Riesz basis property of a second-order hyperbolic system with ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
Riesz Basis Property of a Second-Order Hyperbolic System With Collocated Scalar Input–Output Bao-Zhu Guo and Yue-Hu Luo
Abstract—The closed-loop form of a second-order hyperbolic system with scalar collocated sensor/actuator is considered. The Riesz basis property of the system is verified for diagonal semigroups based on an abstract result of one rank perturbation of discrete-type operators in Hilbert spaces. A simplified proof for the abstract result is presented. Index Terms—Collocated input–output, perturbation of linear operators, Riesz basis, second-order system.
I. INTRODUCTION In the case of infinite-dimensional systems of the form x_ (t) = Ax(t), where A is the generator of a C0 -semigroup in a Hilbert space H , it is well-known that stability of the spectrum of A (i.e., real parts
x(t). However, if for instance, the eigenfunctions f n g1 1 which associate with eigenvalues fn g11 form a Riesz basis (a basis which is equivalent to an orthonormal basis in H , see, e.g., [19]), then the
are negative) does not imply any stability properties in the solutions
solution can be represented as
x(t) =
1 n=1
an e t n ;
8 x(0) =
1 n=1
where the control u(t) is a scalar function and k a constant. A : D(A)( X ) ! X is an unbounded positive selfadjoint operator in X . b is an element of the dual space [D(A1=2 )]0 of [D(A1=2 )], the graph space of D(A1=2 ). System (1) is a special case of the following generic infinite-dimensional system with one rank perturbation in a Hilbert space H :
x_ = Ax(t) + f (x(t))b
A
(2)
where is a discrete type operator (see the definition in Appendix) and an element of the dual space [D( 3 )]0 , where 3 denotes the adjoint of . is bounded if 2 H , otherwise, it is unbounded. f is a functional in H , which is usually -bounded. Two basic problems are discussed mostly in the literature: For fixed f or , find conditions so that a) the system (2) is a Riesz spectral system ([18]), that is, there is a sequence of generalized eigenvectors of system (2), which forms a Riesz basis for H ; b) the system (2) has specified eigenvalues. The second problem is usually referred to as pole assignment problem. Its significance is guaranteed by the first one because of the spectrum-determined growth condition which may be not valid for generic infinite dimensional systems. The first paper concerning these questions is contributed to the works of [15], where the problem is completely resolved for the case where both f and are bounded. In the past two decades, much effort has been concentrated along the same direction to the solution of these problems under variant conditions for example [6], [8] and [18], name just a few. Usually, these works are based on the basic assumption that either f or is bounded. In [11] and [12], although both f and are assumed to be unbounded, is assumed to be the infinitesimal generator of an analytic semigroup. This limits its applications to such systems as beam equation without consideration of viscous damping. In [9], the author developed a general result for the case where neither f nor is required to be bounded without the analyticity assumption the semigroup generated by . In this note, we shall apply this abstract result to answer the question a) for a special case of the system (1), namely, the diagonal semigroup case. It is shown that under some spectrum assumption of , the system (1) is a Riesz spectral system and hence the spectrum-determined growth condition holds true. The result can be applied directly to a connected Euler-Bernoulli beam equation under joint feedback control. However, it avoids the estimation for the eigenfunctions for the beam equation as was done in [4] and, more importantly, it can give a complete answer for some cases that appear to be impossible by the approach of [4]. The remaining of this note is organized as follows. In Section II, we give the Riesz basis property for the system (1). In the Appendix, we give an outline of a simplified proof of the abstract result, which was first obtained in [9].
b
Ab
b
A
A
A
b
b
an n 2 H:
In this case not only stability of the system is determined by the spectrum of the system operator, which is usually referred as spectrum-determined growth condition, but also the dynamic behavior of the system can be described by eigenpairs f(n ; n )g. Riesz basis is also the basis of so called method of moment, a powerful method in the study of controllability of hyperbolic systems; we refer the readers for earlier works on moment method to [13] and a recent systematic summary to [2]. General Riesz spectral systems are discussed in [14]. A nice recent result on the relation of exact controllability and Riesz basis can be found in [7]. The general Riesz basis theory is developed in the context of nonharmonic Fourier series, which is originated from works of Paley and Wiener in the 1930s and developed extensively later by many mathematicians. Earlier results–are summarized in [19], and a nice recent summary can be found in [2]. Unfortunately, verification of the Riesz basis generation in practice is challenging even for some extensively studied systems. We refer to [10], [20], [3], and [4] for the study of this problem for string and beam equations. Most of these particular examples can be put into the following form of controlled second order hyperbolic system with scalar collocated input–output in a Hilbert space X
ytt + Ay + bu(t) = 0; u(t) = kb3 yt
693
b
A
b
b
A
A
(1)
II. RIESZ BASIS GENERATION Rewrite (1) as
Manuscript received April 19, 2001; revised October 18, 2001. Recommended by Associate Editor M. Reyhanoglu. This work was supported by the National Natural Science Foundation of China. B.-Z. Guo is with the Institute of Systems Science, Academy of Mathematics and System Sciences, Beijing 100080, P.R. China (e-mail: [email protected]). Y.-H. Luo, deceased , was with the Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P.R. China. Publisher Item Identifier S 0018-9286(02)03754-6.
