Stability of a Complex Network of Euler-Bernoulli Beams

Transcription

1 Stability of a Complex Network of Euler-Bernoulli Beams Kui Ting Zhang Tianjin University Department of Mathematics Tianjin, 372 China zhangkt@tjueducn Gen Qi Xu Nikos E Mastorakis Tianjin University Military Institutes of University Education Department of Mathematics Hellenic Naval Academy Tianjin, 372 Terma Hatzikyriakou, Piraeus, China Greece gqxu@tjueducn mastor@hnagr Abstract: A complex network of Euler-Bernoulli beams is studied in this paper As for this network, the boundary vertices are clamped, the displacements of the structure are continuous but the rotations of different beams are not continuous at the interior vertices The feedback controller are designed at the interior nodes to stabilize the elastic system The well-posed-ness of the closed loop system is proved by the semigroup theory By complete spectral analysis of the system operator, the distribution of spectrum, the completeness the Riesz basis property of the roots vectors of the system operator are given As a consequence, the asymptotical stability of the system is derived under certain conditions Key Words: Euler-Bernoulli beam, network, spectral analysis, stability Introduction Many authors have studied the networks of various flexible elements in recent years For a multi-link flexible structures, one studies not only its global dynamic motion but also has to take into account the interaction transmission of elastic elements For examples, Dekoninck Nicaise in [] [2] studied control eigenvalue problems of network of Euler- Bernoulli beams; Ammari Jellouli in [3] [4] studied the stabilization problem of tree-shaped network of strings; Xu et al in [5] studied an abstract second order hyperbolic system applied the result to controlled network of strings Xu Yung in [6] studied a star-shaped network of Euler-Bernoulli beams with boundary dampings Xu Mastorakis in [7] [8] studied the stability spectral distribution of a hybrid network of strings Euler beams with boundary dampings Guo Wang made contributions to the stabilization of a tree-shaped network with three beams in [9], [], [] A general flexible network of beams was investigated by Mercier Regnier in [2] [3] The more general differential equation on networks can refer to [4] In the present paper, we shall study a complex network of Euler-Bernoulli beams which is different from those mentioned above For the network, we at first design the nodal feedback controllers to stabilize the system, then analyze the closed loop system Here we shall carry out a complete spectral analysis for the closed loop system It is well known that the calcu- Corresponding author lation of the spectrum of a elastic network has been a tough work because of its complexity In the present paper, we employ the asymptotic analysis technique to get spectral distribution, the completeness Riesz basis property of the roots vectors of the system operator Further,we derive the asymptotic stability of the system Let us introduce the network under consideration Let G be a plane graph of the form as shown Fig in which the edge set E {γ, γ 2,, γ 5 } The arc γ 2 γ 3 γ 4 γ Figure : A network of Euler-Bernoulli beams length of each edge is We parameterize each edge by its arc length s as π j : [, ] γ j, γ 5 x γj π j (s) so that the graph becomes a metric graph Suppose that the equilibrium position of the elastic structure coincides with G The elastic structure undergoes a small vibration, denote by w(x, t) the displacement depart from its equilibrium position in position x at time t On each edge γ j, j, 2,, 5, ISSN: Issue 3, Volume 8, March 29

