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Systems & Control Letters 51 (2004) 33–50 www.elsevier.com/locate/sysconle
Exponential stability of variable coe#cients Rayleigh beams under boundary feedback controls: a Riesz basis approach Jun-min Wanga;∗, Gen-qi Xub, Siu-Pang Yunga
aDepartment of Mathematics, The University of Hong Kong, Hong Kong, People’s Republic of China bDepartment of Mathematics, Shanxi University, Taiyuan 030006, People’s Republic of China
Received 3 March 2003; received in revised form 26 June 2003; accepted 30 June 2003
Abstract
In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, sti9ness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered. c 2003 Elsevier B.V. All rights reserved.
Keywords: Rayleigh beam; Eigenvalue distributions; Riesz basis; Exponential stability
1. Introduction
In this paper, we study non-homogeneous Rayleigh beam governed by
(x)utt − (I(x)uttx)x +(EI(x)uxx)xx =0; 0 ¡x¡1;t¿0;
u(0;t)=ux(0;t)=0;t¿0;
EI(1)uxx(1;t)+ uxt(1;t)=0;t¿0;
(EIuxx)x(1;t) − I(1)uxtt(1;t) − ÿu5(1;t)=0;t¿0;
u(x; 0) = u0(x);ut(x; 0) = u1(x): (1.1) The situations where the coe#cients are non-constant arise in engineering problems that use non-homogeneous materials, such as smart materials. Here, u(x; t) is the transverse displacement and x; t stand respectively for the position and time, and (x) ¿ 0 is the mass density, EI(x) ¿ 0 is the sti9ness of the beam, I(x) ¿ 0is the mass moment of inertia and ; ÿ ¿ 0, which can be tuned, are constant feedback gains of the actuators
∗ Corresponding author. Tel.: +(852)28592300; fax: +(852)25592225. E-mail address: [email protected] (J. Wang).
0167-6911/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-6911(03)00205-6 34 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 with collocated measurements at the tip ends. A very nice description of the physical background as well as earlier development of (1.1) can be found in [9] and we would like to refer the interested readers there for other details of this model. The constant coe#cient version of (1.1) has been investigated in [8,2] say, where the former established the exponential stability under the condition =1;ÿ¿ 0 and the later proved a Riesz basis property. In this paper, we shall extend these results to the variable coe#cient cases with less restrictive conditions on the parameters. In fact, we show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, sti9ness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. Our investigation concentrates on proving that the generalized eigenfunctions of the feedback system form a Riesz basis in the state Hilbert space. Although there are many works in the literatures dealing with the Riesz basis of vibration systems, for example [3] for Euler-Bernoulli beam and [10,14] for Timoshenko beam, our method to obtain Riesz basis is di9erent from that used in [2,3,10,14], among them, the authors mainly proved that the generalized eigenfunctions of the systems are quadratically close to a reference Riesz basis system, and then used Bari’s Theorem to complete the proof of the Riesz basis generation. Here we treat the veriÿcation of Riesz basis property by the exponential family (see [15]). The advantage of this method lies on which it need not to compute the generalized eigenfunctions as long as the distribution and the separability of the eigenvalues of system (1.1) are known. As a consequence, we also provide an answer for the conjecture in [2]. It is well known that the study of variable coe#cient di9erential equations with (control) parameters is di#cult because explicit solution formula is hard to come by. One of the two crucial steps in this paper to overcome this obstacle is the proving of the Riesz basis property because then the system will satisfy the spectrum determined growth condition and stability can be deduced directly from an spectral analysis, which becomes the second crucial stepbecause the classical treatment [ 7] alone is not working and it has to combine with the method of operator pencil in [1,11,12] to obtain asymptotic expansions for the fundamental solutions, the characteristic determinant and the eigenvalues of problem (1.1). The rest of the paper is organized as follows. In Section 2 we convert system (1.1) into an evolution equation in an appropriate Hilbert space and then prove that the evolutionary system is associated to a C0 semigroup of linear operators whose generator has compact resolvents. Hence, the problem is well-posed. In Section 3, we recall from [13] an asymptotic distribution of the eigenvalues and this forms our foundation to prove the Riesz basis property and the exponential stability of the system in the last section. Finally, a conjecture in [2] is discussed and answered. Throughout this paper, we always assume that
4 (x);I(x);EI(x) ∈ C [0; 1]: (1.2)
2. Hilbert space setup and eigenvalue problem
We start our investigation by formulating the problem in the following Hilbert spaces: V := {f ∈ H 1(0; 1) | f(0)=0} with norm f2 := 1 [(x)|f(x)|2 + I (x)|f (x)|2]dx and W := {f ∈ H 2(0; 1) | f(0) = f (0)=0} V 0 with norm f2 = 1 EI(x)|f (x)|2 dx. Easy to see that W 0 W ⊂ V ⊂ L2[0; 1] ⊂ V ⊂ W ; where W and V are the dual spaces of W and V , respectively. We now deÿne linear operators A; D ∈ L(W; W ) and B; C ∈ L(V; V )by 1 A; = EI(x) (x) (x)dx; ∀; ∈ W; (2.1) 0 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 35
D; = (1) (1); ∀; ∈ W; (2.2)
B; = (1) (1); ∀; ∈ V; (2.3)
1 C; = [(x)(x) (x)+I(x) (x) (x)] dx; ∀; ∈ V: (2.4) 0 Lax–Milgram Theorem [16, p. 92] says that A (resp. C) is a canonical isomorphism of W (resp. V ) onto W (resp. V ). With these operators, Eq. (1.1) can be written into a variational equation
Cutt; + Au; + Dut; + ÿBut; =0; ∀ ∈ W: (2.5)
2 2 2 Let H := W × V with norm (f; g)H := fW + gV . Then we can deÿne a linear operator A on H by D(A):={(f; g) ∈ H | g ∈ W; Af + Dg + ÿBg ∈ V }; (2.6)
A(f; g):=(g; −C−1(Af + Dg + ÿBg)); ∀(f; g) ∈ D(A): (2.7)
Thus, (2.5) can be formulated into an evolution equation in H (with Y (t):=(u; ut)) dY (t) = AY (t);t¿0;Y(0) = Y =(u ;u ): (2.8) dt 0 0 1
Lemma 2.1. Let A be deÿned by (2:6) and (2:7). Then A is a densely deÿned closed dissipative operator with 0 ∈ (A), and so A generates a C0 semigroup of contraction.