ytt + Ay + kbb3 yt = 0
(3)
where A : D(A)( X ) ! X is an unbounded positive selfadjoint operator in a Hilbert space X . b 2 [D(A1=2 )]0 . b3 2 L([D(A1=2 )]; C I) is defined by
b3 x = hx; bi[D(A
0018-9286/02$17.00 © 2002 IEEE
)]2[D(A
)]
; 8x 2 D A1=2 :
(4)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
We assume that A is diagonal, that is, there is an orthonormal basis f n g1 for X such that
1
A n
= !n2 n !n > 0:
To explain (1) in the framework of general well-posed system theory, we usually write (1) as d dt
(5)
=A
By these assumptions, we can write b
1
=
=1
n
2bn n;
1 jbnj2 < 1: 2 !n
=1
n
3
= 2bn :
b n
Define an extension A~ 2 L
~
Ax; z
~ =
~
D
of A by
[D(A )] 2[D(A )] = A1=2 x; A1=2 z X 2X ; 8x; z 2 D
That is Ax
D A1=2
(7)
0 A1=2
1
2
=1
an !n n for any x
n
=
1 =1
an n
n
1=2
A
:
1=2
2
D A
(8)
:
(9)
)] to [D(A1=2 )]0 by Lax-Milgram theorem. It is well-known that for any 2 (A); ( 0 A~) is an isomorphism from [D(A1=2 )] to [D(A1=2 )]0 . ~ + kbb3 yt = 0; in + Ay
1=2 0
D A
y yt
= k (0; b3 )
y yt
0
= 0A~ 0 kbb3
I
y yt
2
f g
I = 00A~ 0kbb 3 g = 0Af ~ 0 kbb3 g
1=2 0 : (11)
=
f g
(12)
jbn j c > 0 f or some c > 0 and all n 1: We are now in a position to state our main result of this note. Theorem 3: In addition to (15), assume that
1 1
2 =1 !n
1:
0.
0 !n M !n +1 ; n = 1; 2; . . .
(15)
:
(18)
For the definition of [D]-class operators we refer to Definition A1 in the Appendix. The proof of Theorem 3 will be split into several lemmas. The following lemma is straightforward. Lemma 1: Under the condition (15)
M 1+ !1 (1 + )01 n1+ ; n > 1:
(19)
Therefore, A01 is compact. Let and be defined by (16). It is seen that
A
Therefore, (3) is actually being considered as d dt
(17)
[D]-class. Moreover, there exists a constant C > 0 such that the eigenpairs f(n ; n )g [ ftheir conjugateg of A satisfy
!n+1
with D
A
Ab
D A
However, we want to consider (11) in the energy state space H [D(A1=2 )] 2 X . To this purpose, we define
A
= kb3 yt :
If b is admissible, then the operator defined by (12)–(13) is of
;
in D A1=2
y yt
b A
n
d dt
()
u t
(16)
b
(10)
which can be considered as the following first-order equation:
0 0b
y yt
It is seen that b 2 [D(A1=2 )]0 if and only if 2 [D( )]0 . We say b is admissible if is admissible with respect to ([16]). Admissibility is a basic assumption for infinite dimensional control system (Ac ; Bc ) to be well-posed ([16]). When Bc is admissible, even if the control space is one dimensional, (Ac ; Bc ) can be exactly controllable ([7]) and the output feedback can make system be exponentially stable ([4]). This is contrast to the bounded control (see [14, Th. 4.1.5]). Theorem 2: Suppose (15) holds. Then i) b is admissible if and only if bn defined by (7) is uniformly bounded; ii) when b is admissible, system (16) is a regular linear system ([17]); iii) system (14) is exponentially stable if and only if ( ; ) is exactly controllable or equivalently
[ (
ytt
t
O t
A is an isometric from D A1=2
With these preparations, we may write (3) further as
= 00A I0 y 0 bu(t) y
( ) = kb3
(6)
It is seen by (4) and (6) that
y yt
b A86n = 6i!n86n = 6n86n; n 1:
f8ng forms an orthonormal basis for H 01 8n = 0i!n n 80n =
01 n i!n n
n
and
b=
1 =1
n
8 +
bn n
1 =1
n
bn
80n :
Let H01 be the completion of H under the norm
kF k01 = A01 F
H
8 2 H:
; F
;n
(20)
1
(21)
(22)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
Then H01
=
=
F
01;n6=0
1
A~ )b 2 D(f ) and
~ (; hence R
1 n=
695
8j
~
an n
jan j
2
0 and all n 1 (it follows from Lemma 1 that (17) is satisfied for any 1=2). Then g is a bounded functional on H . Proof: It suffices to show that
(29) )]
2D(A
)
an bjnj
1
In particular
and f is -bounded: jf (F )j kkbk[D(A lation, we can now write
)
=0
=
where
(8n ) = 02kbn
2D(A
)
kRn bk < 1 1 g (F ) = an bjnj kRn bk ; n 01;n6 an 8n 2 D(g ) 8F = 1 n 01;n6
(26)
Define the functional f in H such that
f
~
bjnj f R ;
n=
D(A
= 1 n=01;n6=0 an 8n 2 H j
F
1
1
+ !n2 ; b
proving (31). In order to apply Theorem A in Appendix, we need the following functional g :
an n
n=
=
For any
n=
for any F
2
= 0 k 22+bn!2 h n ; biD(A n n=1 1 2b = 2 +n!2 f (8n ) n n=1
Let
1
n=1
1
(23)
A~ = 00A~ I0 : ~ is an extension of A and D(A~ ) = H . Then A ~ (; A~ ) is an isomorphism from H0 to H . 2 (A); R 1 an ~ ; A~ F = 8n ; R 0 n
1 2b e n n
0 !k
!k
1 jj
k= n +2
!
0 !k0
1
1
0 !k
2
dx
!
1
!jnj
! !
!jnj
0x
!jnj
1
2
0x
2
dx
:
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where
I2
1
To show the result, it suffices to show that
1
!
M!j j =jnj+2 n k
1
dx
!jnj 0 x 2
!
= M!1 ! 10 ! M 21!2 jnj jnj jnj+1 jnj while as !jnj01 > 2!1 1 jnj02
I1 M
=1
M
1
M
k
=1 !
!
1 M!
!
1
dx x !jnj 0 x 2 =2 1 dx x ! 0x 2
1
1 !jnj 0
1 M!
+
1
2 dx =2 x !jnj 0 x
1
4
!
4
1
+ M!
2
jnj01 !jnj 0 !jnj01
!
jnj01
2 1 + 4 : = M! 1 !jnj M 2 !j2nj01
1
C for some C > 0
f (8k ) bjkj jmjn jkj=n+1;k6=m jm 0 k j
1
jkj=n+1;k6=m
= 0:
2C
1
1
+ f 820!m bjmj jmj
!jmj 0 !k k=n+1;jkj6=jmj
1
+ 2!1
jmj
+ 2!1 ! 0 ! jmj k jmj k=n+1;jk0jmjj2 1 1 + + !jmj+1 0 !jmj
1
1
!jmj 0 !jmj01
2C + 2!1 ! 0 ! jmj j m j j k j k=n+1;jk0jmjj2 + 2 : M!jmj
1
!
