BOUNDARY CONTROL OF THE TIMOSHENKO BEAM

SIAM J. CONTROL AND OPTIMIZATIONVol. 25, No. 6, November 1987

1,987 Society for Industrial and Applied Mathematics004

BOUNDARY CONTROL OF THE TIMOSHENKO BEAM*JONG UHN KIMf AND YURIKO RENARDY

Abstract. It is shown that the Timoshenko beam can be uniformly stabilized by means of a boundarycontrol. A numerical study on the spectrum is also presented.

Key words. Timoshenko beam, uniform stabilization, exponential decay, boundary control, energymethod, Co-semigroup, linear stability, eigenvalues, spectral method

AMS(MOS) subject classifications. 35B37, 35L15, 73K05, 93C20, 93D15, 65F15, 65N25

0. Introduction. The purpose of this paper is to investigate uniform stabilizationofthe Timoshenko beam with boundary control. The motion of a beam can be describedby the Euler beam equation when the cross-sectional dimensions are small in com-parison with the length of the beam. If the cross-sectional dimensions are not negligible,the effect of the rotatory inertia should be considered and the motion is better describedby the Rayleigh beam equation. If the deflection due to shear is also taken into accountin addition to the rotatory inertia, we arrive at a still more accurate model, which iscalled the Timoshenko beam. Its motion is described by the following system ofequations:

02WK

Ozw+K=0,(0.1) P Ot Oxz Ox

02)E1 +K - =0.(0.2) I,

Ot

Here, is the time variable and x is the space coordinate along the beam in itsequilibrium position. We denote by w(x, t) the deflection of the beam from theequilibrium line, which is described by w 0, and by (x, t) the slope of the deflectioncue when the shearing force is neglected; for the precise meaning of, see Timoshenko11 or Traill-Nash and Collar 12]. We assume that the motion occurs in the wx-planeand that 0 x L. The coecients p, I, E and I are the mass per unit length, themass moment of ineia of the cross section, Young’s modulus and the moment ofineia of the cross section, respectively. The coecient K is equal to kGA, where Gis the modulus of elasticity in shear, A is the cross sectional area and k is a numericalfactor depending on the shape of the cross section. The boundary condition we employat x 0 is

(0.3) w(0, t)=0, b(0, t)=0,

which is for the clamped end at x 0, and the boundary control at x L is of the form

(0.4)ow ow

Kc(L, t)- K-f-- (L, t) a-7-? (L, t),ot

(0.5) EIOdp

(L, t)=Ox -- (L, t)

* Received by the editors January 27, 1986; accepted for publication November 9, 1986.? Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

24061. The work of this author was supported by the Air Force Office of Scientific Research under grantAFOSR-86-0085 and by the National Science Foundation grant DMS-8521848.

t Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia24061. The work of this author was supported by National Science Foundation grant DMS-8615203 underthe National Science Foundation Research Opportunities for Women Program.

1417

1418 J.u. KIM AND Y. RENARDY

where c and/3 are positive constants depending on the control device. This boundarycontrol corresponds to a control mechanism which monitors Ow/Ot and Och/Ot at x Land transforms them into the lateral force and moment applied at x L, respectively.Russell [10] and Washizu [13] derived (0.1) and (0.2) through the energy principle byusing the natural energy of the beam given by

(0.6) e(t) = p + I, + K - + E1 dx.

One can also derive an equivalent fourth-order equation in terms of w; see Timoshenko[11] and Traill-Nash and Collar [12]. In particular, [12] discusses various boundaryconditions.

This paper consists of two main parts. In the first part, we show that the energye(t) decays exponentially fast under (0.3), (0.4) and (0.5). For the one-dimensionalwave equation with boundary control, Quinn and Russell [9] established the exponen-tial decay of solutions. Later, Chen 1], [2] obtained the same result for a wave equationin any space dimension under some geometrical conditions on the domain. A veryrestrictive part of these conditions was eliminated by Lagnese [5] with the aid of anew energy estimate. Lagnese [6] also extended the result to linear elastodynamicsystems. In contrast to the above works, Lasiecka and Triggiani [7] employed boundaryfeedback acting in the Dirichlet boundary condition to achieve exponential decay ofsolutions to the wave equation. More recently, Chen et al. [3] discussed the case of achain of Euler beams and obtained a similar result. Our result for the Timoshenkobeam is most closely related to [3]. We use the energy method combined with C0-semigroup theory as in [1]-[3], [5] and [6]. The essence of the method is to constructa suitable energy functional associated with e(t). Details are given in 2.