Proof. For any (f; g) ∈ D(A), we have
1 1 A(f; g); (f; g) = EI(x)g (x)f (x)dx − [Af + Dg + ÿBg]g(x)dx 0 0 = Ag; f −Af; g − Dg; g −ÿBg; g
= Ag; f −Af; g − |g (1)|2 − ÿ|g(1)|2:
So
ReA(f; g); (f; g) = − |g (1)|2 − ÿ|g(1)|2 6 0:
To show that 0 ∈ (A), we let (y; z) ∈ H and consider the resolvent equation
A(f; g)=(y; z); which is equivalent to y = g and z = −C−1[Af + Dg + ÿBg]. So, for any ∈ W ,
Af + Dg + ÿBg; = −Cz; :
Substituting g = y back into the above equation yields
Af; = −Cz + Dy + ÿBy; ; ∀ ∈ W: (2.9) 36 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50
2 Since for any ∈ W , we have A ; = W . So from the Lax–Milgram Theorem, there exists a unique f ∈ W so that (2.9) holds and 0 ∈ (A) by letting g := y. The remaining part of the lemma is a direct consequence from the theory of semigroupof operators(see, [ 4, p. 3, Theorem 1.2.4]).
Lemma 2.2. Let (f; g) ∈ H. Then (f; g) ∈ D(A) if and only if f ∈ W ∩ H 3 and g ∈ W such that
−1 (EI(x)f (x)) |x=1 + I[C (Af; Dg + ÿBg)] |x=1 − ÿg(1)=0; EI(1)f (1) + g (1)=0:
From this we see that A−1 is compact.
Proof. The su#ciency is obvious. To prove the necessity, let (f; g) ∈ D(A) and A(f; g)=(y; z) ∈ H. Then we have g = y ∈ W and
−C−1[Af + Dg + ÿBg]=z:
Since z ∈ V and C : V → V is an isomorphism, so we have
Af + Dy + ÿBy = −Cz; in V ⊂ W and hence
1 1 M M M EIf dx + y (1) (1) + ÿy(1) (1) + [z + Iz ]dx =0; ∀ ∈ W: (2.10) 0 0 Now for any ∈ C∞(0; 1), let (x)= x (s)ds and substitute it into (2.10) yields 0 0 1 1 1 1 1 M M M M EIf dx + ÿy(1) dx + dx z ds + Iz dx =0 0 0 0 x 0 and 1 1 1 M M EIf dx = − ÿy(1) + z ds + Iz dx; 0 0 x for all ∈ W . Thus
1 2 (EIf ) = ÿy(1) + z ds + Iz ∈ L [0; 1]: (2.11) x
4 3 Since EI ∈ C [0; 1] (see (1.2)), so we have f ∈ H (0; 1) ∩ W . In particular, (EIf ) |x=1 = ÿy(1) + Iz |x=1. Inserting g = y and z = −C−1(Af + Dg + ÿBg) into the above yields
−1 (EIf ) |x=1 + I[C (Af + Dg + ÿBg)] |x=1 − ÿg(1)=0: Again, for ∈ V with (1) = 1, we let := x (s)ds, and plug it into (2.10) and conclude from (2.11) 0 that EI(1)f (1) + y (1) = 0. The necessity is also proved because g = y. By Lemma 2.1, A−1 exists and is bounded on H. From the Sobolev Embedding Theorem, A−1 is compact. J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 37
We are now in a position to recall some results for the eigenvalues of A that have been announced in [13]. Let ∈ (A) and (; ) ∈ H be such that A(; )= (; ). Then, = and satisÿes
2 2 (x)(x) − (I(x) (x)) +(EI(x) (x)) =0; 0 ¡x¡1; (0) = (0)=0;EI(1) (1) + (1)=0;
2 (EI ) (1) − I(1) (1) − ÿ (1)=0: (2.12)
Lemma 2.3. Let h1(x);h2(x) be two linearly independent solutions for the second-order linear homogeneous di8erential equation
(I(x) (x)) − (x)(x)=0; (2.13) then we have
D := h1(0)h2(1) − h1(1)h2(0) =0: (2.14)
Lemma 2.4. If + ÿ¿0, then
Re( ) ¡ 0: (2.15)
The proofs of the above two lemmas are just direct veriÿcations (see [13]). To further simplify (2.12), we expand it to yield EI EI I I (4) +2 + − 2 + − =0; EI EI EI EI EI (0) = (0)=0;EI(1) (1) + (1)=0;
2 EI(1) (1) + EI (1) (1) − I(1) (1) − ÿ (1)=0: (2.16)
Introducing a space-scaling transformation (see [3]) 1 x I ( ) 1=2 1 I ( ) 1=2 (x):=f(z);z:= d ;h:= d ; (2.17) h 0 EI( ) 0 EI( ) then Eq. (2.16) can be rewritten as
f(4)(z)+a(z)f (z)+b(z)f (z)+c(z)f (z) − h2 2[f (z)+d(z)f (z) − e(z)f(z)]=0;
f(0) = f (0)=0;b21f (1) + b22f (1) + b23 f (1)=0;
2 b11f (1) + b12f (1) + b13f (1) − b14f (1) − ÿ f(1)=0: (2.18)
Here z 1 EI (x) 1 I (x) 1=2 a(z):=6 xx +2 ;z = ; (2.19) 2 x zx zx EI(x) h EI(x) z2 z z EI (x) 1 EI (x) b(z):=3 xx +4 xxx +6 xx + ; (2.20) 4 3 3 2 zx zx zx EI(x) zx EI(x) 38 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50
z z EI (x) z EI (x) c(z):= xxxx +2 xxx + xx ; (2.21) 4 4 4 zx zx EI(x) zx EI(x) z 1 I (x) d(z):= xx + ; 2 2 3 zx h zx EI(x) 1 (x) e(z):= ; (2.22) 2 4 h zx EI(x)
3 2 b11 := zx (1)EI(1);b12 := 3zx(1)zxx(1)EI(1) + zx (1)EI (1); b14 := I(1)zx(1);b13 := zxxx(1)EI(1) + zxx(1)EI (1);
2 b21 := zx (1)EI(1);b22 := zxx(1)EI(1);b23 := zx(1): (2.23) If we replace by ' := h , then (2.18) changes to