2 we parameterize it by w j (s, t) w(π j (s), t) Then it satisfies the differential equation w j,tt (s, t) + a j w j,ssss (s, t), s (, ) Suppose that the boundary vertices of the structure are clamped, at each interior vertex (ie, common node) the elastic elements are connected through some conditions on w j (s, t) its derivatives w j,s l, l, 2, 3, 4 More precisely, we simply impose the continuity of w j (s, t) w j,s (s, t) (representing the rotation of the beam γ j ) is not continuous at the common nodes There are two exterior forces at the common node 3 γ j, which is parameterized as ; At the common node 5 j3 γ j, here we parameterize it as, there is only one extra force By the action of the extra forces, the motion of the elastic structure are governed by the partial deferential equations w j,tt (s, t) + a j w j,ssss (s, t), j, 2,, 5 w (, t) w 2 (, t) w 3 (, t) a i w i,sss (, t) i w 3 (, t) w 4 (, t) w 5 (, t) a i w i,sss (, t) () i3 w,ss (, t) u (t), w (, t) w,s (, t) w 2,ss (, t) w 2 (, t) w 2,s (, t) w 3,ss (, t) u (t), w 3,ss (, t) u 2 (t) w 4,ss (, t) w 4 (, t) w 4,s (, t) w 5,ss (, t) w 5 (, t) w 5,s (, t) Obviously, this is a complex network of Euler- Bernoulli beams Our purpose in the present paper is to design the feedback controllers to stabilize the system The rest is organized as follows In next section we design the feedback controllers then discuss the well-posed-ness of the closed loop system In section 3, we carry out a complete asymptotic analysis for the spectrum of operator determined by the closed loop system we get the spectral distribution, then prove the completeness the Riesz basis property of root vectors (the eigenvectors generalized eigenvectors) of the system operator Finally, in section 4, we discuss the stability of the system Under the certain conditions we derive the asymptotic stability of the system 2 Design of Controllers Wellposed-ness In this section, we shall design the feedback controllers for the system () then study the wellposed-ness of the closed loop system Let us consider the energy function of (), which is given by E(t) 2 [a j w 2 j,ss(s, t) + w 2 j,t(s, t)]ds Differentiate it formally with respect to time t, we have de(t) a w,ss (, t)w,st (, t) dt a 3 w 3,ss (, t)w 3,st (, t) + a 3 w 3,ss (, t)w 3,st (, t) a u (t)w,st (, t) a 3 u (t)w 3,st (, t) +a 3 u 2 (t)w 3,st (, t) So designing controllers as u (t) w,st (, t), u (t) w 3,st (, t), u 2 (t) w 3,st (, t) yields de(t) dt With these feedback controllers, the model () becomes a closed-loop system w j,tt (s, t) + a j w j,ssss (s, t), j, 2,, 5 w (, t) w 2 (, t) w 3 (, t) a i w i,sss (, t) i w 3 (, t) w 4 (, t) w 5 (, t) a i w i,sss (, t) i3 (2) w,ss (, t) w,st (, t), w (, t) w,s (, t) w 2,ss (, t) w 2 (, t) w 2,s (, t) w 3,ss (, t) w 3,st (, t), w 3,ss (, t) w 3,st (, t) w 4,ss (, t) w 4 (, t) w 4,s (, t) w 5,ss (, t) w 5 (, t) w 5,s (, t) Now we formulate the system (2) into an appropriate Hilbert state space Let L 2 (G) C(G) be defined as usual, we define linear space H k (G a ), k N, by H k (G a ) { f L 2 (G) f γj f j H k (, ) } A function w(x, t) is said to be a solution to (2), it means that, for each t >, w(x, t) H 4 (G a ) w(x, t) is continuously differentiable with respect to t w tt (x, t) exists in L 2 (G) Denote the space W by ( H 2 (, ) L 2 [, ] ) 5 let the state space be H (f, g) W f () f 2 () f 3 (), f () f 2 (), f 3 () f 4 () f 5 (), f 4 () f 5 (), f,s () f 2,s (), f 4,s () f 5,s () ISSN: Issue 3, Volume 8, March 29

