A : D(A) ⊂ X → X defined by Aφ = −∆φ, where D(A) = H 2 (Ω , R) ∩ H01 (Ω , R).

(2.1)

Then the eigenvalues λ j of A have finite multiplicity γ j equal to the dimension of the corresponding eigenspace and 0 < λ1 < λ2 < · · · < λn → ∞. Moreover (a) there exists a complete orthonormal set {φ j,k } of eigenvector of A;


where P j Bu = P j b1 u 1 + · · · + P j bm u m . Here, we will not make distinction between the operator D k P j B and its corresponding matrix representation. The case D = 1, n = 1 follows from the above references and has been studied in Sakawa (1974) where condition (1.6) takes the form (1.3). One of the points that makes this work different from other author’s works is that most of them assume that the diffusion matrix D is diagonal with positive entries. However, cross-diffusion phenomena are not uncommon: in de Oliveira (1998) one can find several mathematical models in which D is not diagonal, even more it is not diagonalizable. In that work many systems of partial differential equations can be rewritten as a system of reaction–diffusion of the form (1.4), like models of thermoelastic plate equation, damped vibration of a string, beam equation (see also Leiva, 2003) and a Lotka–Volterra system with diffusion (see Lopez & Pardo San Gil, 1992). So, our results can be applied to many of these models. In fact, in the same way one can study the approximate controllability of the following systems: The strongly damped wave equation with Dirichlet boundary condition wtt + η(−∆)1/2 wt + γ (−∆)w = d1 (x)u 1 + · · · + dm (x)u m , t 0, x ∈ Ω , (1.7) w(t, x) = 0, t 0, x ∈ ∂Ω .

190

H . LEIVA

(b) for all x ∈ D(A) we have Ax =

∞

λj

j=1

γj

x, φ j,k φ j,k =

∞

k=1

λ j E j x,

(2.2)

j=1

where ·, · is the inner product in X and γj

x, φ j,k φ j,k ,

(2.3)

k=1

so {E j } is a complete family of orthogonal projections in X and x= ∞ x ∈ X; j=1 E j x, (c) −A generates an analytic semigroup {e−At } given by e−At x =

∞

e−λ j t E j x.

(2.4)

j=1

Now, we consider Z = L 2 (Ω , Rn ) and define the following unbounded operator: A D : D(A D ) ⊂ Z → Z ,

A D ψ = −D∆ψ,

(2.5)

where D(A D ) = H 2 (Ω , Rn ) ∩ H01 (Ω , Rn ). With this notation system (1.4) can be written as follows: z = −A D z + Bu,

t > 0,

(2.6)

where B : Rm → Z is a linear bounded operator given by BU =

m

bi Ui ,

U = (U1 , U2 , . . . , Um ) ∈ Rm .

i=1

So, the control u ∈ L 2 (0, t1 ; Rm ). We shall use the following lemma from Leiva (2003) to prove the next theorem. L EMMA 2.1 Let Z be a separable Hilbert space and {An }n 1 , {Pn }n 1 two families of bounded linear operators in Z with {Pn }n 1 being a complete family of orthogonal projections such that An Pn = Pn An , n = 1, 2, 3, . . . .

(2.7)

Define the following family of linear operators: T (t)z =

∞ n=1

Then,

e An t Pn z,

t 0.

(2.8)


Ejx =


e

191

(a) T (t) is a linear bounded operator if

e An t g(t),

n = 1, 2, 3, . . .

