IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 1, JANUARY 2000
On the Spectrum-Determined Growth Condition of a Vibration Cable with a Tip Mass Baozhu Guo and Cheng-Zhong Xu Abstract—In this correspondence, we show that the spectrum-determined growth condition holds for the closed-loop system of a vibration cable with a tip mass and linear boundary feedback control. The optimal decay rate of the energy for a case left unsolved in the [1] is determined, and the asymptotic expansion of the associated semigroup is obtained. As a consequence of the approach, we show, in a different point of view, the lack of uniform exponential stability of the system with only boundary velocity feedback control. Index Terms—Infinite-dimensional systems, optimal decay rate of a -semigroup, spectrum-determined growth condition.
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bound of the underlying semigroup is equal to the spectral bound of the semigroup generator. Because the approach relies on concrete expressions of the eigenvalues and eigenfunctions, however, the case in which 6= (m=a) was left unsolved in [1] because, in this case, eigenfunctions are not explicitly available. The purpose of our correspondence is to show that the spectrum-determined growth condition always holds for the system (1). This fact makes it possible to get the optimal decay rate of the energy directly from the spectral bound of the generator. As a consequence, we can design the feedback gains for assigning the spectrum of the generator to fix the decay rate of the energy. Our approach is different from that of [1] and based on the results of Neves et al. [2] and some standard theorems in spectral theory. For the specific case in which 6= (m=a) = 1, we give explicit expressions for the optimal decay rate and asymptotic decomposition of the semigroup. We consider the Hilbert spaces X = V 2 L2 [0; 1] 2 and H = 2 2 (L [0; 1]) 2 equipped, respectively, with their inner products
I. INTRODUCTION
utt (x; t) 0 uxx (x; t) = 0; 0 < x < 1; t > 0 u(0; t) = 0 ux (1; t) + mutt (1; t) + auxt (1; t) + ut (1; t) = 0:
(1)
1
hf; giX =
Consider the closed-loop system of a pinched vibration cable with a tip mass attached at the free end and with a control force linear feedback applied on the mass in Morgul et al. [1]
0
hf; giH =
E (t) =
0 +
[ut (x; t) + ux (x; t)] 1
m + a
1 0 +
[f1 (x)g1 (x) + f2 (x)g2 (x)]
2 (a + m)
2(m + a )
dx
f3 g3 :
(2)
for some positive constants M; ! and for any initial condition (u(1; 0); 2 L2 [0; 1] and t 0; where the energy E (t) is defined as 2
f3 g3
and
T ( (x); (x); d) = (1=2( (x) 0 x (x)); 1=2( (x) + x (x)); d):
ut (1; 0)) 2 V
2
2(m + a )
dx
Then, the following mapping T from X to H is a unitary mapping:
E (t) Me0!t E (0)
1
2 (a + m)
+
It is shown in [1] that, for all m > 0 and all feedback gains a > 0 and > 0; the dynamic system (1) is exponentially stable in the sense that
[f1x (x)g1x (x) + f2 (x)g2 (x)]
dx
2 [aux (1; t) + mut (1; t)]
and V = fu 2 H 1 [0; 1]; u(0) = 0g: It is meant that the hybrid system is uniformly stabilized by choosing a suitable feedback law for the control force depending on the velocity and the angular velocity at the free end. Moreover, based on the Riesz basis approach, the optimal decay rate of the energy has been obtained in [1] for the case in which = (m=a): The optimal decay rate of the energy is defined to be the infimum of (0! ) such that (2) holds for some M depending on !: In other words, the authors of [1] have shown that the spectrum-determined growth condition is satisfied for that case: the growth Manuscript received March 20, 1997; revised January 6, 1999. Recommended by Associate Editor, O. Egeland. This work was supported by the National Science Foundation of China and the INIRA, under Project CONGE. The work of B. Guo was supported in part by the National Key Project of China. B. Guo was with INIRA-Lorraine, CONGE and ESA CNRS 7035 (MMAS), Université de Metz, 57045 Metz Cedex, France. He is now with the Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China. C.-Z. Xu is with the INRIA-Lorraine, CONGE and ESA CNRS 7035 (MMAS), Université de Metz, 57045, Metz Cedex 01, France (e-mail:
[email protected]). Publisher Item Identifier S 0018-9286(00)00812-6.