3 equipped with inner product (f, g), (u, v) H [a j f j,ss (s)u j,ss (s) + g j (s)v j (s)]ds Obviously, H is a Hilbert space Denote the space V by (H 4 (, ) H 2 (, )) 5 define the operator A in H by (w, z) V, a l w l,sss (), l w,ss () z,s (), w 3,ss () z 3,s () D(A) w 2,ss (), a l w l,sss (), l3 w 3,ss () z 3,s (), w 4,ss () w 5,ss () A(w, z) {(z j, a j w j,ssss )} H, (w, z) D(A) With the help of these notations we can rewrite (2) into an evolutionary equation in H dw (t) AW (t), t > dt (3) W () W H where W (t) {(w j (s, t), w j,t (s, t)}, W H is the initial data given In what follows, we shall discuss the well-poseness of the system (3) Firstly, we have the following result Theorem Let A be defined as above Then A is dissipative, ρ(a) A is compact Hence the spectrum of A consists of all isolated eigenvalues of finite multiplicity Proof: For any real (w, z) D(A), we have A(w, z), (w, z) H [ aj z j,ss (s)w j,ss (s) a j w j,ssss (s)z j (s) ] ds a j w j,sss ()z j () a w,ss ()z,s () a 3 w 3,ss ()z 3,s () +a 3 w 3,ss ()z 3,s () a j w j,sss ()z j () j3 a z 2,s() a 3 z 2 3,s() a 3 z 2 3,s() The dissipatedness of A follows For any (f, g) H, we consider the resolvent equation A(w, z) (f, g), ie, z (s) f (s) z 2 (s) f 2 (s) a w,ssss (s) g (s) a 2 w 2,ssss (s) g 2 (s) w () w 2 () w 2 () w 3 () w,ss () z,s () w 2,ss () w () w 2 () w,s () w 2,s () z 3 (s) f 3 (s) a 3 w 3,ssss (s) g 3 (s) a j w j,sss () (4) w 3,ss () z 3,s () a j w j,sss () j3 w 3,ss () z 3,s () z 4 (s) f 4 (s) z 5 (s) f 5 (s) a 4 w 4,ssss (s) g 4 (s) a 5 w 5,ssss (s) g 5 (s) w 4 () w 3 () w 5 () w 4 () w 4,ss () w 5,ss () w 4 () w 5 () w 4,s () w 5,s () Solving the differential equation in (4) yields w j (s) w j () + sw j,s () + s2 2 w j,ss() + s3 3! w j,sss() s (s r) 3 3!a j w j,s (s) w j,s () + sw j,ss () + s2 2 w j,sss() s (s r) 2 2!a j w j,ss (s) w j,ss () + sw j,sss () s w j,sss (s) w j,sss () s Thus g j (r) a j dr w j () w j () + w j,s () + 2 w j,ss() + 6 w j,sss() ( r) 3 3!a j w j,s () w j,s () + w j,ss () + 2 w j,sss() ( r) 2 2!a j w j,ss () w j,ss () + w j,sss () w j,sss () w j,sss () g j (r) a j dr (5) s r a j r a j Substituting above into the boundary conditions in (4), we can determine the w j (), w j,s (), w j,ss () w j,sss () Substituting them into (5), we can get unique functions w j (s), j, 2,, 5 Set z f, then we have (w, z) D(A) A(w, z) (f, g) The Inverse Operator Theorem asserts ρ(a) Note that D(A) H 2 (G a ) H 4 (G a ) The Sobolev s embedding Theorem ensures that A is compact on H ISSN: Issue 3, Volume 8, March 29

4 As a direct consequence of Theorem Lumer-Phillips Theorem (eg see,[9]), we have the following result Corollary 2 Let A be defined as above Then the system (3) is well-posed in H 3 Spectral Analysis In order to investigate the properties of the semigroup T (t) generated by A, we need to find out some spectral properties of A From Theorem, we know that σ(a) σ p (A) In this section, we shall discuss the eigenvalue problem of A give its asymptotical distribution Let λ C, we consider the solvability of nonzero solution of the equation A(w, z) λ(w, z) in H For the sake of simplicity, we assume a i, i, 2,, 5 From the representation of A, we know that A(w, z) λ(w, z) implies that z λw w satisfy the differential equations w j,ssss (s) λ 2 w j (s), j, 2, 3, 4, 5 w () w 2 () w 3 () w i,sss () i w 3 () w 4 () w 5 () w i,sss () i3 w,ss () λw,s (), w () w,s () w 2,ss () w 2 () w 2,s () w 3,ss () λw 3,s (), w 3,ss () λw 3,s () w 4,ss () w 4 () w 4,s () w 5,ss () w 5 () w 