The second part of this paper is concerned with a numerical study. Since thenature of the spectrum is an important question in the investigation of the stability ofa linear system, we carried out numerical experiments on the spectrum of (0.1) and(0.2) under (0.3)-(0.5). We express the temporal variation of the eigenfunction innormal modes ofthe form ea’, transforming (0.1)-(0.5) to ordinary differential equationswith boundary conditions. The Chebyshev-tau method [4], [8] is used to discretize thespatial variation of the eigenfunctions, thus yielding a matrix eigenvalue problem withdiscrete complex-valued eigenvalues A. These are computed using a NAG routine inquadruple precision on a VAX 11/785. Results of numerical experiments are presentedin3.

1. Notation and preliminaries. We shall use the notation

f,=o,f =Of A=Oxf Of Ax=O,,,,f O2fat’ ox’ OX2’ etc.

L always denotes L2(O, 1) and we write

gin= f, fL,k=l, ,m

Our basic function space 03 is the set of all quadruplets

W1

BOUNDARY CONTROL OF TIMOSHENKO BEAM 1419

satisfying

wiH1, wl(0)=0, wzL2,(])1 f H’, bl(0)= 0, b2 Le,

equipped with the inner product

(1.1) (z, z) (oxwl)(a.l)+w2.,2+-i-o(axd,)(oxl)+O2e dx.

We shall also use the function space 5e which is the set of all quadruplets

W1

W2

satisfying

Wl He, W1(0 0, w2 Hl, we(0) 0,, e He, (DI(0) 0, 2 H’, 4,2(0) 0,

K61(1)- K OxW,(1)= awe(l), EI 0,,b (1) -/34,2(1),

equipped with the inner product induced by H2 H H2 H1. Here, K, a, E, I and/3 are the same as in the previous section. It is easy to show that 0 is dense in

We define the operator A in "

i iid 0 0

(1.2) A=(K/ 0,,, 0 -(K/p) O,, 0

0 0 id

[(K/Io) Ox 0 (EI/Io) O,,,,-(K/Ip) id 0

where id is the identity mapping and the domain of A is taken to be be. It is then easyto see that (0.1), (0.2) with (0.3), (0.4) and (0.5) can be put in the abstract form"

dz(1.3) --=Az wherez=

dtand L is taken to be 1.

We also observe the following.LEMMA 1.1. A is an infinitesimal generator of a Co-semigroup in .Proof Let us write A Ao+ A1, where

(1.4) Ao

with @(Ao) and

(1.5) A

0 id 0 0

(K/p) Ox, 0 0 0

0 0 0 id

0 0 (EI/Io)0,, 0

0 0 0 0

0 0 -(K/p)O, 0

0 0 0 0

(K/Ip) O, 0 -(K/Ip) id 0


It is apparent that A1 is a bounded linear operator on . Thus, it is enough to showthat Ao is an infinitesimal generator of a Co-semigroup. By virtue of the Lumer-Phillipstheorem, it is enough to show that

(1.6) (Aoz, z) <- cllzll for all z 5,

where c is a positive constant, and that

(1.7) Range of (hid- Ao)= for some h > c.

By integration by parts using the boundary condition of z e , we find that

K E1(Aoz, z) w.2(1) a,,w,(1 + 4_(1) a,,tbl

)=l (K6(1)- 62(1

1(1.8) -< K24,(1)2

4pa

1 K2fo )2dx_-< (axe,4pa

g2

-4paEI I llzll ,

from which (1.6) follows. We next prove (1.7) for any A > O. Let A > 0 and

f,f2 eg

Then, we have to find

W1

14"2

such that

(1.9) IW1-- W2"-- flK

(1.10) hWz---O,,,,wl=f2,P

(1.11) a4,1 42 gl,

EI(1.12) A42 --}7 O,,x4l g2

We can find (1 H2 such that

(1.13)

(1.14)

Oxx g2 q-Agl,

cb,(O) O, ,BAch,( + EI O,,cb,( fig,( ).


In fact, tl is given by

(1.15) cb(x)=csinhlxxtzEI

(g(s)+,g(s))sinhtz(x-s) ds

where tz=,(Io/EI)/ and c is uniquely determined from [3Adp(1)+FIOxdl(1)flgl(1).