! ! !
!
dx
! !
x0 dx x 0 !jmj
x0 dx + 1 !jmj 0 x M x0 dx !jmj 0 x
1
!
dx x x + !jmj
0!
x0 dx x 0 !jmj
(37)
By Lebesgue’s dominated convergence theorem
1
1
dx ! 0! x x + !jmj 1 dx M1 ! 0; as jmj ! 1: M! x x + !jmj
S2 = M
!n+1 and jmj n ! 1
1
f (8k ) bjkj jkj=n+1;jkj6=jmj !jmj 0 !jkj
1
=jmj+2 M
dx x 0 !jmj
x !jmj 0 x
!
=S1 + S2 :
S1 = M
1
!
!
When !jmj01=2
n
f (8k ) bjkj jm 0 k j
2C
k
1 + M1
4 1 +2 4 kRnbk2 C !12 + M 26!2 + M! ; M 2 !j2nj01 1 !jnj jnj jnj (35) !jnj01 >2!1 :
Proof: For any integer n, as jmj
1
M
lim sup n!1
1
=1
The proof is complete by the assumption (17). Lemma 4: Suppose (15). Assume that jkbn j2 and all n 1. Then
1
M k=n+1
M1
Therefore
1
!jmj 0 x
!
!k 0 !k01 k=jmj+2 jmj02 !
+
+ M 2 !2
1 !jnj 0 2
1
= +1 !k+1 0 !k
k n
!
(36)
1 1 + ! 0 ! ! k k 0 !jmj j m j k=n+1 jmj+2 jmj02 ! 1 dx
=
dx x !jnj 0 x 2
jnj
! 0 (jmj n ! 1) :
1
! 0 !k k=n+1;jk0jmjj2 jmj jmj02
!
+ M1
1
!
!
Now
1
!
1
!jmj 0 !jkj k=n+1;jk0jmjj2
!
dx
x !jmj 0 x 2 x0 dx 1 ! + M ! !jmj 0 x M ! M!!jmj02!1 0 20!n!+1 jmj jmj01 n+1 ! dx + M!2 =2 !jmj 0 x jmj01 !
=1
! !
x0 dx =2 !jmj 0 x
=
M!1 + M!2
log 1 + ! !0jm!j01 jmj jmj01 jmj01 10 ! 2 log 1 + jmj01 ! 0
+1
n
M!1 + M! +1
n
jmj01
while as !n+1 > !jmj01 =2 and jmj
M
n!1
x0 dx 1 !jmj 0 x M ! !10 M!2 log 1 + jmMj01 jmj01
1
S1 = M
!
! !
! 0:
x0
2 !jmj 0 x
=
dx
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
697
[ ]
Therefore
S1 + S2 ! 0 as jmj n ! 1: (36) is thus proved and Lemma 4 follows. Proof of Theorem 3: By Lemmas 2–4, f and b are compatible with respect to A according to definition A2 in Appendix. Hence, all conditions of Theorem A in Appendix are satisfied. So A is of D -class. Equation (18) follows from (29), (35) and Theorem A1 in Appendix.
[ ]
APPENDIX For the sake of completeness, we will give here an outline of a simplified proof of perturbation result which we cite for the proof of Theorem 3. The result stated below is a special case of [9, Th. 2.6]. Let us first introduce a definition of D -class operators. 1) Definition A1: A linear operator A in a Hilbert space H is called discrete type, or D -class in short, if there are Riesz basis f n g1 of H , a sequence of complex numbers fn g1 and a nonnegative integer N such that i) n!1 jn j 1, n 6 m , as n > N , m , n 6 m; ii) A n n n , n > N ; iii) Aspanf 1 ; 2 ; ; N g spanf 1 ; 2 ; ; N g and A has spectrum fi g1N in spanf 1 ; 2 ; ; N g, where spanf 1 ; 2 ; ; N g denotes the linear subspace spanned by all f i g1N . It is well-known that A is a discrete operator in H (that is, its resolvent is compact) when it is of D -class. And for the D -class operator A, the semigroup eAt generated by A satisfies the spectrum-determined growth condition. Let A be a discrete operator in H . We denote by H01 the completion of H with respect to the norm
[ ]
[ ]
1
lim
= ... ...
=
1
=
1 = ... ...