(2.9)

for some continuous real-valued function g(t); (b) under the condition (2.9) {T (t)}t 0 is a C0 -semigroup in the Hilbert space Z whose infinitesimal generator A is given by ∞

z ∈ D(A)

An Pn z,

(2.10)

n=1

with D(A) = {z ∈ Z :

∞

An Pn z 2 < ∞};

(2.11)

n=1

(c) the spectrum σ (A) of A is given by σ (A) =

∞

σ ( A¯ n ),

(2.12)

n=1

where A¯ n = An Pn : R(Pn ) → R(Pn ). The proof of the following theorem follows from Lemma 2.1. T HEOREM 2.1 The operator −A D is the infinitesimal generator of an analytic semigroup {T (t)}t 0 given by T (t)z =

∞

e−λ j Dt P j z,

z ∈ Z,

t 0

(2.13)

j=1

where P j j 0 is a complete orthogonal projections in the Hilbert space Z given by P j = diag E j , E j , . . . , E j n×n , j 1. Proof. Let us compute A D z:   ∞     z1 ∆z 1 j=1 λ j E j z 1  ∞ λ j E j z2  z2  ∆z 2    j=1      AD  ·  = D  ·  = D  ·    ··   ··  ··   ∞ zn ∆z n λ E z j j n j=1     E j z1 Ej ∞ ∞   E j z 2  0     = D λj  λj D  ·  =  j=1  ···  j=1  ·· E j zn 0 ∞ = λ j D P j z. j=1

0 Ej ·· · 0

... ... ··

· ...

  0 z1 0  z2    · ··  ·   ··  zn Ej

(2.14)


Az =

192

H . LEIVA

Therefore, −A D z =

∞

A j P j z, A j = −λ j D.

(2.15)

j=1

σ (A j ) = {−λ j ρ1 , −λ j ρ2 , . . . , −λ j ρn },

(2.16)

where ρi are the eigenvalues of D. Since the eigenvalues of the matrix D are semi-simple with positive real part, there exists a complete family of complementary projections {qs }ns=1 on Rn such that e Dt =

n

eρs t qs .

s=1

Then,

e A j t Me−βt ,

t 0,

j = 1, 2, . . . ,

where β = min{Re(ρs ) : s = 1, 2, . . . , n}. Moreover, applying Lemma 2.1 we obtain that −A D generates a semigroup given by (2.13) and

T (t) Me−βt ,

t 0.

Moreover, this semigroup is analytic.

Now, we shall give the definition of approximate controllability in terms of system (2.6). To this end, for all z 0 ∈ Z and a control u ∈ L 2 (0, t1 ; Rm ) the equation (2.6) has a unique mild solution given by t z(t) = T (t)z 0 + T (t − s)Bu(s) ds, 0 t t1 . (2.17) 0

D EFINITION 2.1 We say that (2.6) is approximately controllable in [0, t1 ] if for all z 0 , z 1 ∈ Z and > 0, there exists a control u ∈ L 2 (0, t1 ; Rm ) such that the solution z(t) given by (2.17) satisfies

z(t1 ) − z 1 . The following theorem holds in general and can be found in Curtain & Pritchard (1978). T HEOREM 2.2 (2.6) is approximately controllable on [0, t1 ] iff B ∗ T ∗ (t)z = 0,

∀t ∈ [0, t1 ],

⇒ z = 0.

3. Main theorem Now, we are ready to formulate the main result of this work. Under the above conditions we will prove the following theorem.

(2.18)


It is clear that A j P j = P j A j . Now, we need to check condition (2.9) from Lemma 2.1. To this end, we have to compute the spectrum of the matrix A j . The spectrum σ (A j ) of A j is given by


e

193

T HEOREM 3.1 (2.6) is approximately controllable on [0, t1 ] iff the finite-dimensional systems are controllable on [0, t1 ]: y = −λ j D P j y + P j Bu, i.e.

y ∈ R(P j );

j = 1, 2, . . . , ∞,

(3.1)



j = 1, 2, . . . , ∞.