5,s () (6) In order to discuss the existence of nonzero solution of (6) distribution of the eigenvalues of A, we divide the discussion into the following several subsections 3 Fundamental matrix distribution of eigenvalues In this subsection we discuss existence of nonzero solution of (6) distribution of the eigenvalues of A To this end, we formulate equations (6) into the normalized boundary value problem Let { Wj [w j, w j,s, w j,ss, w j,sss ] T, j, 2,, 5 W ( ) [W, W 2, W 3, W 4, W 5 ] T (7) Then (6) can be written as dw (s) A(λ)W (s) ds B (λ)w () + B (λ)w () (8) where A (λ) A 2 (λ) A(λ) A 3 (λ) A 4 (λ) A 5 (λ) B(λ) C B2(λ) C B (λ) D D 2 B3(λ) B4(λ) B5(λ) B(λ) B2(λ) B (λ) B3(λ) D 4 D 5 C 2 B4(λ) C 2 B5(λ) with A j (λ), j, 2,, 5 λ 2 B(λ) λ, B (λ) B2(λ), B 2(λ) B3(λ) λ, B 3(λ) λ B4(λ), B 4(λ) B5(λ), B 5(λ) C, D i, i, 2 ISSN: Issue 3, Volume 8, March 29

5 C 2, D j, j 4, 5 Lemma 3 Let A be defined as before, then the spectrum of A distributes symmetrically with respect to the real axis Moreover λ σ(a) if only if there is a non-trivial solution W (s) to (8) Thanks to Lemma 3, we consider only λ that are located in the first second quadrant of the complex plane Set λ iρ 2, ρ C If argλ < π 2, choosing 3π 4 argρ < π, then we have R(ρ) <, R( ρ) >, R(iρ) <, R( iρ) > (9) if π 2 argλ < π, taking argρ < π 4, we have R(ρ) >, R( ρ) <, R(iρ), R( iρ) () Define invertible matrices by P (ρ) where P j (ρ) Then, we have P (ρ) P 2 (ρ) P 3 (ρ) P 4 (ρ) P 5 (ρ) ρ ρ ρ ρ ρ 2 ρ 2 iρ 2 iρ 2 ρ 3 ρ 3 ρ 3 ρ 3 ρ 4 ρ 4 iρ 4 iρ 4 P j (ρ)a j (iρ 2 )P j (ρ), j, 2,, 5 ρ ρ iρ iρ Lemma 4 Let P (ρ) be defined as above, P (ρ)e(s) is a fundamental solution matrix to system (8), where E (s) E 2 (s) E(s) E 3 (s) () E 4 (s) E 5 (s) with e ρs E j (s) e ρs e iρs, j, 2,, 5 e iρs Proof: The inverse matrix of P (ρ) is given by P (ρ) Set (ρ) P2 (ρ) P3 (ρ) P4 (ρ) P5 (ρ) P Ŵ (s) P (ρ)w (s) (2) Â(iρ2 ) P (ρ)a(iρ 2 )P (ρ), then we have Since with dŵ (s) ds Â(iρ2 )Ŵ (s) (3) Â(iρ 2 )P (ρ)a(iρ 2 )P (ρ) Â (iρ 2 ) Â 2 (iρ 2 ) Â 3 (iρ 2 ) Â 4 (iρ 2 ) Â 5 (iρ 2 ) Â j (iρ 2 ) P j (ρ)a(iρ 2 )P j (ρ), j, 2, 5 Obviously, Â(iρ 2 ) is a diagonal matrix Hence, the fundamental matrix to (3) can be written as E(s) And by (2), we can assert that P (ρ)e(s) is a fundamental matrix to system (8) Set Ω(ρ) B (iρ 2 )P (ρ)e()+b (iρ 2 )P (ρ)e() (4) Before calculating the determinant of Ω(ρ), we put forward some decompositions of E() E() as E() E E 2, E() E E 2 where E, E, E 2 are diagonal matrices similar to E(s) with their elements Ej, E j, E2 j, j, 2,, 5, they will be defined according to the sign of R(λ), respectively When R(λ) >, according to (9), we have e ρ, e iρ as R(λ) + ISSN: Issue 3, Volume 8, March 29

6 the entries Ej i, i,, 2, j, 2,, 5 are defined as E 2 j E j E j e ρ e iρ e ρ e iρ e ρ e iρ When R(λ) <, we have we define,,, j, 2,, 5 e ρ, e iρ as R(λ) E 2 j E j E j Thus we get e ρ e iρ e ρ e iρ e ρ e iρ,,, j, 2,, 5 Ω(ρ) [B (iρ 2 )P (ρ)e +B (iρ 2 )P (ρ)e ]E 2 (5) Obviously,(8) has a non-zero solution if only if det Ω(ρ) Now we set (ρ) det[ω(ρ)]/ det[e 2 ] det[b (iρ 2 )P (ρ)e + B (iρ 2 )P (ρ)e ] In order to calculate (ρ), we need some transformations Let [ ] [ ] B ˆB (iρ 2 ) (iρ 2 ) B I, ˆB (iρ 2 ) (iρ 2 ) [ ] [ ] I E Ê E E3, Ê [ ] E3 P (ρ) ˆP (ρ) P 3 (ρ) where ˆΩ(ρ) ˆB (iρ 2 ) ˆP (ρ)ê + ˆB (iρ 2 ) ˆP (ρ)ê (6) I, I Set ˆ (ρ) det[ˆω(ρ)], obviously (ρ) ˆ (ρ)/ det[i P 3 (ρ)e3 + I P 3 (ρ)e3 ] ˆ (ρ)/[2iρ + O(ρ 9 )] After some elementary transformation to ˆΩ(ρ), we get where M L M 2 L 2 ˆ (ρ) M 3 L 3 M 4 L 4 M 5 L 5 N N 2 N 3 N 4 N 5 Q M j Bj (iρ2 )P j (ρ)ej + B j (iρ2 )P j (ρ)ej, j, 2, 4, 5 M 3 iρ 2 P 3(ρ)E3 + P 3(ρ)E3 iρ 2 N j P j(ρ)ej + P j(ρ)ej, j, 2 N 3 P 3(ρ)E3 + P 3(ρ)E3 N j P j (ρ)ej + P j(ρ)ej, j 4, 5 ISSN: Issue 3, Volume 8, March 29