[ ]
b=
(38)
According to [16], H01 is equivalent to D A3 0 , the dual space of the graph space D A3 , where A3 is the adjoint operator of A. For any 2 A , R ; A 0 A 01 has a natural extension R ; A from H01 to H : R ; A x R ; A x for x 2 H . Certainly A can be extended to whole H by
[ ( )]
~ yi = hx; A3 yi; for any x 2 H; y 2 D (A3 ) : hAx;
~(
For any functional f define Af;b by
~ + f (x)b 2 H ; D (Af;b ) = x 2 H Ax ~ Af;b x = Ax + f (x)b:
(41)
We say f is A-bounded, if jf (x)j M kAxk for some M > 0 and all x 2 D(A). When b 2 H , Af;b = A + f (1)b, which has been studied
extensively in [8] and [18]. 2) Lemma A1: Let b 2 H01 . Suppose that f is A-bounded and there exists an 2 A (and hence for all 2 A ) such that R ; A b 2 D f . Then Af;b is a discrete operator in H . Moreover, 2 Af;b for any satisfying 0 f R ; A b 6 and
( )
( ) ~( ) () ( ) 1 ( ~( ) ) = 0 f (R~ (; A)1) ~ R(; Af;b ) = R(; A) + 1 0 f (R~ (; A)b) R(; A)b:
(42)
(43)
( )
x=
1 ak k 2 H01 k=1
1
j j2 jn 0 k j02 < 1
(44)
(
:
~
)
It is obvious that Rn 2 L H01 ; H . Moreover, the restriction Rn of Rn on H is a bounded operator on H with the norm
kR~n k M ; 8n > N n
(45)
where
n =
inf 6= jn 0 m j ; n 1:
(46)
Next define a linear functional gf;b in H by
1 an n 2 H n>N
D (gf;b ) = x =
(x) =
g
1 jan f ( n )jkRn bk < 1 ; n=1 1 an f ( n ) kRn bk ;
(47)
f;b n>N 8x = 1 n=1 an n 2 D (gf;b ) :
3) Definition A2: Suppose f is an A-bounded functional, b 2 H01 is given by (43). f and b are called compatible with respect to A if i) there exists an 2 A such that R ; A b 2 D f and
( )
f (R~ ( ; A)b) =
1
n=1
~(
)
()
bn f (R~ ( ; A) n );
(48)
ii) the functional gf;b defined by (47) is bounded on H ; iii)
(39)
(1) in H; D(f ) D(A) and b 2 H01 , we can
jbnj2 < 1: 2 n=1 j 0 0 n j
k=1;k6=n ak
)
(40)
n=1
1
bn n in H01 with
= = D (Rn ) =
lim sup n!1 mn
A~ is an isomorphism from H to H01 . By [16, Prop. 3.3], for any L 2 L(H ) which is permutable with A, there is an extension L~ on H01 :
~ = R ( 0 ; A)01 LR~ ( 0 ; A) : L
1
For any positive integer n > N , define Rn as 1 a Rn x k=1;k6=n 0 k ; 1 8x k=1 ak k 2 D Rn ;
[ ]
k 1 k01 = k ( 0 0 A)01 1 k for some 0 2 (A):
[ ( )] ( ) ( )=( ) ~( ) = ( )
In the sequel, we always suppose that A is of D -class satisfying the conditions of Definition A1 and the notations are kept throughout the discussions. For any given b 2 H01 , we write
1
jf ( k ) bk j = 0: j 0 k j k=n+1;k6=m m
(49)
For the given f and b as previously shown, denote
1n = 2 CI 0 n rn ; when f ( n ) bn = 0 (50) rn = 3 2 jf ( n ) bnj ; while f ( n ) bn 6= 0. 4) Lemma A2: Let 1n be given as (50). If f and b are compatible with respect to A, then there exists an integer K > N such that Af;b possesses an eigenvalue n in 1n for each n K and jn 0 nj 2 jf ( n ) bnj 13 n (51) and
1 10f 2 0
R (n ; AN )
1
N k=1
f ( k ) bk 0 k k=N +1;k6=n n
k bk
32 for all n K
(52)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
0 ;
b
f ( n )
n =
1
0 f R (n; AN ) Nk bk k f b 0 ; 0 1 k N ;k6 n 0
when f ( n ) bn = 6 0, (55)
=1
=
+1
=
(
)
1
otherwise
and
yn = R (n ; AN )
N k=1
bk k +
AN is the restriction of A on the subspace spanf 1 ; 2 ; . . . ; N g. 5) Theorem A: Suppose that f is an A-bounded functional and b 2 H01 . If f and b are compatible with respect to A, then Af;b is of [D]-class. Moreover, the eigenvectors f n g of Af;b and that of A where
satisfy
k n 0 n k C jf ( n )j jn 0 j0
1
0
+
kRn bk
(53)
for some C > and all sufficiently large n, where Rn is defined by (44). Proof: Suppose that K is the constant in Lemma A2. In the proof of Lemma A2, the eigenvectors n of Af;b corresponding to n can be chosen as n = n + n yn
(54)
where (55), shown at the top of the page, holds. By (51) and (52)
j n j 2 jf ( n )j
n K:
(56)
Next, it can be shown that
kyn k C jn 0 j0 0
1
+
kRn bk
;
n K:
(57)
Finally, since f is A-bounded and b are compatible with respect to A, it follows from (54), (56), and (57) that
1 n=K
k n 0 n k 2C 2
C
1
1 n=K
jf ( n )j jn 0 j
2
0
2
f R~ ( 0 ; A)
+ 2
jf ( n )j kRn bk
+
2
kgf;bk
2
2
0. By [3, Th. 6.3], Af;b is of [D]-class. The proof is complete. ACKNOWLEDGMENT The first author would like to express his sorrow in memory of his coauthor Y.-H. Luo, who passed away suddenly while this note was under review. REFERENCES [1] K. Ammari and M. Tucsnak, “Stabilization of second order evolution equations by a class of unbounded feedbacks,” ESAIM Control, Optim. Calc.Var., vol. 6, pp. 361–386, , 2001. [2] S. A. Avdonin and S. A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge, U.K.: Cambridge Univ. Press, 1995. [3] B. Z. Guo, “Riesz basis approach to the stabilization of a flexible beam with a tip mass,” SIAM J.Control Optim., vol. 39, pp. 1736–1747, 2001.