(3)
Setting (p(x; t); q (x; t); d) = T (u(x; t); ut (x; t); d); we can write (1) under the equivalent form
@ p(x; t) + K @ p(x; t) = 0 @t q(x; t) @x q(x; t) d q(1; t) + m 0 a p(1; t) = 1 0 p(1; t) 0 1 + q(1; t); dt m+a m+a m+a p(0; t) = 0q(0; t) (4) where
K=
1 0
0 01 :
System (4) enters the class of systems investigated in Neves et al. [2]. Take H as our state space, and define the operator A by
D(A) =
( (x);
d=
2 (H 1 [0; 1])2 2 ; (0) = 0 m 0 a (1) (1) +
A( (x); (x); d) =
0018–9286/00$10.00 © 2000 IEEE
(x); d)
m+a 0 0 (x); 0 (x);
(0)
10 (1) 0 1 + m+a m+a 8( (x); (x); d) 2 D(A):
(1)
;
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 1, JANUARY 2000
Setting W (t) = (p(1; t); q (1; t); q (1; t) + (m 0 a)=(m + a)p(1; t)); we rewrite (4) as an evolution equation on H
dW (t) = AW (t): dt
Proposition 1: For each t > 0; the set f(eAt 0 eBt 5)W ; kW k 1; W 2 H g is precompact in H and
ess (eAt ) = ess (eBt ) s(B ) = !0 (B ):
(5)
As in [2], we associate with the system (4) a reduced system defined on (L2 [0; 1])2 (equipped with the usual inner product)
@ p(x; t) + K @ p(x; t) = 0 @t q(x; t) @x q(x; t) m 0 a p(1; t) = 0 q(1; t) + m+a p(0; t) = 0q(0; t):
(6)
Correspondingly, define B by
D(B ) = ( (x); (x)) 2 (H 1 [0; 1])2 ; (0) = 0 (0) (1) + mm0+aa (1) = 0 B ( (x); (x)) = (0 0 (x); 0 (x)): An element (u; v ) 2 (L2 [0; 1])2 is identified with (u; v; 0) 2 that (L2 [0; 1])2 is a subspace of H: Define the projector 5 by
H
so
(10)
Remark 1: Proposition 1 follows from the result of [2], which is valid for far more general hyperbolic systems than ours. Otherwise, taking the right hand-side of the second equation in (4) as a rank-one relatively bounded perturbation, we prove the first assertion with the help of [5, Lemma 5]. From [1], it is clear that the spectrum-determined growth condition holds for the reduced system (6). We prefer applying the result of [2], however, which has inspired our work here. From our notations and [2, Lemma 5], et! (B ) = ess (etB ) = (etB ) = et! (B) = ets(B) : Thus, !ess (B ) = !0 (B ) = s(B ): Our main result is the following. Theorem 1: The spectrum-determined growth condition holds for the system (4): !0 (A) = s(A): Proof of Theorem 1: From (10), we have !ess (A) = !ess (B ): Thus
s(A) !0 (A) = maxfs(A); !ess (A)g = maxfs(A); !ess (B )g (11) maxfs(A); !0 (B)g = maxfs(A); s(B)g: We claim that s(B ) s(A): The claim together with (11) implies that
5( (x); (x); d) = ( (x); (x); 0):
s(A) !0 (A) s(A):
We claim that A and B are the generators of C0 -semigroups of contractions, respectively, on H and (L2 [0; 1])2 : Refer to the Appendix for a proof of our claim. Let us recall briefly some notations that will be used. (Refer to Clément et al. [3] for more details.) Let K(H ) stand for the closed ideal of all compact operators in L(H ): Given a 3 2 L(H ); the essential spectral radius of 3 is defined as
This proves our Theorem 1. Let us prove our claim. From [2], both A and B have compact resolvents so that only isolated eigenvalues with finite algebraic multiplicity are possible in the spectrum of these two operators. It is checked that (see also [1]) the eigenvalues of operator A consist of the zero points of an order-one entire function
ess (3) =: n! lim +1
k k
1=n 3n ess
(
k3 0 K kess = k3kess and so ess (3 0 K ) = ess (3): Let s(B ) be the spectral bound of B : s(B ) = supfRe(); 2 (B )g: The growth bound !0 (B ) of the corresponding semigroup is defined as
ln ketB k : t
Denoting by (3) the spectral radius of 3 2
L(H ), we have
(etB ) = et! (B) : (7) Similarly, the essential growth bound !ess (1) of a C0 -semigroup gen-
erated by (1) is well defined (see [3, p. 200]) such that
ess (et(1) ) = et!