7 L j P 3(ρ)E3 + P 3 (ρ)e3, j, 2 L 3 P 3(ρ)E3 + P 3(ρ)E3 L j P 3 (ρ)e3 + P 3(ρ)E3, j 4, 5 Q P 3(ρ)E3 + P 3(ρ)E3 Furthermore, M ˆ (ρ) 5 det(m j ) ˆ (ρ) M2 M5 I I L 5 I L 2 det(m j ) I L 5 N M N 2M2 N 5 M5 Q I L I L 2 5 det(m j ) I L 5 Q 5 N j M j L j 5 det(m j ) det[q 5 A direct computation shows So Thus N j M j L j ] ˆ (ρ) 5 det(m j) det[q 5 { N jm j L j ] 28( i)ρ 55 + O(ρ 54 ), R(λ) >, 256( + i)ρ 55 + O(ρ 54 ), R(λ) < (ρ) { ˆ (ρ)/[2iρ + O(ρ 9 )] 64( + i)ρ 45 + O(ρ 44 ), R(λ) >, 28( i)ρ 45 + O(ρ 44 ), R(λ) < det[ω(ρ)] (ρ) det(e 2 ) (ρ) 5 { det(e2 j ) [64( + i)ρ 45 +O(ρ 44 )]e 5(+i)ρ, R(λ)>, [ 28( i)ρ 45 +O(ρ 44 )]e 5( i)ρ, R(λ)< Therefore, we have lim R(λ) + lim R(λ) (7) det[ω(ρ)] ρ 45 64( + i) (8) e 5(+i)ρ det[ω(ρ)] ρ 45 28( i) (9) e5( i)ρ Hence, there are positive constants c, c 2 h such that c ρ 45 e 5(i±)ρ det[ω(ρ)] c 2 ρ 45 e 5(i±)ρ (2) where ± signr(λ), R(λ) h Eqs(2) shows that det[ω(ρ)] is a sine type function on C (see [6, Definition II, 27, pp6]) Levin theorem (see [6, Proposition II, 28]) asserts that the set of zeros of det[ω(ρ)] is a union of finitely many separable sets From the above analysis, we have the following result Theorem 5 Let A be defined as before, then the spectrum of A distributes in a strip parallel to the imaginary, that is, there is a positive constant h such that σ(a) { λ C h R(λ) } In particular, σ(a) is an union of finite many separated sets Proof: Under the assumption, we always have inequality (2) This together with dissipatedness of A asserts that the spectrum of A distributes in a strip parallel to the imaginary axis In particular, the Levin s Theorem says that the set of zero of det[ω(ρ)] is an union of finite separated sets 32 Completeness basis property of eigenvectors In this subsection, we shall discuss the completeness Riesz basis property of (generalized) eigenvectors of A, Firstly, we establish the completeness of (generalized) eigenvectors of operator A then use the spectral distribution of A to obtain the Riesz property We need the following lemma Lemma 6 Let A be the generator of a C -semigroup in a Hilbert space H Assume that A is discrete for λ ρ(a ), R(λ, A ) is of the form R(λ, A )x G(λ)x F (λ), x H ISSN: Issue 3, Volume 8, March 29

8 where G(λ)x for each x H is a H-valued entire function with order less than or equal to ρ F (λ) is a scalar entire function of order ρ 2 Let ρ max{ρ, ρ 2 } < an integer n so that n ρ < n If there are n + rays γ j, j,,, n, on the complex plane argγ π 2 < argγ argγ 2 argγ n 3π 2 with arg γ j+ argγ j π n, j n so that R(λ, A )x is bounded on each ray argγ j, j n as λ for any x H, then Sp(A) Sp(A ) H where Sp(A) be the closed subspace spanned by all generalized eigenvectors of A Now we can prove the completeness of generalized eigenvectors of A Theorem 7 Let A be defined as before Then the generalized eigenvectors of A is complete in H, ie, Sp(A) H Proof: we prove the assertion by the following four steps: Step The adjoint operator of A, A, has the form: A (w, z) {(z j, a j w j,ssss )}, (w, z) D(A ) (w, z) V, a l w l,sss (), l w,ss () z,s (), w 2,ss (), D(A w ) 3,ss () z 3,s (), (2) a l w l,sss (), l3 w 3,ss () z 3,s (), w 4,ss () w 5,ss () This is a direct verification, we omit the detail Due to σ(a ) σ(a) the symmetry of σ(a) with respect to the real axis, we have σ(a ) σ(a) Step 2 Let A be the operator defined by A (w, z) {(z