1
bk k 0 k : n k=N +1;k6=n
[4] B. Z. Guo and K. Y. Chan, “Riesz basis generation, eigenvalues distribution and exponential stability for an Euler-Bernoulli beam with joint feedback control,” Revista Matemática Complutense, vol. 14, pp. 205–386, 2001. [5] B. Z. Guo and Y. H. Luo, “Controllability and stability of a second order hyperbolic system with collocated sensor/actuator,” Syst. Control Lett.. [6] L. F. Ho, “Spectral assignability of systems with scalar control and application to a degenerate hyperbolic system,” SIAM J. Control Optim., vol. 24, pp. 1212–1231, 1985. [7] B. Jacob and H. Zwart, “Exact controllability of C -groups with onedimensional input operators,” in Advances in Mathematical Systems Theory, F. Colonius et al., Eds. Boston, MA: Birkhäuser, 2000, pp. 221–242. [8] Y. H. Luo, “On the perturbation of (D )-class operators” (in Chinese), Syst. Sci. Math. Sci., vol. 13, no. 3, pp. 83–89, 1993. , “One rank perturbation and pole assignment of distributed param[9] eter system with unbounded single input” (in Chinese), Syst. Sci. Math. Sci., vol. 18, no. 2, pp. 179–189, 1999. [10] B. P. Rao, “Optimal energy decay rate in a damped Rayleigh beam,” in Contemporary Mathematics 209, S. Cox and I. Lasiecka, Eds. Providence, RI: American Mathematical Society, 1997, pp. 221–229. [11] R. Rebarber, “Spectral assignability for distributed parameter systems with unbounded scalar control,” SIAM J.Control Optim., vol. 27, pp. 148–169, 1989. [12] , “Exponential stability of coupled beams with dissipative joints: A frequency domain approach,” SIAM J.Control Optim., vol. 33, pp. 1–28, 1995. [13] D. L. Russell, “Nonharmonic Fourier series in control theory of distributed systems,” J. Math. Anal. Appl., vol. 18, pp. 542–559, 1967. [14] R. F. Curtain and H. Zwart, “An introduction to infinite-dimensional linear systems theory,” in Texts in Applied Mathematics 21. New York: Springer-Verlag, 1995. [15] S. H. Sun, “On spectrum distribution of completely controllable linear systems,” SIAM J.Control Optim., vol. 19, pp. 730–743, 1981. [16] G. Weiss, “Admissibility of unbounded control operators,” SIAM J.Control & Optim., vol. 27, pp. 527–545, 1989. [17] R. Curtain and G. Weiss, “Well posedness of triples of operators (in the sense of linear systems theory),” in Control and Estimation of Distributed Parameter Systems, F. Kappel, K. Kunisch, and W. Schappacher, Eds. Boston, MA: Birkhäuser, 1989, vol. 91, pp. 41–59. [18] C. Z. Xu and G. Sallet, “On spectrum and Riesz basis assignment of infinite dimensional linear systems by bounded linear feedbacks,” SIAM J.Control Optim., vol. 34, pp. 521–541, 1981. [19] R. M. Young, An introduction to Nonharmonic Fourier Series. London, U.K.: Academic, 1980. [20] S. Cox and E. Zuazua, “The rate at which energy decays in a damped string,” Commun. Partial Diff. Eqs., vol. 19, pp. 213–243, 2001.
Riesz Basis Property of a Second-Order Hyperbolic System With Collocated Scalar Input–Output Bao-Zhu Guo and Yue-Hu Luo
Abstract—The closed-loop form of a second-order hyperbolic system with scalar collocated sensor/actuator is considered. The Riesz basis property of the system is verified for diagonal semigroups based on an abstract result of one rank perturbation of discrete-type operators in Hilbert spaces. A simplified proof for the abstract result is presented. Index Terms—Collocated input–output, perturbation of linear operators, Riesz basis, second-order system.
I. INTRODUCTION In the case of infinite-dimensional systems of the form x_ (t) = Ax(t), where A is the generator of a C0 -semigroup in a Hilbert space H , it is well-known that stability of the spectrum of A (i.e., real parts
x(t). However, if for instance, the eigenfunctions f n g1 1 which associate with eigenvalues fn g11 form a Riesz basis (a basis which is equivalent to an orthonormal basis in H , see, e.g., [19]), then the
are negative) does not imply any stability properties in the solutions
solution can be represented as
x(t) =
1 n=1
an e t n ;
8 x(0) =
1 n=1
where the control u(t) is a scalar function and k a constant. A : D(A)( X ) ! X is an unbounded positive selfadjoint operator in X . b is an element of the dual space [D(A1=2 )]0 of [D(A1=2 )], the graph space of D(A1=2 ). System (1) is a special case of the following generic infinite-dimensional system with one rank perturbation in a Hilbert space H :
x_ = Ax(t) + f (x(t))b
A
(2)
where is a discrete type operator (see the definition in Appendix) and an element of the dual space [D( 3 )]0 , where 3 denotes the adjoint of . is bounded if 2 H , otherwise, it is unbounded. f is a functional in H , which is usually -bounded. Two basic problems are discussed mostly in the literature: For fixed f or , find conditions so that a) the system (2) is a Riesz spectral system ([18]), that is, there is a sequence of generalized eigenvectors of system (2), which forms a Riesz basis for H ; b) the system (2) has specified eigenvalues. The second problem is usually referred to as pole assignment problem. Its significance is guaranteed by the first one because of the spectrum-determined growth condition which may be not valid for generic infinite dimensional systems. The first paper concerning these questions is contributed to the works of [15], where the problem is completely resolved for the case where both f and are bounded. In the past two decades, much effort has been concentrated along the same direction to the solution of these problems under variant conditions for example [6], [8] and [18], name just a few. Usually, these works are based on the basic assumption that either f or is bounded. In [11] and [12], although both f and are assumed to be unbounded, is assumed to be the infinitesimal generator of an analytic semigroup. This limits its applications to such systems as beam equation without consideration of viscous damping. In [9], the author developed a general result for the case where neither f nor is required to be bounded without the analyticity assumption the semigroup generated by . In this note, we shall apply this abstract result to answer the question a) for a special case of the system (1), namely, the diagonal semigroup case. It is shown that under some spectrum assumption of , the system (1) is a Riesz spectral system and hence the spectrum-determined growth condition holds true. The result can be applied directly to a connected Euler-Bernoulli beam equation under joint feedback control. However, it avoids the estimation for the eigenfunctions for the beam equation as was done in [4] and, more importantly, it can give a complete answer for some cases that appear to be impossible by the approach of [4]. The remaining of this note is organized as follows. In Section II, we give the Riesz basis property for the system (1). In the Appendix, we give an outline of a simplified proof of the abstract result, which was first obtained in [9].
b
Ab
b
A
A
A
b
b
an n 2 H:
In this case not only stability of the system is determined by the spectrum of the system operator, which is usually referred as spectrum-determined growth condition, but also the dynamic behavior of the system can be described by eigenpairs f(n ; n )g. Riesz basis is also the basis of so called method of moment, a powerful method in the study of controllability of hyperbolic systems; we refer the readers for earlier works on moment method to [13] and a recent systematic summary to [2]. General Riesz spectral systems are discussed in [14]. A nice recent result on the relation of exact controllability and Riesz basis can be found in [7]. The general Riesz basis theory is developed in the context of nonharmonic Fourier series, which is originated from works of Paley and Wiener in the 1930s and developed extensively later by many mathematicians. Earlier results–are summarized in [19], and a nice recent summary can be found in [2]. Unfortunately, verification of the Riesz basis generation in practice is challenging even for some extensively studied systems. We refer to [10], [20], [3], and [4] for the study of this problem for string and beam equations. Most of these particular examples can be put into the following form of controlled second order hyperbolic system with scalar collocated input–output in a Hilbert space X
ytt + Ay + bu(t) = 0; u(t) = kb3 yt
693
b
A
b
b
A
A
(1)
II. RIESZ BASIS GENERATION Rewrite (1) as
Manuscript received April 19, 2001; revised October 18, 2001. Recommended by Associate Editor M. Reyhanoglu. This work was supported by the National Natural Science Foundation of China. B.-Z. Guo is with the Institute of Systems Science, Academy of Mathematics and System Sciences, Beijing 100080, P.R. China (e-mail: [email protected]). Y.-H. Luo, deceased , was with the Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P.R. China. Publisher Item Identifier S 0018-9286(02)03754-6.
