TY - JOUR

T1 - Decay rates for a beam with pointwise force and moment feedback

AU - Ammari, Kais

AU - Liu, Zhuangyi

AU - Tucsnak, Marius

PY - 2002/9/18

Y1 - 2002/9/18

N2 - We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0, π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π = p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler-Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

AB - We consider the Rayleigh beam equation and the Euler-Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0, π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π = p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler-Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

KW - Exponential decay

KW - Observability inequality

KW - Pointwise control

KW - Polynomial decay

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U2 - 10.1007/s004980200009

DO - 10.1007/s004980200009

M3 - Article

AN - SCOPUS:0036036612

SN - 0932-4194

VL - 15

SP - 229

EP - 255

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

IS - 3

ER -