j, a j w j,ssss )}, (w, z) D(A ) (w, z) V, a l w l,sss () l w D(A ),ss () w 2,ss () w 3,ss () (22) a l w l,sss () l3 w 3,ss () w 4,ss () w 5,ss () Then A is a skew adjoint operator, ie, A A Step 3 For λ ρ(a) F H, R(λ, A)F R(λ, A )F are meromorphic function of finite exponential type, ie, ρ, ρ 2 since R(λ, A )F, R(λ, A)F consist of functions sinh iλ ω j, cosh iλ ω j, sin iλ ω j, cos iλ ω j as well as their integrations Step 4 The root vectors of A is complete in H In fact, we can assume without loss of generality that R ρ(a) Let λ R, for any given F H, set Y R(λ, A )F, Y 2 R(λ, A )F Φ Y Y 2 Denote Φ (w, z), Y 2 (u, v) Then Φ + Y 2 satisfies (2), Φ (w, z) satisfies z λw w satisfying w j,ssss (s) λ 2 w j (s), j, 2,, 5 w () w 2 () w 3 () w i,sss () i w 3 () w 4 () w 5 () w i,sss () i3 (23) w,ss () λw,s () v,s () w () w,s () w 2,ss () w 2 () w 2,s () w 3,ss () λw 3,s () v 3,s () w 3,ss () + λw 3,s () v 3,s () w 4,ss () w 4 () w 4,s () w 5,ss () w 5 () w 5,s () Therefore, we have [ Φ w 2 j,ss (s) + zj 2(s)] ds [ w 2 j,ss (s) + λ 2 wj 2(s)] ds w,ss ()w,s () w 3,ss ()w 3,s () +w 3,ss ()w 3,s () λw 2,s() + w,s ()v,s () + λw 2 3,s() +w 3,s ()v 3,s () + λw 2 3,s() + w 3,s ()v 3,s () To calculate w,s (), w 3s () w 3,s (), we assume that P (ρ)e(s) is the fundamental matrix, then Φ P (ρ)e(s)c Substitute it into the boundary conditions of (6) use (4), we get [ B (iρ 2 )P (ρ)e + B (iρ 2 )P (ρ)e ] E 2 C F with F [V, V 2, V 3, V 4, V 5 ] T, in which V [, v,s (),, ],V 3 [, v 3,s (),, v 3,s ()] V 2 V 4 V 5 [,,, ] Thus C [E 2 ] [B (iρ 2 )P (ρ)e +B (iρ 2 )P (ρ)e ] F Set ρ a + bi, if λ are located in the first quadrant of the complex plane, ρ satisfies that 3π argρ < π 4 It s obvious that a < b > Therefore, e ρ, e iρ as λ + e 5(+i)ρ e 5(b a) 5(a+b)i e 5( b + a ) e 5 ρ ISSN: Issue 3, Volume 8, March 29

9 If λ are located in the second quadrant of the complex plane, ρ satisfies that argρ < π We have 4 e ρ, e iρ as λ + e 5( i)ρ e 5(a+b) 5(a b)i e 5( b + a ) e 5 ρ Similar to (7), there exists positive constants h M such that Ω(ρ) M ρ 45 e 5 ρ where λ > h Hence, Φ P (ρ) E(s) Ω(ρ) F 65 ρ 5 i M ρ 45 e 5 ρ F M F where λ > h M is a positive constant Since the operator V defined by V : F F V (R(λ, A )F ) C 2 is a bounded linear operator on H, there exists a constant M 2 such that F M 2 F So we have Φ M M 2 F Therefore, R(λ, A )F Y Y 2 + Φ R(λ, A )F + M M 2 F ( λ + M ) M 2 F, λ > h This means that R(λ, A ) is bounded on the negative real axis By now all conditions in Lemma 6 are verified The desired result follows from Lemma 6 To obtain the basis property of generalized eigenvectors of A, we need the following lemma, which from [7] [8] Lemma 8 Let A be the generator of a C -semigroup in a Hilbert space H Suppose that the following conditions are satisfied: ) The spectrum of A has the decomposition σ(a) σ (A) σ 2 (A); 2) There exists a real number α R such that sup{rλ; λ σ (A)} α inf{rλ; λ σ 2 (A)} 3) The set σ 2 (A) {λ k } k N consists of eigenvalues of A is essential space finite separated (or equivalent saying that it is an union of finitely many separated sets) Then there exists two T (t)-invariant closed subspaces H H 2 H {f H : E(λ, A)f, λ σ 2 (A)} H 2 span{ m E(λ k, A)H, m N} k H H2 {} with property that σ(a H ) σ (A) σ(a H2 ) σ 2 (A) Moreover, there exists a finite collection Ω k of elements in σ 2 (A), the corresponding Riesz projector E(Ω k, A) λ Ω k E(Ω, A) such that {E(Ω k, A)H 2 } k N forms a subspaces Riesz basis for H 2 As a direct application of Lemma 8, we have the following result Theorem 9 Let A be defined