ytt + Ay + kbb3 yt = 0
(3)
where A : D(A)( X ) ! X is an unbounded positive selfadjoint operator in a Hilbert space X . b 2 [D(A1=2 )]0 . b3 2 L([D(A1=2 )]; C I) is defined by
b3 x = hx; bi[D(A
0018-9286/02$17.00 © 2002 IEEE
)]2[D(A
)]
; 8x 2 D A1=2 :
(4)
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We assume that A is diagonal, that is, there is an orthonormal basis f n g1 for X such that
1
A n
= !n2 n !n > 0:
To explain (1) in the framework of general well-posed system theory, we usually write (1) as d dt
(5)
=A
By these assumptions, we can write b
1
=
=1
n
2bn n;
1 jbnj2 < 1: 2 !n
=1
n
3
= 2bn :
b n
Define an extension A~ 2 L
~
Ax; z
~ =
~
D
of A by
[D(A )] 2[D(A )] = A1=2 x; A1=2 z X 2X ; 8x; z 2 D
That is Ax
D A1=2
(7)
0 A1=2
1
2
=1
an !n n for any x
n
=
1 =1
an n
n
1=2
A
:
1=2
2
D A
(8)
:
(9)
)] to [D(A1=2 )]0 by Lax-Milgram theorem. It is well-known that for any 2 (A); ( 0 A~) is an isomorphism from [D(A1=2 )] to [D(A1=2 )]0 . ~ + kbb3 yt = 0; in + Ay
1=2 0
D A
y yt
= k (0; b3 )
y yt
0
= 0A~ 0 kbb3
I
y yt
2
f g
I = 00A~ 0kbb 3 g = 0Af ~ 0 kbb3 g
1=2 0 : (11)
=
f g
(12)
jbn j c > 0 f or some c > 0 and all n 1: We are now in a position to state our main result of this note. Theorem 3: In addition to (15), assume that
1 1
2 =1 !n
1:
0.
0 !n M !n +1 ; n = 1; 2; . . .
(15)
:
(18)
For the definition of [D]-class operators we refer to Definition A1 in the Appendix. The proof of Theorem 3 will be split into several lemmas. The following lemma is straightforward. Lemma 1: Under the condition (15)
M 1+ !1 (1 + )01 n1+ ; n > 1:
(19)
Therefore, A01 is compact. Let and be defined by (16). It is seen that
A
Therefore, (3) is actually being considered as d dt
(17)
[D]-class. Moreover, there exists a constant C > 0 such that the eigenpairs f(n ; n )g [ ftheir conjugateg of A satisfy
!n+1
with D
A
Ab
D A
However, we want to consider (11) in the energy state space H [D(A1=2 )] 2 X . To this purpose, we define
A
= kb3 yt :
If b is admissible, then the operator defined by (12)–(13) is of
;
in D A1=2
y yt
b A
n
d dt
()
u t
(16)
b
(10)
which can be considered as the following first-order equation:
0 0b
y yt
It is seen that b 2 [D(A1=2 )]0 if and only if 2 [D( )]0 . We say b is admissible if is admissible with respect to ([16]). Admissibility is a basic assumption for infinite dimensional control system (Ac ; Bc ) to be well-posed ([16]). When Bc is admissible, even if the control space is one dimensional, (Ac ; Bc ) can be exactly controllable ([7]) and the output feedback can make system be exponentially stable ([4]). This is contrast to the bounded control (see [14, Th. 4.1.5]). Theorem 2: Suppose (15) holds. Then i) b is admissible if and only if bn defined by (7) is uniformly bounded; ii) when b is admissible, system (16) is a regular linear system ([17]); iii) system (14) is exponentially stable if and only if ( ; ) is exactly controllable or equivalently
[ (
ytt
t
O t
A is an isometric from D A1=2
With these preparations, we may write (3) further as
= 00A I0 y 0 bu(t) y
( ) = kb3
(6)
It is seen by (4) and (6) that
y yt
b A86n = 6i!n86n = 6n86n; n 1:
f8ng forms an orthonormal basis for H 01 8n = 0i!n n 80n =
01 n i!n n
n
and
b=
1 =1
n
8 +
bn n
1 =1
n
bn
80n :
Let H01 be the completion of H under the norm
kF k01 = A01 F
H
8 2 H:
; F
;n
(20)
1
(21)
(22)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
Then H01
=
=
F
01;n6=0
1
A~ )b 2 D(f ) and
~ (; hence R
1 n=
695
8j
~
an n
jan j
2
0 and all n 1 (it follows from Lemma 1 that (17) is satisfied for any 1=2). Then g is a bounded functional on H . Proof: It suffices to show that
(29) )]
2D(A
)
an bjnj
1
In particular
and f is -bounded: jf (F )j kkbk[D(A lation, we can now write
)
=0
=
where
(8n ) = 02kbn
2D(A
)
kRn bk < 1 1 g (F ) = an bjnj kRn bk ; n 01;n6 an 8n 2 D(g ) 8F = 1 n 01;n6
(26)
Define the functional f in H such that
f
~
bjnj f R ;
n=
D(A
= 1 n=01;n6=0 an 8n 2 H j
F
1
1
+ !n2 ; b
proving (31). In order to apply Theorem A in Appendix, we need the following functional g :
an n
n=
=
For any
n=
for any F
2
= 0 k 22+bn!2 h n ; biD(A n n=1 1 2b = 2 +n!2 f (8n ) n n=1
Let
1
n=1
1
(23)
A~ = 00A~ I0 : ~ is an extension of A and D(A~ ) = H . Then A ~ (; A~ ) is an isomorphism from H0 to H . 2 (A); R 1 an ~ ; A~ F = 8n ; R 0 n
1 2b e n n
0 !k
!k
1 jj
k= n +2
!