f(4)(z)+a(z)f (z)+b(z)f (z)+c(z)f (z) − '2[f (z)+d(z)f (z) − e(z)f(z)]=0;
−1 f(0)=0;f(0)=0;b21f (1) + b22f (1) + b23 h 'f (1)=0;
−2 2 −1 b11f (1) + b12f (1) + b13f (1) − h ' b14f (1) − ÿh 'f(1)=0; (2.24) which is equivalent to Eq. (2.16). In summary, we have the following result.
Theorem 2.1. ∈ (A) if and only if Eq. (2:24) has a nonzero solution f(z) for ' := h . In addition, the function (x) in the corresponding eigenfunction (; ) of A is given by (2:17).
3. Asymptotic expressions of eigenfrequencies
In this section, we shall obtain asymptotic expansions for the eigenvalues of A. The main trick is to treat the fundamental solutions of (2.24) ÿrst, and then use them to expand the characteristic determinant of A and obtain the asymptotic eigenfrequency. To begin, we use a standard technique of Naimark [7] (see also [3]) and divide the complex plane into four sectors k) (k +1)) S := z ∈ C: 6 arg z 6 ;k=0; 1; 2; 3 (3.1) k 2 2 and for each Sk , we will pick !1 and !2 to be the square roots of −1 so that
Re(!1) 6 Re(!2); ∀ ∈ Sk : (3.2)
i)=2 i(3=2)) In particular, we will choose !1 := e ;!2 := e in sector S0 and re-shuOe them in each the remaining sectors so that (3.2) holds. Writing ' := !1 for in each sector Sk , we have the following result on the fundamental solutions of (2.24) from [12, Theorem 3] (see also [1]).
Lemma 3.1. In each sector Sk , for ∈ Sk with || su (j) (j) −2 ys (z; )=hs (z)+O( );s=1; 2; (3.4) (j) j !s−2x −1 ys (z; )=(!s−2) e [y0(z)+O( )];s=3; 4; (3.5) −(1=2) z (a(t)−d(t)) dt y0(z):=e 0 : (3.6) Here, h1(z):=h1(x(z)), h2(z):=h2(x(z)) are the two linearly independent solutions of (2.13) after the transformation x(z):=z(x)−1. That is, they are two linearly independent solutions of f (z)+d(z)f (z) − e(z)f(z)=0: From (3.4) and (3.5), we can obtain asymptotic expansions for the boundary conditions of system (2.24). For brevity, we shall use the following notation in the sequel: −1 [a]1 := a + O( ): Theorem 3.1. Denote the boundary conditions of system (2:7) respectively by U1, U2, U3 and U4. Then, for ∈ S0 with || su b22 b23 −1 U2(ys; )= ys (1; )+ ys(1; )+ i h ys(1; ) b21 b21 b23 −1 −1 i h xz(1)hs(1) + O( ) ;s=1; 2; b21 = b 2 !s−2 2 23 −1 −1 e y0(1)!s−2 +i h y0(1)!s−2 + O( ) ;s=3; 4; b21 b i 23 h−1x (1)h (1) ;s=1; 2; z s b21 1 := (3.9) b 2 !s−2 2 23 −1 e y0(1)!s−2 +i h y0(1)!s−2 ;s=3; 4; b21 1 b b b ÿ U (y ; )=y (1; )+ 12 y (1; )+ 13 y (1; )+ 2 14 y (1; ) − i y (1;) 1 s s s s 2 s s b11 b11 h b11 hb11 40 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 2 b14 −2 −1 h xz(1)hs(1) + O( ) ;s=1; 2; b11 = b 3 !s−2 3 14 −2 −1 e y0(1)!s−2 + h y0(1)!s−2 + O( ) ;s=3; 4; b11 2 [xz(1)hs(1)]1;s=1; 2; := (3.10) 3 !s−2 3 e [y0(1)!s−2 + y0(1)!s−2]1;s=3; 4: Proof. The proof is just a direct substitution of the fundamental solutions (3.4) and (3.5) into the boundary conditions and makes use of the fact that in (3.10), b I (1)z (1) 14 = x = h2: 3 b11 zx (1)EI(1) Since the zeros of the characteristic determinant U4(y1;) U4(y2;) U4(y3;) U4(y4;) U3(y1;) U3(y2;) U3(y3;) U3(y4;) -():= (3.11) U2(y1;) U2(y2;) U2(y3;) U2(y4;) U1(y1;) U1(y2;) U1(y3;) U1(y4;) are the eigenvalues of (2.12) (see. [7, pp. 13–15]), to estimate the eigenvalues, we substitute (3.7)–(3.10) into -() and obtain the following asymptotic expansion for them. Theorem 3.2. In sector S0, the characteristic determinant -() of the characteristic Eq. (2:24) has an asymptotic expansion 5 −i i −1 -()=−i y0(1)xz(1)D{e (1 − .)+ e (1 + .)