as above, then there exist a collection of generalized eigenvectors of A such that it forms a Riesz basis with parentheses for H Proof: Set σ 2 (A) σ(a) σ (A) { }, Theorem 5 ensures that all conditions in Lemma 8 are satisfied By Lemma 8, there is a finite collection Ω k of elements in σ(a) such that {E(Ω k, A)H 2 } k N forms a subspaces Riesz basis for H 2 Theorem 7 reads that H H 2 Therefore {E(Ω k, A)H} k N is a subspace Riesz basis 4 Stability Analysis In this section, we discuss the stability of the system Based on Theorem 9, we need only to determine the location of the eigenvalues of A Firstly we study eigenvalues of A on the imaginary axis Let λ ir, λ Since for any (w, z) D(A), we have R(A(w, z), (w, z)) a z 2,s() a 3 z 2 3,s() a 3 z 2 3,s() this implies that A(w, z) λ(w, z) has a nonzero solution if only if z,s () z 3,s () z 3,s (), z(x) λw(x), w(x) satisfy a j w j,ssss (s) λ 2 w j (s), j, 2,, 5 w () w 2 () w 3 () a i w i,sss () i w 3 () w 4 () w 5 () a i w i,sss () (24) i3 w,ss () w,s () w () w,s () w 2,ss () w 2 () w 2,s () w 3,ss () w 3,s () w 3,ss () w 3,s () w 4,ss () w 4 () w 4,s () w 5,ss () w 5 () w 5,s () ISSN: Issue 3, Volume 8, March 29

10 Set ω j 4 /a j, λ iρ 2, define functions: v j (s) 2 [cosh ω jρs + cos ω j ρs], v j2 (s) 2 [sinh ω jρs + sin ω j ρs], v j3 (s) 2 [cosh ω jρs cos ω j ρs], v j4 (s) 2 [sinh ω jρs sin ω j ρs] Then the general solution to the differential equation in (24) has the form w j (s) b j v j (s)+b j2 v j2 (s)+b j3 v j3 (s)+b j4 v j4 (s) (25) Substituting (25) into the boundary condition of w (s) yields b [v 4 ()v (s) v ()v 4 (s)], w (s) as sinh ω ρ sin ω ρ ; (26), otherwise substituting (25) into the boundary condition of w 3 (s) yields b 3 [v 33 ()v 3 (s) v 34 ()v 34 (s)], w 3 (s) as cosh ω 3 ρ cos ω 3 ρ ; (27), otherwise Similarly, we can get w 2 (s) b 23 [v 22 ()v 23 ( s) v 2 ()v 24 ( s)] w 4 (s) b 43 [v 42 ()v 43 (s) v 4 ()v 44 (s)] w 5 (s) b 53 [v 52 ()v 53 (s) v 5 ()v 54 (s)] We divide them into the following cases: Case I w (s) (or w 3 (s) ) From the connective condition w () w 2 () w 3 (), we can get a j w j,sss () b v 4 () b 3 v 34 () b 23 [v 22 ()v 23 () v 2 ()v 24 ()] (28) b v ()/ω +b 23 v 2 ()/ω 2 b 3 v 34 ()/ω 3 (29) If w (s), then w 3 () w () According to (28), b 3, that is w 3 (s) ; Similarly, if w 3 (s), w (s) holds And if w (s) or w 3 (s), using (29), we have b 23, that is w 2 (s) Further, using the connective conditions w 3 () w 4 () w 5 (), a j w j,sss () j3 we get b 43 [v 42 ()v 43 () v 4 ()v 44 ()] (3) b 53 [v 52 ()v 53 () v 5 ()v 54 ()] (3) b 43 [ v 42 ()v 44 () v 4 ()v 4 ()]/ω 4 +b 53 [v 52 ()v 54 () v 5 ()v 5 ()]/ω 5 (32) Obviously, if b 43 or b 53, then w 4 (s) w 5 (s) Hence, if v j2 ()v j3 () v j ()v j4 () 2 [sin ω jρ cosh ω j ρ cos ω j ρ sinh ω j ρ], j 4, 5, there exist nonzero solutions Define the solution set D {θ i tan θ i tanh θ i, i, 2, } If there are no θ m θ n such that θ m /θ n ω 4 /ω 5, eqs (24) has only zero solution Case II ρ R satisfies sinh ω ρ sin ω ρ, cosh ω 3 ρ cos ω 3 ρ In this case, one may have that w (s) w 3 (s) Then there exists a k N such that ω ρ kπ cosh ω 3 ω kπ cos ω 3 ω kπ From discussion we see that the following assertion holds true Theorem Let A ω j, D be defined as before, if the following conditions are satisfied: ) There are no θ m θ n such that θ m /θ n ω 4 /ω 5 2) There is no k N such that cosh ω 3 ω kπ cos ω 3 ω kπ Then there exist no eigenvalues on the imaginary axis, hence the system is asymptotically stable 5 Conclusion In this paper, we design the feedback controllers for a complex network of Euler-Bernoulli beams then proved that the closed loop system is well posed We show further that the root vectors of the system operator A are completeness, there exists a sequence of the root vectors that forms a Riesz basis for the state space H By the detailed analysis we get the asymptotic stability of the system Note that the Riesz basis property implies that the system satisfies the spectrum ISSN: Issue 3, Volume 8, March 29