Then, b2 is determined by (1.13). It is obvious that b2 H1, b2(0)-0 andEI 0xbl (1) -flb(1). Similarly, we can find Wl HE such that

(1.16) (h :z K )---Oxx w=f+ Aft,P

(1.17) Wl(0 =0, aAwl(a) + K 0xWl(1)= Kb,(1) + afl(1).

Then, w2 is determined from (1.19). It is easy to see that

W1

w zO

and (1.9)-(1.12) hold.We shall use the following elementary inequality later on:

Io’ Io’2(1.18) wxdx<-2 (b wx)2 dx+2 dP dx

for all b HI, w H satisfying b(0) 0.

2. Statement and proof of the main result. In this section, we take L 1 withoutloss of generality. Let S(t) be the Co-semigroup in c generated by A in the previoussection. We assert the following.

THEOREM 2.1. The operator norm of S(t) satisfies

(2.1) Ils(t)ll<-Me-’ for all t>-O

where M and r are positive constants.Before giving details of the proof, we shall outline our arguments. Let us fix any

Zo 5e. Using e(t) associated with S(t)Zo, we define

(2.2) F(t) Ite(t)+ G(S(t)Zo)

where/z is a positive constant depending only on the coefficients of (0.1), (0.2), andG(.) is a suitable functional on such that

(2..3) [G(z)]_-< C[[z][ for all z .With the aid of (1.18), we can derive that

(2.4) dllS(t)zo[12 <- e(t) <-_ d[[ S( t)zoll

holds for all t->0, where d and d2 are positive constants depending only on thecoefficients of (0.1) and (0.2). We then show that

(2.5) F(t) <= Toll


holds for all 0, where M1 is a positive constant depending only on a, /3 and thecoefficients of (0.1) and (0.2). Formulae (2.3) and (2.4) imply

1(2.6) IIs(t)Zoll<-_TM2llZolla for all t>0

where M2 is a positive constant depending only on a,/3 and the coefficients of (0.1)and (0.2). Since is dense in , (2.6) implies

(2.7) IIs(t)zll [Izoll for all z

Finally, we use the semigroup property of S(t) to arrive at (2.1).Proof of Theorem 2.1. Fix any z0 5. Then,

(2.8) S(t)Zo6 C([0, oo); )tq C1([0, oo); )

and

d(2.9)

dtS(t)Zo AS(t)Zo for every t-> 0.

Hence, we can write

(2.10) S(t)Zo=

w(x, t) ]O,w(x,b(x, t) IO,c(x, t)J

where w(x, t) and th(x, t) satisfy (0.1)-(0.5).We now construct F(t):

F(t) =-- { pw2t + Ipdp2t + K qb wx)2 + Eldp2x} dx

(2.11) + p xW,Wx dx + Io xcb,Cbx dx+Io cbcb, dxo 2+r/

p ww, dx2+7

where and are positive constants which will be determined later on. By virtue of(2.8) and (2.10), we can differentiate (2.11) to obtain

9

(2.12) dF_ IJ1 + ItJ2 + Y Jdt n=3

where

J1 e(t)=- {pw2, + lo62,+ K(cb wx)2+ ElcbZx} dx,

J2 {pw,w,, + Iodptcb,, + K(d Wx)(Cb,- Wx,)+ EI4’xCbx,} dx,o


J p xw,w, dx, & I. x..G dx.

1J4 p xw,,Wx dx, J7 2 + "0

P 2 Pw dx wwu dx.2+’0 2+’0

Using (0.1)-(0.5), we can integrate by parts to arrive at

(2.13) J2 -aw,(1, t)z- fld,(1, t)2,

1 )2 1 2(2.14) J3=- Ow,(a, - p wt dx,o

(2.15).14 XWx Kwxx Kthx dx

o

z K w dx K XWxbx dx,

(2.16)1

J5--" Ipt(1, t)2-- Ip ch2t dx,

J6 x&x EI&x Kdp + Kwx dx

(2.17)1Elc/bx(1 t) 2

1 Io 1=- -- EI 6 dx-- Kth(1, t)2

1K 2 dx + K xchxwx dx,

b Eldp)x Kqb + Kwx dxJr-2+’0

1EI

1EI4)(1, t)6,(1, t)-2 / ,q2+’0

(2.18)11

K cb2dx+2+’0 2+,/

Kch(1, t)w(1, t)

Kwx(1, t)w(1, t)

1K wxdx.