+O( )}; (3.12) −1=2 where . := (I(1)EI(1)) ;D:= (h2(1)h1(0) − h1(1)h2(0)) the non-zero determinant deÿned in (2.14). The asymptotic expansion (3.12) also holds in the other sectors as well. Furthermore, the boundary problem (2.24) is strongly regular in the sense of [11, p. 259] if and only if the following condition holds: −1=2 1 − (I(1)EI(1)) =0 (i:e: 1 − . =0): (3.13) Therefore when (3.13) holds, the zeros of -() are simple when their modulus are su Proof. In sector S0, with !1 := i;!2 := −i, we conclude that 3 !s−2 U1(ys;)= e [0]1;s=3; 4; (3.14) b 2 !s−2 s 23 −1 U2(ys;)= e −y0(1)+(−1) h y0(1) ;s=3; 4: (3.15) b21 1 Substituting (3.7)–(3.10), (3.14) and (3.15) into the characteristic determinant (3.11), we have [h (0)] [h (0)] 1 1 2 1 [x (0)h (0)] [x (0)h (0)] z 1 1 z 2 1 -()= [i b23 h−1x (1)h (1)] [i b23 h−1x (1)h (1)] b z 1 1 b z 2 1 21 21 2 2 [xz(1)h1(1)]1 [xz(1)h2(1)]1 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 41 [1] [1] 1 1 i[1] −i[1] 1 1 2e!1 [ − y (1) − b23 h−1y (1)] 2e!2 [ − y (1) + b23 h−1y (1)] 0 b 0 1 0 b 0 1 21 21 3 !1 3 !2 e [0]1 e [0]1 b b 5 !2 23 −1 !1 23 −1 = −i y0(1)xz(1)D e 1 − h +e 1+ h : b21 1 b21 1 Combining with (2.17) and (2.23), we have b z (1) 23 = x = h(I (1)EI(1))−1=2 = h.; (3.16) 2 b21 zx (1)EI(1) which yields (3.12). The strong regularity deÿned in [11, Deÿnition 2.7] together with the simplicity of the zeros with large enough moduli can be veriÿed directly from the fact that y0(1);xz(1) ¿ 0 and (2.14), (3.13). Remark 3.1. In Theorem 3.2, we show that the zeros of -() are simple provided || large enough. The theory of ordinary di9erential equation asserts that the algebraic multiplicity for an non-zero eigenvalue of an ordinary di9erential operator L with appropriate boundary condition is equal to the multiplicity for it as a zero of the characteristic determinant -() (see, [5, pp. 92–95]). So we can say that the eigenvalues of A are geometrically simple provided their moduli are large enough. On the other hand, it can be veriÿed that the eigenvalues of A with large moduli are also algebraically simple, which can be concluded from the general formula: ma 6 p · mg (see [6, p. 148]), where p denotes the pole of the resolvent operator and ma;mg denote the algebraic and geometric multiplicities, respectively. Theorem 3.3. Suppose that condition (3:13) is fulÿlled, then there exists an integer N¿0 such that the eigenvalues k of problem (2:12) have the following asymptotic behavior: 1 = 1 3 + k)i + O(k−1); |k| ¿ N; k ∈ Z; (3.17) k h 2 0 where .−1 1 1=2 ln ; .¿1; I( ) .+1 h := d and 30 := 1− . 0 EI( ) ln 1+ . + )i; .¡1: Also, . − 1 1 Re 3 =ln ¡ 0 and Re → Re 3 ¡ 0 as k →∞: (3.18) 0 . +1 k 2h 0 Proof. Since Eq. (2.12) is equivalent to (2.24), in sector S0, we see from (3.12) and (2.14) that equation -() = 0 becomes e−i(1 − .)+ei(1 + .)+O(−1)=0: (3.19) Solving the equation with lower order terms, e−i(1 − .)+ei(1 + .)=0 42 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 will yield solutions 1 '˜k =ik = 2 30 + k)i;k=1; 2;::: : (3.20) Applying Rouche’s theorem to (3.19), then there exists an integer N¿0 such that its solutions are expressed by 1 −1 'k = 2 30 + k)i+O(k );k¿ N; k ∈ N: (3.21) Note that the eigenvalues of A are distributed symmetrically with respect to the real axis. So the dual eigenvalues are 1 −1 'k = 2 30 − k)i+O(k );k¿ N; k ∈ N: (3.22) Hence, we can conclude from (3.21), (3.22) and ' = h that 1 1 = ' = 1 3 + k)i + O(k−1); |k| ¿ N; k ∈ Z: (3.23) k h k h 2 0 So all eigenvalues of A are given by (3.23) and the proof is then completed. Remark 3.2. Form Theorem 3.3, we know that for |k| ¿ N, the eigenvalues are simple and the asymptotic expressions can be given by (3.23). But we cannot say that the low frequency of A has 2(N − 1) number of elements. Indeed, we do not know any exact number of the eigenvalues and the algebraic multiplicity of each eigenvalue in ÿnite domain. Even, so it is not an obstacle for the veriÿcation of Riesz basis property because only the asymptotic eigenfrequencies matter. All the above discussions can be summarized into the following result on the spectrum of A. Theorem 3.4. Let A be deÿned as in (2:6) and (2.7). Then each ∈ (A) is an eigenvalue and, when (3.13) holds, each is algebraically simple when | | is large enough, and has an asymptotic expression given by (3.17). 