0 !k0
1
1
0 !k
2
dx
!
1
!jnj
! !
!jnj
0x
!jnj
1
2
0x
2
dx
:
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where
I2
1
To show the result, it suffices to show that
1
!
M!j j =jnj+2 n k
1
dx
!jnj 0 x 2
!
= M!1 ! 10 ! M 21!2 jnj jnj jnj+1 jnj while as !jnj01 > 2!1 1 jnj02
I1 M
=1
M
1
M
k
=1 !
!
1 M!
!
1
dx x !jnj 0 x 2 =2 1 dx x ! 0x 2
1
1 !jnj 0
1 M!
+
1
2 dx =2 x !jnj 0 x
1
4
!
4
1
+ M!
2
jnj01 !jnj 0 !jnj01
!
jnj01
2 1 + 4 : = M! 1 !jnj M 2 !j2nj01
1
C for some C > 0
f (8k ) bjkj jmjn jkj=n+1;k6=m jm 0 k j
1
jkj=n+1;k6=m
= 0:
2C
1
1
+ f 820!m bjmj jmj
!jmj 0 !k k=n+1;jkj6=jmj
1
+ 2!1
jmj
+ 2!1 ! 0 ! jmj k jmj k=n+1;jk0jmjj2 1 1 + + !jmj+1 0 !jmj
1
1
!jmj 0 !jmj01
2C + 2!1 ! 0 ! jmj j m j j k j k=n+1;jk0jmjj2 + 2 : M!jmj
1
!
! ! !
!
dx
! !
x0 dx x 0 !jmj
x0 dx + 1 !jmj 0 x M x0 dx !jmj 0 x
1
!
dx x x + !jmj
0!
x0 dx x 0 !jmj
(37)
By Lebesgue’s dominated convergence theorem
1
1
dx ! 0! x x + !jmj 1 dx M1 ! 0; as jmj ! 1: M! x x + !jmj
S2 = M
!n+1 and jmj n ! 1
1
f (8k ) bjkj jkj=n+1;jkj6=jmj !jmj 0 !jkj
1
=jmj+2 M
dx x 0 !jmj
x !jmj 0 x
!
=S1 + S2 :
S1 = M
1
!
!
When !jmj01=2
n
f (8k ) bjkj jm 0 k j
2C
k
1 + M1
4 1 +2 4 kRnbk2 C !12 + M 26!2 + M! ; M 2 !j2nj01 1 !jnj jnj jnj (35) !jnj01 >2!1 :
Proof: For any integer n, as jmj
1
M
lim sup n!1
1
=1
The proof is complete by the assumption (17). Lemma 4: Suppose (15). Assume that jkbn j2 and all n 1. Then
1
M k=n+1
M1
Therefore
1
!jmj 0 x
!
!k 0 !k01 k=jmj+2 jmj02 !
+
+ M 2 !2
1 !jnj 0 2
1
= +1 !k+1 0 !k
k n
!
(36)
1 1 + ! 0 ! ! k k 0 !jmj j m j k=n+1 jmj+2 jmj02 ! 1 dx
=
dx x !jnj 0 x 2
jnj
! 0 (jmj n ! 1) :
1
! 0 !k k=n+1;jk0jmjj2 jmj jmj02
!
+ M1
1
!
!
Now
1
!
1
!jmj 0 !jkj k=n+1;jk0jmjj2
!
dx
x !jmj 0 x 2 x0 dx 1 ! + M ! !jmj 0 x M ! M!!jmj02!1 0 20!n!+1 jmj jmj01 n+1 ! dx + M!2 =2 !jmj 0 x jmj01 !
=1
! !
x0 dx =2 !jmj 0 x
=
M!1 + M!2
log 1 + ! !0jm!j01 jmj jmj01 jmj01 10 ! 2 log 1 + jmj01 ! 0
+1
n
M!1 + M! +1
n
jmj01
while as !n+1 > !jmj01 =2 and jmj
M
n!1
x0 dx 1 !jmj 0 x M ! !10 M!2 log 1 + jmMj01 jmj01
1
S1 = M
!
! !
! 0:
x0
2 !jmj 0 x
=
dx
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
697
[ ]
Therefore
S1 + S2 ! 0 as jmj n ! 1: (36) is thus proved and Lemma 4 follows. Proof of Theorem 3: By Lemmas 2–4, f and b are compatible with respect to A according to definition A2 in Appendix. Hence, all conditions of Theorem A in Appendix are satisfied. So A is of D -class. Equation (18) follows from (29), (35) and Theorem A1 in Appendix.
[ ]
APPENDIX For the sake of completeness, we will give here an outline of a simplified proof of perturbation result which we cite for the proof of Theorem 3. The result stated below is a special case of [9, Th. 2.6]. Let us first introduce a definition of D -class operators. 1) Definition A1: A linear operator A in a Hilbert space H is called discrete type, or D -class in short, if there are Riesz basis f n g1 of H , a sequence of complex numbers fn g1 and a nonnegative integer N such that i) n!1 jn j 1, n 6 m , as n > N , m , n 6 m; ii) A n n n , n > N ; iii) Aspanf 1 ; 2 ; ; N g spanf 1 ; 2 ; ; N g and A has spectrum fi g1N in spanf 1 ; 2 ; ; N g, where spanf 1 ; 2 ; ; N g denotes the linear subspace spanned by all f i g1N . It is well-known that A is a discrete operator in H (that is, its resolvent is compact) when it is of D -class. And for the D -class operator A, the semigroup eAt generated by A satisfies the spectrum-determined growth condition. Let A be a discrete operator in H . We denote by H01 the completion of H with respect to the norm
[ ]
[ ]
1
lim
= ... ...
=
1
=
1 = ... ...