2+’0 o

Let us choose ’0 > 0 such that

(2.19)1

K2 1)12+’0 =

1424 J.U. KIM AND Y. RENARDY

and then, choose/ > 0 such that

(2.20)

(2.21) 2/x<=2 2+7

Then, we find that

dt- 2

1EI 62x dx

2+r/ - 2 2+r/ wx dx

oqxtwt(1, t)-- fltxtd,bt(1, t)2 +1/2pwt(1, t)2

(2.22) +1/2Kwx(1, t)2+1/2Io6t(1, t)2+1/2EI4b(1, t)2

1-1/2 Kb(1, t)2+EId)(1, t) t;b,,(1, t)

2+/

1 1K6( l’ t)w( l’ t) 2 ++ r/

Kwx(1, t)w(1, t).

By means of (0.4) and (0.5) and the inequalities

Io’ )2(2.23) tk(1, t)2 px(X, dx,

(2.24) )w(1, t)2 Wx(X, dx

we deduce from (2.22) that for all >_- T,

dF(2.25)

dt

where T is a positive constant depending only on a,/3 and the coefficients of (0.1)and (0.2). Consequently, we arrive at (2.5). The argumem following (2.5) completesthe proof.

Remark 2.2. Finally, we remark that our arguments with the same energy func-tional also yield exponential stabilization for a hinged boundary condition at x 0"

04,(2.26) w(O, t) O, xx (0, t) O.

However, in this case it seems necessary to impose the zero mean condition o b dx 0in order to avoid some technical difficulties (see [1]).

3. Numerical study of the spectrum. We present numerical results on the linearstability of our system. We use normal mode analysis and set

(3.1) w(x,t)=e;’tp(x),

(3.2) 6(x, t)= ea’Q(x).


Thus, (0.1)-(0.5) become the following system of ordinary differential equations withboundary conditions:

(3.3)

(3.4)

(3.5)

-KP,, + KQ. + ApP O,

-EIQxx + K Q P) + AEIpQ O,

P=Q-0 atx=0,

(3.6) EIQx + AflQ 0 at x L,

(3.7)

This system is of the form Ao+A1A 4-AzA2=0, where

(3.8) Ao

(3.9) A1

(3.10) A2

K(Q-P,,)-AaP=O atx=L.

We rewrite this in the form

-KPx + KQx-EIQxx+K(Q-P)

P(O)Q(O)

EIQx(L)KQ(L) KP(L)

0

0

0

0

flQ(L)uaP(L).

loP

0

0

0

0

1 AI+AA2

so that our system takes on the customary form A AB, where

(3.12) A=0

1 A

(3.13) B0 -Az

We discretize P(x) and Q(x) by the Chebyshev-tau method [4], [8]. This is aspectral method where the expansion functions are the Chebyshev polynomials T, (z)defined by T,(cos 0)=cos nO when z=cos 0. This method approximates discreteeigenvalues belonging to C eigenfunctions with infinite-order accuracy.

1426 J.U. KIM AND Y. RENARDY

We rescale the spatial variable to z (2x)/L-1, so that -1 <-z-< 1. We set

N

(3.14) P(z)= Y p,,T,,(z),

N

(3.15) Q(z)= E q,,T,,(z)n=O

and substitute into (3.3)-(3.7). Thus, there are 2N+2 unknowns. In the differentialequations, we equate coefficients of like powers of the Chebyshev polynomials. Sincethe first equation (3.3) contains Px,, it yields equations for the coefficients up to degreeN-2 in the polynomials, thus "giving N-1 equations. Similarly, (3.4) yields N-1equations. In the tau approximation, the expansion functions T,,(z) are not requiredto satisfy the boundary conditions individually. The four boundary conditions areimposed as part of the conditions determining the coefficients Pn and q,. The totalnumber of equations is 2N + 2. The size of the final matrix equation A AB is 4N+ 4square. Our computer program uses the NAG routine F02GJF to compute the eigen-values in complex quadruple precision on a VAX 11/785.

