SIAM J. CONTROL AND OPTIMIZATION Vol. 25, No. 6, November 1987 1,987 Society for Industrial and Applied Mathematics 004 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 boundary control. A numerical study on the spectrum is also presented. Key words. Timoshenko beam, uniform stabilization, exponential decay, boundary control, energy method, 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 stabilization of the Timoshenko beam with boundary control. The motion of a beam can be described by 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 described by the Rayleigh beam equation. If the deflection due to shear is also taken into account in addition to the rotatory inertia, we arrive at a still more accurate model, which is called the Timoshenko beam. Its motion is described by the following system of equations: 02 W K Ozw +K=0, (0.1) P Ot Ox z Ox 02) E1 +K - =0. (0.2) I, Ot Here, is the time variable and x is the space coordinate along the beam in its equilibrium position. We denote by w(x, t) the deflection of the beam from the equilibrium line, which is described by w 0, and by (x, t) the slope of the deflection cue when the shearing force is neglected; for the precise meaning of , see Timoshenko 11 or Traill-Nash and Collar 12]. We assume that the motion occurs in the wx-plane and that 0 x L. The coecients p, I, E and I are the mass per unit length, the mass moment of ineia of the cross section, Young’s modulus and the moment of ineia of the cross section, respectively. The coecient K is equal to kGA, where G is the modulus of elasticity in shear, A is the cross sectional area and k is a numerical factor depending on the shape of the cross section. The boundary condition we employ at 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) EI Odp (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 grant AFOSR-86-0085 and by the National Science Foundation grant DMS-8521848. t Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. The work of this author was supported by National Science Foundation grant DMS-8615203 under the National Science Foundation Research Opportunities for Women Program. 1417