[ ]
b=
(38)
According to [16], H01 is equivalent to D A3 0 , the dual space of the graph space D A3 , where A3 is the adjoint operator of A. For any 2 A , R ; A 0 A 01 has a natural extension R ; A from H01 to H : R ; A x R ; A x for x 2 H . Certainly A can be extended to whole H by
[ ( )]
~ yi = hx; A3 yi; for any x 2 H; y 2 D (A3 ) : hAx;
~(
For any functional f define Af;b by
~ + f (x)b 2 H ; D (Af;b ) = x 2 H Ax ~ Af;b x = Ax + f (x)b:
(41)
We say f is A-bounded, if jf (x)j M kAxk for some M > 0 and all x 2 D(A). When b 2 H , Af;b = A + f (1)b, which has been studied
extensively in [8] and [18]. 2) Lemma A1: Let b 2 H01 . Suppose that f is A-bounded and there exists an 2 A (and hence for all 2 A ) such that R ; A b 2 D f . Then Af;b is a discrete operator in H . Moreover, 2 Af;b for any satisfying 0 f R ; A b 6 and
( )
( ) ~( ) () ( ) 1 ( ~( ) ) = 0 f (R~ (; A)1) ~ R(; Af;b ) = R(; A) + 1 0 f (R~ (; A)b) R(; A)b:
(42)
(43)
( )
x=
1 ak k 2 H01 k=1
1
j j2 jn 0 k j02 < 1
(44)
(
:
~
)
It is obvious that Rn 2 L H01 ; H . Moreover, the restriction Rn of Rn on H is a bounded operator on H with the norm
kR~n k M ; 8n > N n
(45)
where
n =
inf 6= jn 0 m j ; n 1:
(46)
Next define a linear functional gf;b in H by
1 an n 2 H n>N
D (gf;b ) = x =
(x) =
g
1 jan f ( n )jkRn bk < 1 ; n=1 1 an f ( n ) kRn bk ;
(47)
f;b n>N 8x = 1 n=1 an n 2 D (gf;b ) :
3) Definition A2: Suppose f is an A-bounded functional, b 2 H01 is given by (43). f and b are called compatible with respect to A if i) there exists an 2 A such that R ; A b 2 D f and
( )
f (R~ ( ; A)b) =
1
n=1
~(
)
()
bn f (R~ ( ; A) n );
(48)
ii) the functional gf;b defined by (47) is bounded on H ; iii)
(39)
(1) in H; D(f ) D(A) and b 2 H01 , we can
jbnj2 < 1: 2 n=1 j 0 0 n j
k=1;k6=n ak
)
(40)
n=1
1
bn n in H01 with
= = D (Rn ) =
lim sup n!1 mn
A~ is an isomorphism from H to H01 . By [16, Prop. 3.3], for any L 2 L(H ) which is permutable with A, there is an extension L~ on H01 :
~ = R ( 0 ; A)01 LR~ ( 0 ; A) : L
1
For any positive integer n > N , define Rn as 1 a Rn x k=1;k6=n 0 k ; 1 8x k=1 ak k 2 D Rn ;
[ ]
k 1 k01 = k ( 0 0 A)01 1 k for some 0 2 (A):
[ ( )] ( ) ( )=( ) ~( ) = ( )
In the sequel, we always suppose that A is of D -class satisfying the conditions of Definition A1 and the notations are kept throughout the discussions. For any given b 2 H01 , we write
1
jf ( k ) bk j = 0: j 0 k j k=n+1;k6=m m
(49)
For the given f and b as previously shown, denote
1n = 2 CI 0 n rn ; when f ( n ) bn = 0 (50) rn = 3 2 jf ( n ) bnj ; while f ( n ) bn 6= 0. 4) Lemma A2: Let 1n be given as (50). If f and b are compatible with respect to A, then there exists an integer K > N such that Af;b possesses an eigenvalue n in 1n for each n K and jn 0 nj 2 jf ( n ) bnj 13 n (51) and
1 10f 2 0
R (n ; AN )
1
N k=1
f ( k ) bk 0 k k=N +1;k6=n n
k bk
32 for all n K
(52)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
0 ;
b
f ( n )
n =
1
0 f R (n; AN ) Nk bk k f b 0 ; 0 1 k N ;k6 n 0
when f ( n ) bn = 6 0, (55)
=1
=
+1
=
(
)
1
otherwise
and
yn = R (n ; AN )
N k=1
bk k +
AN is the restriction of A on the subspace spanf 1 ; 2 ; . . . ; N g. 5) Theorem A: Suppose that f is an A-bounded functional and b 2 H01 . If f and b are compatible with respect to A, then Af;b is of [D]-class. Moreover, the eigenvectors f n g of Af;b and that of A where
satisfy
k n 0 n k C jf ( n )j jn 0 j0
1
0
+
kRn bk
(53)
for some C > and all sufficiently large n, where Rn is defined by (44). Proof: Suppose that K is the constant in Lemma A2. In the proof of Lemma A2, the eigenvectors n of Af;b corresponding to n can be chosen as n = n + n yn
(54)
where (55), shown at the top of the page, holds. By (51) and (52)
j n j 2 jf ( n )j
n K:
(56)
Next, it can be shown that
kyn k C jn 0 j0 0
1
+
kRn bk
;
n K:
(57)
Finally, since f is A-bounded and b are compatible with respect to A, it follows from (54), (56), and (57) that
1 n=K
k n 0 n k 2C 2
C
1
1 n=K
jf ( n )j jn 0 j
2
0
2
f R~ ( 0 ; A)
+ 2
jf ( n )j kRn bk
+
2
kgf;bk
2
2
0. By [3, Th. 6.3], Af;b is of [D]-class. The proof is complete. ACKNOWLEDGMENT The first author would like to express his sorrow in memory of his coauthor Y.-H. Luo, who passed away suddenly while this note was under review. REFERENCES [1] K. Ammari and M. Tucsnak, “Stabilization of second order evolution equations by a class of unbounded feedbacks,” ESAIM Control, Optim. Calc.Var., vol. 6, pp. 361–386, , 2001. [2] S. A. Avdonin and S. A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge, U.K.: Cambridge Univ. Press, 1995. [3] B. Z. Guo, “Riesz basis approach to the stabilization of a flexible beam with a tip mass,” SIAM J.Control Optim., vol. 39, pp. 1736–1747, 2001.
1
bk k 0 k : n k=N +1;k6=n
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