The accuracy of our numerical results was established in the following way. Theeigenvalues must satisfy the characteristic equation:

[m,l(A) m12(A) m13(A) m14(A)

/m21(A) m22(A) m23(A) m24(A)=0(3.16) det

/ m31(A m32(A m33(A m34(L m41(A m42(A m43(A m44(/

where, for j 1, 2, 3, 4

mlj 1, m2j I,A 2rlj Elrl., m3j =(EIrb + flA en,(3.17)

m4j EIA}- IA3- IoA+ EIn e,and are the roots of

1 [ ()(A4( )2 )1/2(3.18) 2= A2 I0+

___4pEIA2

In order to check that our computed eigenvalues satisfy the determinant equation(3.16), we have chosen moderate-sized parameters so that the evaluation of the deter-minant avoids cancellation between large numbers. We choose p 1, K 1.5, I 2,E =2.5, I 3, L=0.1, a =3.5 and fl =4.1. Computations at N= 15, 20, 25 and 30showed that a few eigenvalues are already converged to about 15 digits at N= 15.About 12 eigenvalues at N 15 are converged to at least 5 digits, and satisfy (3.16)to that accuracy. All converged eigenvalues have negative real pas and are either realor complex conjugates. The number of digits to which each pair is a complex conjugateis an indication of the amount of roundoff error present. The eigenvalues consist oftwo groups. One group is lined up approximately along the line -4.47 and the imaginarypas are almost multiples, staing with A =-4.4709 and then A -4.4769 38.475 i,-4.4770 76.951i and so on. The other group is lined up approximately along -34.45,and the imaginary pas are almost multiples, staing with A =-34.505 and thenA =-34.44660.829i, -34.452 121.68i and so on.


Computations were done at the following set of parameters to model a solidaluminum bar: p =400 g/cm, K =2.8 1013 g" cm/sec2, Ip =3,332 g" cm, E7.6 1011 g/cm/sec2, I 833 cm4 and L= 200 cm. We allow a and fl to be

(i) a 10 g/sec, fl 10 g. cm2/sec;(ii) a 2 x 10 g/sec,/3 2 x 10 g. cm2/sec;(iii) a 5 10 g/sec, fl 5 107 g" cm/sec.By comparing the results of N 40 and N 45, we conclude that about 20 complex

conjugate pairs have converged to 5 digits at N =40, and there are no real-valuedeigenvalues. The results for case (i) are plotted in Figs. 1 and 2. Figure 2 is a

magnification of Fig. 1 close to the origin. All eigenvalues have negative real parts.Figure 1 does not indicate that the ratio Im (A)/Re (A) approaches a constant for largeIAI. Results of cases (i)-(iii) are displayed in Table 1. Essentially, the real parts of Aare approximately proportional to a or/3 and the imaginary parts of the three cases

IHGINRY 10J3) 10.00 20.00 BO.O0 40.00 50.00 60.00 70.00 80.00

FIG. 1. The graph displays the upper half quadrant of the first 23 complex conjugate pairs of case (i).

FIG. 2. The graph is a magnification of Fig. close to the origin to clarify the location of the eigenvaluesthere.

1428 J.u. KIM AND V. RENARDY

TABLEThis table displays to 5 digits the first 22 to 23 complex conjugate pairs for cases (i)-(iii) described in 3.

Case (i) Case (ii) Case (iii)

-0.36717E-01 +/-0.11039E+03 -0.73434E-01 +/-0.11039E+03 -0.18359E+00-0.16364E+00 +0.68450E+03 -0.32729E+00 +0.68450E+03 -0.81822E+00-0.38376E+00 +0.18853E+04 -0.76753E+00 +0.18853E+04 -0.19188E+01-0.69283E+00 +/-0.36108E+04 -0.13857E+01-0.10607E+01 +0.58051E+04 -0.21214E+01-0.14663E + 01 +0.84010E+ 04 -0.29326E + 01-0.18925E+01 +0.11335E+05 -0.37851E+01-0.23278E+01 +0.14548E+05 -0.46555E+01-0.27652E+01 +0.17990E+05 -0.55303E+01-0.32017E+01 +0.21620E+ 05 -0.64033E+01-0.36372E+01 +0.25401E+05 -0.72743E+01-0.40737E+01 +0.29306E+05 -0.81472E+01-0.45146E+01 +0.33310E+05 -0.90290E+01-0.49647E + 01 +0.37395E+ 05 -0.99293E+ 01