1:
( )
(8)
From Proposition 8.6 of ([3, p. 201]), for any generator A, we have
!0 (A) = maxfs(A); !ess (A)g:
g() = (m + a)e2 0 m + a:
)
K , we have
!0 (B ) = t!lim +1
(12)
and those of the operator B consist of the zeros of the entire function
k3kess = dist (3; K(H )) =: inf fk3 0 K kL H : K 2 K(H )g is the norm of 3 in L(H )=K(H ): For every compact operator
where
f () = ((1 0 ) + (a 0 m))e02 + (1 + ) + (a + m)
(9)
Case 1: m vertical line
(13)
6= a: In this case, all solutions of g() = 0 lie on the 0a : 0 t > 0: ux (1; t) = 0ut (1; t);
Combining the above lemmas with the asymptotic expression of a compact C0 -semigroup (see Yu et al. [4]) allows us to obtain our main result. Theorem 2: Suppose that a = m and 6= 1: Let the complex conjugate eigenvalues fi ; i gi=1;2;111 of A be arranged such that 1; then keAt k < Me0t log ( +1+a)=( 01+a) ; for all t 0: If < 1p ; and a > 1 0 ; then keAt k < a=(10 );01=(a0(10 ))g ; for all t 0: t maxf0 log Me p If < 1; and a 1 0 ; then keAt k < Me0t log 1=(10 ) ; for all t 0: For each integer m 1 and each > 0; there is a positive constant M (m; ) such that for any t 0
i) If ii) iii) iv)
eAt I 0
iii)
for t 2:
0 a + 0 1 < 0 < 0 21 log +0 11 ++ aa : (17) If < 1; then at most two negative real eigenvalues f01 ;
02 g exist. Three cases are possible as follows. a) if a 1 0 and no real eigenvalue exists, then all eigenvalues satisfy
f (0) = 2 for all < 0: Let = + i : Using ex < 1 0 x for x < 0 and (24) with < 0, we get
0
(25)
whose associated semigroup has essential spectral radius equal to one. Because the original system (1) is essentially a compact perturbation of (25), its essential spectral radius is also equal to one so that no exponential stability of (1) exists for a = 0: APPENDIX Proof of the Claim: We prove the claim just for B because the same reasoning is valid for A: Set X = (L2 [0; 1])2 : Indeed, for all
2 D(B) 0 0 cos 2 = 1 + + 2a 2 0 (1 0 )e0 2hBf; f iX = 0[1 0 (m 0 a) (m + a) ]f (1) 0 ( 0 1)e and so B is dissipative. The domain D(B ) is clearly dense in X beor (1 0 )e0 1 and cause C [0; 1] is contained in D(B ): Because je (m 0 a) (m + 0 a ) j < 1 ; for each g 2 X , a unique solution f 2 D(B) exists such 0: For any complex eigenvalue = + i ; from (24), that (I 0 A)f = g: Therefore, the range of I 0 A is the whole space sin(2 ) = (2 a=(1 0 ))e : So, (a=(1 0 ))e 3 X : It follows from the theorem of Lumer-Phillips (see [6, p. 14]) that 1 and