4. Completeness and Riesz basis property for system (2.8) In this section we will ÿrst establish the completeness of the generalized eigenfunctions of A, and then show that they form a Riesz basis. We begin with the following lemma. Lemma 4.1. Let A be deÿned as in (2:6) and (2:7) and 5k (k ∈ Z) be an numeration of all eigenvalues of A, and let 6¿0. Then there exists a constant M¿0 such that, for any ∈ (A) with | − 5k | ¿6, k ∈ Z, it holds that R( ; A) 6 M| |2: (4.1) Proof. Let ∈ (A) and (; ) ∈ H, we consider the resolvent equation [ I − A](f; g)=(; ); i.e., f − g = ; g + C−1(Af + Dg + ÿBg)= : (4.2) Simplifying the second equation in (4.2), we have [g − (Ig ) ]+(EIf ) =[ − (I ) ] J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 43 with boundary conditions f(0) = f (0)=0, EIf |x=1 + g (1) = 0 and (EIf ) |x=1 + I( − g) |x=1 − ÿg(1)=0: Thus g = f − and f satisÿes the following equations: 2 [f − (If ) ]+(EIf ) = F(·; ); (4.3) f(0) = f (0)=0; EIf |x=1 + f (1) = (1); (4.4) 2 (EIf ) (1) − I(1)f (1) − ÿ f(1) = −v( ); (4.5) where F(x; ):=[(x)( (x)+ (x)) − (I(x)[ (x)+ (x)]) ]; v( ):=I(1)( (1) + (1)) + ÿ(1): (4.6) Let yj(x; )(j =1; 2; 3; 4) be the fundamental solutions of the homogenous characteristic equation (2.12). Then any solution f(x)of(4.3)–(4.5) can be expressed by the formula 1 2 f(x; )= G(x; 3; )(F(3; )+ [(3)’(3) − (I(3)’ (3)) ]+ (EI(3)’ (3)) )d3 − ’(x); (4.7) 0 2 3 where ’(x):=c1x + c2x with c1;c2 being bound and given by c 2 − c − 6c c 2 + c + c c := 11 12 13 ;c:= 21 22 23 ; 1 c∗ 2 c∗ c11 := 3 I(1)((1) − (1));c12 := 2 ÿ(1) + I(1)(6 (1)EI(1)+3 (1)); c13 := (1)(EI(1) + EI (1)) + EI(1)(I(1) (1) + ÿ(1)); c21 := 2 I(1)( (1) − (1)); c22 := 2I(1)(EI(1) (1) + (1)) + ÿ(1); c23 := 2EI(1)(I(1) (1) + ÿ(1)) + 2 (1)EI (1); ∗ 2 2 c := (6EI(1)I(1) + ÿ)+ (4ÿEI(1)+6 EI (1)+12 EI(1)) + 12EI (1) and G(x; 3; ) is the Green’s function given by G(x; 3; ):=(1=-())H(x; 3; ) with y1(x; ) y2(x; ) y3(x; ) y4(x; ) Á(x; 3; ) U1(y1) U1(y2) U1(y3) U1(y4) U1(Á) H(x; 3; ):= U2(y1) U2(y2) U2(y3) U2(y4) U2(Á) ; (4.8) U3(y1) U3(y2) U3(y3) U3(y4) U3(Á) U4(y1) U4(y2) U4(y3) U4(y4) U4(Á) 4 1 Á(x; 3; ):=2 sign(x − 3) yj(x; )zj(3; ); (4.9) j=1 44 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 zj(x; ):=(Wj(x; )=W (x; )), W (x; ) the Wronskian determinant of {y; y2;y3;y4} and Wj(x; ) the cofactor of yj(x; )inW (x; ). Substituting the asymptotic expressions (3.4), (3.5) and (3.7)–(3.10) into (4.8) and (4.9), we obtain that for ∈ (A) with | | large enough, there exists a constant M independent of x; 3 ∈ [0; 1] so that @j H(x; 3; ) 6 M| |5+je| |;j=0; 1; 2; (4.10) @xj where we have used the relation !1 = h for all ∈ C. Also keeping in mind that ∈ (A) with | − 5k | ¿ 6; ∀k ∈ Z,by(3.12) we have @j G(x; 3; ) 6 M | |j;j=0; 1; 2; (4.11) @xj 1 where M1 is some constant that is independent of x; 3 ∈ [0; 1]. These will in turn yield estimates for f(x) and its derivatives, for j =0; 1; 2, 1 @j |f(j)(x)| 6 G(x; 3; )(F(3; )+ 2[(3)’(3) − (I (3)’ (3)) ]) d3 + |’(j)(x)| j 0 @x 1 j 2 (j) 6 M1| | |F(3; )+ [(3)’(3) − (I(3)’ (3)) ]| d3 + |’ (x)|: 0 Eventually, we have the following estimate on the resolvent operator: 1 1 2 2 2 2 (f; g) = EI(x)|f (x)| dx + (x)|g(x)| + I(x)|g (x)| dx 0 0 1 1 2 2 2 2 6 EI(x)|f (x)| dx + | | [(x)|f(x)| + I(x)|f (x)| ]dx +2V 0 0 2 4 2 2 6 M2 | |[W + v ]; (4.12) 2 where M2 is some constant independent on | |.SoR( ; A) 6 M2| | . Theorem 4.1. Let A be deÿned as in (2:6) and (2:7). If condition (3:13) is fulÿlled, then the system of the generalized eigenfunctions of A is complete in Hilbert space H. Proof. Let (A)={5n: n ∈ N} and Pn be the Riesz projection associated with 5n. Denote N ∗ Sp(A)= Pk y : y ∈ H; ∀N ∈ N and Q = {y ∈ H: Pk y =0; ∀k ∈ N}: k=1 So H has an orthogonal decomposition H = Sp(A) ⊕ Q and hence the system of generalized eigenfunctions of A is complete in H if and only if Q = {0}. Now for any z ∈ Q, R( ; A∗)z is an entire function on the complex plane C valued in H. Since R( ; A∗)= R∗( ;M A) = R( ;M A), by Lemma 4.1, R( ; A∗)z has growth of order at most 2. Hence, R( ; A∗)z is a polynomial in of degree at most two, that is, ∗ −1 2 ∗ ( I − A ) z = d0 + d1 + d2 with dj ∈ D(A );j=0; 1; 2: J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 45 Thus, ∗ 2 z =( I − A )(d0 + d1 + d2) ∗ ∗ 2 ∗ 3 = −A d0 + (d0 − A d1)+ (d1 − A d2)+ d2; ∀ ∈ C: j Comparing the coe#cients of , we see that d0 = d1 = d2 = 0. Therefore z = 0, which implies Q = {0}. Theorem 4.2. Let X be a separable Hilbert space, and A be the generator of a C0 semigroup T(t). Suppose that the following three conditions hold: ∞ (1) We can decompose (A)= 1(A)∪ 2(A) so that 2(A)={ k }k=1 consists of only isolated eigenvalues of ÿnite multiplicity. (2) For ma( k ):=dimE( k ; A)X , where E( k ; A) is the Riesz projector associated with k , we have sup ma( k ) ¡ ∞: (4.13) k¿1 (3) There is a constant such that sup{Re | ∈ 1(A)} 6 6 inf {Re | ∈ 2(A)} (4.14) and inf | n − m| ¿ 0: (4.15) n=m Then the following assertions are true: (i) There exist two T(t)-invariant closed subspaces X and X such that (A| )= (A), (A| )= (A), 1 2 X1 1 X2 2 ∞ and {E( k ; A)X2}k=1 forms a Riesz basis of subspaces for X2. Furthermore, X = X1 ⊕ X2: (ii) If supk¿1 E( k ; A) ¡ ∞, then D(A) ⊂ X1 ⊕ X2 ⊂ X: (4.16) (iii) X has the topological direct sum decomposition X = X ⊕ X if and only if 1 2 n sup E( ; A) ¡ ∞: (4.17) k n¿1 k=1 Remark 4.1. Theorem 4.2 is a result from [15] which has not been appeared publicly. A brief outline of proof for Theorem 4.2 will be shown in the appendix for the sake of completeness. Combining Theorems 4.1, 4.2 together with Theorem 3.4, we have the following result. Theorem 4.3. Assume that (3:13) be fulÿlled. System (2:8) is a Riesz system (in the sense that its generalized eigenfunctions form a Riesz basis in H) and hence it satisÿes the spectrum determined growth condition. Proof. For system (2.8), from Theorems 3.3 and 3.4, we may take 2(A)= (A), 1(A)={∞}. Theorem 3.4 shows that conditions (2) and (3) in Theorem 4.2 are true. Finally, Theorem 4.1 implies that X1 = {0}. Therefore, the ÿrst assertion of Theorem 4.2 says that there is a sequence of generalized eigenfunctions of A that forms a Riesz basis for H. Since the spectrum determined growth condition is a direct consequence of the existence of a Riesz basis, the proof is completed. As a consequence of Theorem 4.3, we have a stability result for system (2.8). 46 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 Corollary 4.1. Let condition (3:13) be fulÿlled with ¿0 and ÿ ¿ 0. Then the system (2:8) is exponentially stable. Proof. Theorem 4.3 ensures the spectrum-determined growth condition !(A) = sup{Re ; ∈ (A)}, Lemma 2.4 says that Re ¡0 provided ∈ (A) and Theorem 3.3 shows that imaginary axis is not an asymptote of (A). Therefore sup{Re : ∈ (A)} ¡ 0. Remark 4.2. The special case that (x)=EI(x) ≡ 1 and I(x) ≡ .1 ¿ 0 was discussed in [2,8]. In this special case, expression (3.17) then becomes 1 1 −1 k = √ 31 + k)i + O(k ); |k| ¿ N; k ∈ Z (4.18) .1 2 with √ − . 1 √ ln √ ; ¿.1; + .1 3 = √ 1 .1 − √ ln √ + )i; ¡.1 + .1 and √ 1 − .1 Re k → √ ln √ ¡ 0;k→∞: 2 .1 + .1 √ So the closer to ∗ := . the larger the damping rate for system (1.1) which is the conjecture made in [2]. 1 √ However, when we set the control gain = .1, then (A) may contain at most ÿnitely many eigenvalues. This is because A has compact resolvent, so there are at most ÿnitely many eigenvalues inside each bounded set and (3.12) tells us that there are no eigenvalues at all when their moduli are large enough. It should point out that the asymptotic expression of the eigenvalues of A does not include explicitly the parameter ÿ, which means that the parameter ÿ does not take an essential role for the high frequency part of the spectrum of A. However, it plays an important role for the low frequency part of the spectrum of system (1.1). For instance, ∀ ¿0, we can tune the parameter ÿ appropriately so that the decay rate of the system is less than 5 := (1=2h)30, where 5 is the asymptote of the spectrum of A. Acknowledgements This work is supported by an RGC grant of code HKU 7133/02P and the authors would like to thank the referees for their very helpful suggestions and comments. AppendixA. An outline of a proof for Theorem 4.2 Proof of