+0.36108E+04 -0.34641E+01+0.58051E+04 -0.53033E+01+0.84010E + 04 -0.73313E+01+0.11335E+05 -0.94624E + 01+0.14548E+05 -0.11638E+02+0.17990E+05 -0.13825E+02+0.21620E+05 -0.16007E+02+0.25401E+05 -0.18184E+02+/-0.29306E+05 -0.20366E+02+0.33311E+05 -0.22570E + 02+0.37395E+05 -0.24820E+02

-0.54301E+01 +/-0.41544E+ 05-0.59180E+01 +0.45745E+ 05-0.64370E +01 +0.49986E +05-0.69979E+01 +/-0.54258E+05

-0.76138E +01 +0.58553E +05-0.83016E+01 +0.62865E+05-0.90827E+01 +/-0.67186E+05

-0.99853E+01 +0.71509E+05-0.11047E+02 +0.75828E+ 05

-0.10860E+02 +/-0.41544E+05 -0.27146E+02-0.11836E+02 +/-0.45745E+05 -0.29584E+02-0.12874E+02 +0.49986E+05 -0.32178E+02-0.13995E+02 +/-0.54258E+05 -0.34982E+02-0.15227E+02 +0.58553E+05 -0.38061E+02-0.16603E+02 +0.62865E+05 -0.41499E+02-0.18165E+02 +/-0.67186E+05 -0.45405E+02-0.19970E+02 +0.71509E+05 -0.49919E+02-0.22093E + 02 +0.75828E + 05

+0.11039E+03+0.68450E + 03+0.18853E +04+0.36108E + 04+/-0.58051E + 04+0.84011E +04+0.11335E+05+0.14548E+ 05+0.17990E+ 05+0.21620E+ 05+/-0.25402E + 05+0.29306E + 05+0.33311E+05+/-0.37396E+ 05+0.41545E + 05+0.45745E+ 05+0.49987E + 05+0.54259E+ 05+0.58554E+05+0.62866E + 05+0.67187E+05+0.71510E+05

are approximately equal. This indicates that when a and/3 are zero, the real parts arezero and the eigenvalues are purely imaginary. The relative importance of the dampingterms in our computations is seen from the dimensionless equations. There are 6dimensionless parameters: C LQ/P, where Q is the scale of Q and P is the scaleof P, C2 h:pL/ K, C3 KL:Z/ El, C4 h :zI,L:Z/ El, C5 hilLE1 and C6 haL/K. Ourvalues for case (i) are: C2=h2610-7, C3=1.8103, C4=h2210-7, C5h x 3 x 10-6 and C6 h x 7 x 10-9. The dimensionless equations are

(3.19) -Pxx+CQx+C,P=O,

C3(3.20) -Q,,, + C3Q -- P+ C4Q O,

(3.21) P=Q=0 atx-0,

(3.22) Qx + C5Q 0 at x 1,

(3.23) C, Q P C6P 0 at x 1

where P, Q and x have been made dimensionless. The largest eigenvalues in Fig.are O(10) so that C5 is O(10-1) and C6 is O(10-3), indicating that the damping termsare not large. For the smallest eigenvalues, the damping terms are small so that theproperty of proportionality of Re (h) to a or/3 may be an asymptotic behavior forsmall damping.

We note that the qualitative features obtained for the computation with moderatedata are very different from those of the model of an aluminium bar. This is reminiscentof the qualitative differences in Figs. 5 and 7 of Chen et al. [3].


Acknowledgments. We are very grateful to Professors K. Hannsgen and R. Wheelerfor suggesting this problem and for useful discussions. We are also much indebted toProfessor J. Burns whose help was indispensable in revising the previous manuscript.

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L2(0 c; L2(F)) boundary feedback acting in the Dirichlet boundary conditions, Proc. of the 24thIEEE Conference on Decision and Control, Fort Lauderdale, FL, 1985.

[8] S. A. ORSZAG, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, (1971),pp. 689-703.

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[10] D. L. RUSSELL, Mathematical models for the elastic beams and their control-theoretic implications,preprint.

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bending vibrations of beams, Quart. J. Mech. Appl. Math., 6 (1953), pp. 186-222.[13] K. WASHZU, Variational Methods in Elasticity and Plasticity, Pergamon, London-New York, 1968.

BOUNDARY CONTROL OF THE TIMOSHENKO BEAM

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