(3.2)

The following proposition can be proved in the same way as Lemma 1 from Leiva & Zambrano (1999). P ROPOSITION 3.1 The following statements are equivalent: (a) system (3.1) is controllable on [0, t1 ], ∗ (b) B ∗ P j∗ e−λ j D t y = 0, ∀t ∈ [0, t1 ], ⇒ y = 0,   · · · · · · (c) Rank  P j B ·D P j B ·D 2 P j B · · · · D nγ j −1 P j B  = nγ j . For the proof of Theorem 3.1 we will use the following lemma from Curtain & Pritchard (1978, p. 62). L EMMA 3.1 Let {α j } j 1 and {βi, j : i = 1, 2, . . . , m} j 1 be two sequences of complex numbers such that: α1 > α2 > α3 · · · . Then ∞ eα j t βi, j = 0, ∀t ∈ [0, t1 ], i = 1, 2, . . . , m j=1

iff βi, j = 0,

i = 1, 2, . . . , m; j = 1, 2, . . . , ∞.

Proof of Theorem 3.1. Suppose that each system (3.1) is controllable in [0, t1 ]. Next, we compute B ∗ T ∗ (t). B ∗ : Z → Rm , B ∗ z = (b1 , z, . . . , bm , z), and T ∗ (t)z =

∞

∗

e−λ j D t P j∗ z,

z ∈ Z,

t 0.

j=1

Therefore,

B ∗ T ∗ (t)z = (b1 , T ∗ (t)z, . . . , bm , T ∗ (t)z).

Hence, system (2.6) is approximately controllable on [0, t1 ] iff bi , T ∗ (t)z = 0,

∀t ∈ [0, t1 ],

i = 1, 2, . . . , m,

⇒ z = 0.

Now, we shall check condition (3.3): bi , T ∗ (t)z =

∞ j=1

∗

bi , e−λ j D t P j∗ z = 0,

i = 1, 2, . . . , m;

t ∈ [0, t1 ].

(3.3)


 · · ·   · · 2 · nγ j −1 Rank  Pj B  P j B ·D P j B ·D P j B · · · · D  = nγ j ,

194

H . LEIVA

Since the eigenvalues of the matrix D are semi-simple with positive real part, there exists a complete family of complementary projections {qs }ns=1 on Rn such that eD

∗t

=

n

eρs t qs∗ .

s=1

j=1

=

j=1

n ∞

s=1

e−λ j ρs t bi , Ps,∗ j z = 0 i = 1, 2, . . . , m;

t ∈ [0, t1 ],

j=1 s=1

where Ps, j = qs P j = P j qs . Applying Lemma 3.1, we conclude that bi , Ps,∗ j = 0 i = 1, 2, . . . , m; Then, iff

∗

j = 1, 2, . . . , ∞,

bi , e−λ j D t P j∗ z = 0 i = 1, 2, . . . , m; ∗

B ∗ e−λ j D t P j∗ z = 0;

t ∈ [0, t1 ].

j = 1, 2, . . . , ∞,

j = 1, 2, . . . , ∞,

t ∈ [0, t1 ],

t ∈ [0, t1 ].

Since P j∗ D ∗ = D ∗ P j∗ and (P j∗ )2 = P j∗ , we get that ∗

(P j B)∗ e−λ j D t P j∗ z = 0;

j = 1, 2, . . . , ∞,

t ∈ [0, t1 ].

From the controllability of system (3.1), we get that P j∗ z = 0, j = 1, 2, . . . , ∞. Since {P j∗ } j 1 is a complete family of orthogonal projections on Z , we conclude that z = 0. Conversely, assume that system (2.6) is approximately controllable on [0, t1 ] and there exists J such that the system y = −λ J PJ Dy + PJ Bu, y ∈ R(PJ ), is not controllable on [0, t1 ]. Then, there exists V J ∈ R(PJ ) such that ∗

(PJ B)∗ e−λ J D t V J = 0,

t ∈ [0, t1 ]

and

V J = 0.

Letting z = PJ∗ V J , we obtain B ∗ T ∗ (t)z = (b1 , T ∗ (t)z, . . . , bm , T ∗ (t)z) ∗

∗

= (b1 , e−λ J D t V J , . . . , bm , e−λ J D t V J ) ∗ ∗ = B ∗ e−λ J D t V J = (PJ B)∗ e−λ J D t V J = 0, which contradicts the assumption.