Theorem 4.2. Let X be a separable Hilbert space, and A be the generator of C0 semigroup T(t) and satisfy conditions (1–3) of Theorem 4.2. For any k ∈ 2(A), let mk denotes its ÿnite algebraical multiplicity and m := m ( ). Deÿne k a k m Sp 2 (A):= E( k ; A) |∀ ∈ X ; ∀m ∈ N (A.1) k=1 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 47 and X := Sp (A); (A.2) 2 2 then it is easy to see that X2 is a T(t)-invariant closed subspace, and hence it is also A-invariant. Note that (k) (k) t t m −1 t (4.13)–(4.15) imply that the family {E ( 2)}, where E ( 2):={e k ;te k ;:::;t k e k } with k ∈ N, forms k 2 a Riesz basis in space U := span{E ( 2); k ¿ 1}⊂L [0;T] for su#cient large T¿0. Since for any f ∈ X and ∈ Sp 2 (A), we have (T(t); f) ∈ U, so for any ∈ X2, we also have that (T(t); f) ∈ U and ∞ m k −1 tj (T(t); f)= e k t ((A − )jE( ; A); f): j! k k k=1 j=0 (k) By the property of Riesz basis of {E ( 2)} in U, there exists constants C1 and C2 such that ∞ m k −1 ((A − )jE( ; A); f) 2 T C k k 6 |(T(t); f)|2 dt 1 j! k=1 j=0 0 ∞ m k −1 ((A − )jE( ; A); f) 2 6 C k k : 2 j! k=1 j=0 Let T(t) satisÿes T(t) 6 Me!t, then we conclude from the left-hand side of above inequalities that ∞ m k −1 ((A − )jE( ; A); f) 2 e2!T − 1 C k k 6 M 2 2f2; (A.3) 1 j! 2! k=1 j=0 which implies that (by taking mk =1) ∞ e2!T − 1 C |(E( ; A); f)|2 6 M 2 2f2 1 k 2! k=1 and ∞ e2!T − 1 C (E( ; A))2 6 M 2 2: (A.4) 1 k 2! k=1 Since T(t) is also a C 0 semigroupon X2 and its generator is A|X2 with domain D = D(A) ∩ X2, so each † † k ∈ 2(A) is an isolated eigenvalue of A|X2 with ÿnite algebraical multiplicity mk . Let A and T (t)be the adjoint operators of A and T(t) restricted to X2, respectively. Note that X2 endowed with ·X is a † † M † Hilbert space, so T (t)isaC0 semigroupand its generator is A . Then each k ∈ (A ) is also an isolated † † M † eigenvalue of A with ÿnite algebraical multiplicity mk . Moreover, we have that E ( k ; A)=E( k ; A ) and † for any f; g ∈ X2,(T (t)g; f)=(g; T(t)f) ∈ U with ∞ m k −1 j t M (T †(t)g; f)= e k t((A† − M )jE( M ; A†)g; f): j! k k k=1 j=0 Under the same argument as (A.4), we conclude that ∞ e2!T − 1 C E†( ; A)g2 6 M 2 g2: (A.5) 1 k 2! k=1 48 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 N 2 For any ∈ Sp (A), = E( k ; A), we conclude from E ( k ; A)=E( k ; A) that 2 k=1 N 2 =(; )d = E( k ; A); k=1 N † = (E( k ; A); E ( k ; A)) k=1 N 1=2 N 1=2 2 † 2 6 E( k ; A) E ( k ; A) k=1 k=1 1=2 N 2 2!T 1=2 2 M (e − 1) 6 E( k ; A) ; 2!C1 k=1 i.e., 2 2!T m 2 M (e − 1) 2 6 E( k ; A) : 2!C1 k=1 By the limit process, we obtain 2 2!T ∞ 2 M (e − 1) 2 6 E( k ; A) ; ∀ ∈ X2: 2!C1 k=1 Combining this inequality and (A.4) yields that ∀ ∈ X2, 2!C ∞ M 2(e2!T − 1) ∞ 1 E( ; A)2 6 2 6 E( ; A)2: (A.6) 2 2!T k k M (e − 1) 2!C1 k=1 k=1 Therefore we conclude from (A.6) that {E( k ; A)X2}k¿1 is a Riesz basis of subspaces in X2, f = ∞ E( k ; A)f converges unconditionally in X2, and k=1 n sup E( ; A) ¡ ∞; (A.7) k n¿1 k=1 X2 where ·X2 denotes the norm in X2. Deÿne a subspace X1 = Q(A)ofX by X1 = Q(A):={g ∈ X | E( k ; A)g =0; ∀k ¿ 1}: Evidently, we have that X1 ∩X2 ={0} and X1 is a T(t)-invariant closed subspace. In order to prove X =X1 ⊕ X2, ∗ we only need to show that X1 + X2 is dense in X . For this we now consider the conjugate A of A because ∗ ∗ ∗ M T (t) is also a C0 semigroupon Hilbert space X with its generator A . Note that (A )={ | ∈ (A)} = M ∗ 1(A) ∪ 2(A), k ∈ 2(A) is an isolated eigenvalue of A with ÿnite algebraical multiplicity mk , and ∗ M ∗ E ( k ; A)=E( k ; A ), then we deÿne ∗ ∗ X2 := span{E ( k ; A)X; k ¿ 1} and ∗ ∗ Q(A ):={g ∈ X | E ( k ; A)g =0; ∀k ¿ 1}; J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 49 ∗ ∗ ∗ ∗ ∗ so it is easy to see that X2 and Q(A ) are T (t)-invariant closed subspace and Q(A )∩X2 ={0}. Furthermore, we have ∗ ˙ ˙ ∗ X = Q(A )+X2 and X = Q(A)+X2 ; where sign “+”˙ denotes the orthogonal sum. ∗ ∗ So, for any h ∈ X such that h ⊥ X1 + X2, we have h ⊥ X1 and h ⊥ X2. Hence, we obtain that h ∈ Q(A )∩X2 and h = 0, which shows X = X1 ⊕ X2. Obviously, (A|X2 )= 2(A), (A|X1 )= 1(A) and the ÿrst assertion (i) is concluded. Secondly, we suppose that supk¿1 E( k ; A)X = M1 ¡ ∞. Without loss of generality we assume that 0 ∈ (A). Then, for each f ∈ X , we have mk j−1 −1 j+1 (A − k ) E( k ; A)f E( k ; A)A f = (−1) j j=1 k and 2 mk j−1 2 mk −1 2 (A − k ) E( k ; A)f (j − 1)! E( k ; A)A f 6 : (j − 1)! j j=1 j=1 k Note that we conclude from (A.3) and (4.13) respectively that m k −1 (A − )jE( ; A)f 2 e2!T − 1 C k k 6 M 2 E( ; A)f2 1 j! 2! k j=0 and 2 mk (j − 1)! (N !)2 6 2 a ; for | | ¿ 2; j 2 k | k | j=1 k where Na := supk¿1 mk ¡ ∞. Hence we obtain that M 2(e2!T − 1) (N !)2 E( ; A)A−1f2 6 a E( ; A)f2 k 2 k !C1 | k | M 2(e2!T − 1) (N !)2 6 a M 2f2 2 1 !C1 | k | and by condition (4.15), the series ∞ E( ; A)A−1f2 is unconditionally convergent in X . Furthermore, k=1 k ∞ −1 we conclude from the ÿrst assertion (i) that k=1 E( k ; A)A f is an element of X2. Setting ∞ −1 −1 g = A f − E( k ; A)A f; k=1 we have g ∈ Q(A). So D(A) ⊂ X1 ⊕ X2 and the second assertion (ii) is followed. Finally, we suppose that X has a direct sum decomposition X = X1 ⊕ X2. Deÿne projection P from X onto X2 along X1, then P is a bounded operator on X . For each f ∈ X , we have Pf ∈ X2 and n n n E( ; A)Pf 6 E( ; A) Pf 6 E( ; A) Pf: k k k k=1 k=1 k=1 X2 X2 50 J. Wang et al. / Systems & Control Letters 51 (2004) 33–50 So, together with (A.7) we have n sup E( k ; A) ¡ ∞: (A.8) n¿1 k=1 X Conversely, if (A.8) holds, then (A.4) and (A.8) imply that for any n ∈ N;f∈ X , n n 2 (e2!T − 1) E( ; A)f2 6 E( ; A)f k −2 k 2M C1! k=1 k=1 n 2 (e2!T − 1) 6 E( ; A) f2: −2 k 2M C1! k=1 X Therefore, ∞ 2 E( k ; A)f ¡ ∞; k=1 ∞ which shows that for any f ∈ X , the series k=1 E( k ; A)f converges unconditionally in X2. Taking ∞ f1 = f − E( k ; A)f; k=1 then, f1 ∈ Q(A)=X1. Hence X = X1 ⊕ X2 and the proof is completed. References [1] W. Eberhard, G. Freiling, Stone-regulXare Eigenwertprobleme, Math. Z. 160 (1978) 139–161. [2] B.Z. Guo, Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl. 112 (3) (2002) 529–547. [3] B.Z. Guo, Riesz basis property and exponential stability of controlled Euler–Bernoulli beam equations with variable coe#cients, SIAM J. Control Optim. 40 (6) (2002) 1905–1923. [4] Z. Liu, S. Zheng, Semigroups Associated with Dissipative System, in: Research Notes in Mathematics, Vol. 398, Chapman & Hall/CRC, London/Boca Raton, FL, 1999. [5] J. Locker, Spectral Theory of Non-self-adjoint Two-point Di9erential Operators, American Mathematical Society, Providence, RI, 2000. [6] Z.H. Luo, B.Z. Guo, O. Morgul,X Stability and Stabilization of Inÿnite Dimensional Systems with Applications, Springer, London, 1999. [7] M.A. Naimark, Linear Di9erential Operators, I, Frederick Ungar, New York, 1967. [8] B.P. Rao, A compact perturbation method for the boundary stabilization of the Rayleigh beam equation, Appl. Math. Optim. 33 (1996) 253–264. [9] D.L. Russell, Mathematical models for the elastic beam and their control-theoretic implications, in: H. Brezis, M.G. Crandall, F. Kapell (Eds.), Semigroups, Theory and Applications, Vol. II, Longman Sci. Tech., Harlow, 1986, pp. 177–216. [10] M. Shubov, Exact controllability of damped Timoshenko beam, IMA J. Math. Control Inform. 17 (2000) 375–395. [11] C. Tretter, Kamke problems. Properties of the eigenfunctions, Math. Nachr. 170 (1994) 251–275. [12] C. Tretter, On fundamental systems for di9erential equations of Kamke type, Math. Z. 219 (1995) 609–629. [13] J.M. Wang, Q.P. Vu, S.P. Yung, Asymptotic frequency distributions for variable coe#cients Rayleigh beams under boundary feedback control, Proceedings of the ACC’03, 2003, pp. 113–118. [14] G.Q. Xu, D.X. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application, IMA J. Appl. Math. 67 (2002) 357–390. [15] G.Q. Xu, S.P. Yung, The expansion of semigroup and criterion of Riesz basis, Research Report, Mathematics Department, University of Hong Kong, July 2003. [16] K. Yosida, Functional Analysis, 6th Edition, Springer, Berlin, 1999.