11 determined growth condition Hence the system is exponentially stable if only if the imaginary axis is not the asymptote of the eigenvalues In further work we shall analyze the exponential stability of the system Acknowledgements: This research was supported by the Natural Science Foundation of China (grant NSFC ) References: [] B Dekoninck, S Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim Calc Var, Vol4, pp 57 8(999) [2] B Dekoninck, S Nicaise, The eigenvalue problem for networks of beams, Linear Algebia Appl Vol34, pp 65 89(2) [3] K Ammari M Jellouli, Stabilization of star-shaped tree of elastic strings, Differential Integral Equations, Vol7, pp 395 4(24) [4] K Ammari, M Jellouli M Khenissi, Stabilization of generic trees of strings, J Dynamical Control Systems, Vol, 2, pp77 93 (25) [5] G Q Xu, D Y Liu, Y Q Liu, Abstract second order hyperbolic system applications to controlled network of strings, SIAM J Conrtrol & Opthim Vol47, Issue 4, pp (28) [6] GQ Xu, S P Yung, Stability Riesz basis property of a Star-shaped network of Euler- Bernoulli beams with joint damping, Networks Heterogeneous Media, Vol 3, No4 28,pp [7] G Q Xu, N E Mastorakis, Stability of a starshaped coupled network of strings beams, Proceedings of the th WSEAS International Conference on Mathematical Methods Computational Techniques in Electrical Engineering, Sofia, Bulgaria Pages 48 54(28) [8] G Q Xu, N E Mastorakis, Spectral distribution of a star-shaped coupled network, WSEAS Transactions on Applied Theoretical Mechanics Vol3, Issue 4, pp (28) [9] B Z Guo, J M Wang, Dynamic stabilization of an Euler-Bernoulli beam under boundary control non-collocated observation, Systems & Control Letters, Vol 57, No, pp (28) [] B Z Guo, Riesz basis property exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM J Control Optim, Vol 4, No 6, pp (22) [] JM Wang, BZ Guo, Riesz basis stabilization for the flexible structure of a symmetric tree-shaped beam network, Math Method Appl Sci, Vol 3, No 8, pp (28) [2] D Mercier, V Regnier, Spectrum of a network of Euler-Bernoulli beams, J Math Anal Appl Vol 337, No, pp 74 96(28) [3] D Mercier, V Regnier, Control of a network of Euler-Bernoulli beams, J Math Anal Appl Vol 342, No, pp (28) [4] Yu V Pokornyi, A V Borovskikh, Differential equations on networks (Geometric Graphs), Journal of Mathematical Sciences, Vol 9, No 6, pp 69 78(24) [5] V Chiroiu, On the eigenvalues optimization of beams with damping patches, WSEAS Transactions on Applied Theoretical Mechanics, Vol3, Issue 6, pp (28) [6] S A Avdonin, S A Ivanov, Families of exponentials: The method of moments in controllability problems for distributed parameter systems, Cambridge University Press, Cambridge, 995 [7] GQ Xu SP Yung, The expansion of semigroup criterion of Riesz basis, J Differ Equ, Vol 2, No, pp 24(25) [8] GQ Xu, ZJ Han SP Yung, Riesz basis property of serially connected Timoshenko beams, INT J Control, Vol 8, No 3, pp (27) [9] A Pazy, Semigroups of linear operators applications to partial differential equations, Springer-Verlag, Berlin, 983 ISSN: Issue 3, Volume 8, March 29