Without loss of generality, we can assume that the eigenvalues of D are real and ρ1 ρ2 · · · ρn . Therefore, ∞ ∞ n ∗ bi , T ∗ (t)z = bi , bi , e−λ j D t P j∗ z = e−λ j ρs t Ps,∗ j z


e

195

In order to apply the algebraic condition (3.2) we shall compute the matrix representation of each operator Dr P j B. To this end we shall prove the following lemma. Dr P j B : Rm → R(P j ) is given by n  ······ s=1 d1s bs,m , φ j,1  ·· ··   · · n  · · · · · · s=1 d1s bs,m , φ j,γ j   n  ······ s=1 d2s bs,m , φ j,1   ·· ··  · ·  n  ······ s=1 d2s bs,m , φ j,γ j    ·· ··  · ·    ·· ··  · ·  n  ······ d b , φ ns s,m j,1  s=1  ·· ··   · · n ······ s=1 dns bs,m , φ j,γ j nγ

where j = 1, 2, . . . , ∞, r = 0, 1, 2, . . . , nγ j − 1 and  b1,i b2,i    bi =  ·  ,  ··  bn,i 

i = 1, 2, . . . , m,

Dr = (dl,s (r ))n×n .

Proof. Consider the operator Dr P j B : Rm → R(P j ), and the canonical basis B = {e1 , e2 , . . . , em , } in Rm and the following basis in R(P j ): B j = {φ 1jl , where

φ 2jl , . . . φ njl : l = 1, 2, . . . , γ j },    0 0 φ jl  0     0 0 φ 2jl =   , . . . . . . , φ njl =   .  ·   ·   ··   ··  0 φ jl

 φ jl 0   0 1 φ jl =   ,  ·   ··  0 



On the other hand, for all x ∈ X we have Ejx =

γj k=1

x, φ j,k φ j,k .

j ×m


L EMMA 3.2 The matrix representation of the operator n  n s=1 d1s bs,1 , φ j,1 s=1 d1s bs,2 , φ j,1  ·· ··   · · n n  s=1 d1s bs,1 , φ j,γ j d b 1s s,2 , φ j,γ j s=1 n  n  s=1 d2s bs,1 , φ j,1 d b 2s s,2 , φ j,1 s=1   · ·  ·· ··  n  n  s=1 d2s bs,1 , φ j,γ j s=1 d2s bs,2 , φ j,γ j   ·· ··  · ·    ·· ··  · ·  n  n d b , φ d b ns s,2 , φ j,1  s=1 ns s,1 j,1 s=1  ·· ··   · · n n s=1 dns bs,1 , φ j,γ j s=1 dns bs,2 , φ j,γ j

196

H . LEIVA

This yields  γ j



k=1 b1i , φ j,k φ j,k  γ j   b , φ j,k φ j,k   r  k=1 2i



d11 (r )

 ··   · γ j k=1 bni , φ j,k φ j,k

γ j

k=1 b1i , φ j,k φ j,k   d21 (r ) γ j b1i , φ j,k φ j,k  k=1

=  

+ · · · + d1n (r )

γ j



k=1 bni , φ j,k φ j,k  γ j + · · · + d2n (r ) k=1 bni , φ j,k φ j,k  

 ··   · γ j + · · · + dnn (r ) k=1 bni , φ j,k φ j,k

γ j dn1 (r ) k=1 b1i , φ j,k φ j,k γj n = d1s (r )bsi , φ j,k φ 1j,k k=1

+

s=1

γj n k=1

d2s (r )bsi , φ j,k φ 2j,k

s=1

+ ........................ + ........................ γj n + dns (r )bsi , φ j,k φ nj,k . k=1

s=1

From here the proof can be completed.

C OROLLARY 3.1 If D = d I with d > 0 and I the identity matrix, then system (2.6) is approximately controllable on [0, t1 ] iff

(P j B)∗ y = 0,

∀t ∈ [0, t1 ],

j = 1, 2, . . . , ∞,

⇒ y = 0,

iff

Rank(P j B) = nγ j ,

j = 1, 2, . . . , ∞,

(3.4)


Dr P j Bei = Dr P j bi = D   


where



197

 b1,m , φ j,1  ··   ·  b1,m , φ j,γ j   b2,m , φ j,1    ··  ·   b2,m , φ j,γ j    ··  ·    ··  ·  bn,m , φ j,1    ··   · bn,m , φ j,γ j nγ

b1,2 , φ j,1 · · · · · · ·· ·· · · b1,2 , φ j,γ j · · · · · · b2,2 , φ j,1 · · · · · · ·· ·· · · b2,2 , φ j,γ j · · · · · · ·· ·· · · ·· ·· · · bn,2 , φ j,1 · · · · · · ·· ·· · · bn,2 , φ j,γ j · · · · · ·

.

(3.5)

j ×m

C OROLLARY 3.2 If D = [d]1×1 with d > 0, then system (2.6) is approximately controllable iff   b1 , φ j,1 b2 , φ j,1 · · · · · · bm , φ j,1   ·· ·· ·· ··  = γj, Rank  (3.6)   · · · · b1 , φ j,γ j b2 , φ j,γ j · · · · · · bm , φ j,γ j where j = 1, 2, . . . , ∞. R EMARK 3.1 From Lemma 3.2 we can see that the number of controls required for the approximate controllability of (2.6) must be at least that of the highest multiplicity of the eigenvalues multiplied by the number of the equations n: i.e. m nγ j , j = 1, 2, . . . , ∞. E XAMPLE 3.1 Consider the following controlled system of two parabolic equation with Dirichlet boundary conditions:  vt = d11 vx x + d12 wx x + b11 (x)u 1 + b12 (x)u 2    wt = d21 vx x + d22 wx x + b21 (x)u 1 + b22 (x)u 2 (3.7) t 0, 0 x 1    v(t, 1) = w(t, 0) = w(t, 1) = v(t, 0) = 0.

In this case D=

d11 d21

d12 , d22

b11 , b21

b1 =

b12 , b22

b2 =

u=

u1 , u2

and γ j = 1,

λ j = j 2π 2

and

φ j (x) = sin jπ x.

Therefore, the system (3.7) is approximately controllable iff b1,1 , φ j b1,2 , φ j d11 b1,1 , φ j + d12 b2,1 , φ j Rank b2,1 , φ j b2,2 , φ j d21 b1,1 , φ j + d22 b2,1 , φ j d11 b1,2 , φ j + d12 b2,2 , φ j = 2. d21 b1,2 , φ j + d22 b2,2 , φ j

(3.8)


b1,1 , φ j,1  ··   ·  b1,1 , φ j,γ j   b2,1 , φ j,1   ··  ·   b2,1 , φ j,γ j Pj B =   ··  ·    ··  ·   b , φ  n,1 j,1  ··   · bn,1 , φ j,γ j

e

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H . LEIVA

There are several possibilities for condition (3.8) to be satisfied in this case. For example, ! ! !b1,1 , φ j b1,2 , φ j ! ! ! (a) !b2,1 , φ j b2,2 , φ j ! = b1,1 , φ j b2,2 , φ j − b1,2 , φ j b2,1 , φ j = 0. i.e. 1

b11 (x) sin jπ x dx

0

−

1

1


0 1


0


= 0,

j = 1, 2, . . . , ∞.

0

(b)

! !b1,1 , φ j ! !b2,1 , φ j

! d11 b1,1 , φ j + d12 b2,1 , φ j !! = 0. d21 b1,1 , φ j + d22 b2,1 , φ j !

E XAMPLE 3.2 As a special case we can consider the scalar parabolic equation with a single control zt = z x x + b(x)u t 0, 0 x 1 (3.9) z(t, 1) = z(t, 0) = 0. In this case λ j = − j 2 π 2 and φ j (x) = sin jπ x. Therefore, (3.9) is approximately controllable iff

1

b, φ j =

b(x) sin jπ x dx = 0,

j = 1, 2, . . . , ∞.

0

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