Stabilization of the Timoshenko Beam System with Restricted Boundary Feedback Controls
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Stabilization of the Timoshenko Beam System with Restricted Boundary Feedback Controls Dongyi Liu1 · Liping Zhang1 · Zhongjie Han1 · Genqi Xu1
Received: 24 April 2014 / Accepted: 20 January 2015 © Springer Science+Business Media Dordrecht 2015
Abstract This paper concerns with the stabilization of a Timoshenko beam with bounded constraints on boundary feedback controls. Since the resulting controlled system is nonlinear, the weak well-posedness is proven by theories of the nonlinear monotone operators and the optimization. Then, the asymptotical stability of the controlled beam is analyzed by the weak topology, and its exponential stability is also proven by the Lyapunov’s second method. In the end, the numerical experiment indicates that the control design is feasible. Keywords Timoshenko beam · Nonlinear monotone operator · Weak sequentially lower semi-continuity · Stability · Boundary control
1 Introduction It is well known that the beam-type structures such as Euler–Bernoulli, Rayleigh and Timoshenko beams are very important flexible structures in mechanical systems. Especially, the Timoshenko beam is a more accurate beam model, which takes into account not only the rotary inertial energy but also its deformation due to shear. So, the Timoshenko beam model had been studied extensively in the last decades (see [8, 10, 11, 15, 20, 22, 23, 26, 27] and references therein).
This research is supported by the Natural Science Foundation of China grant NSFC-61174080, 61104130.
B D. Liu
[email protected] L. Zhang [email protected] Z. Han [email protected] G. Xu [email protected]
1
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China
D. Liu et al.
This research effort focuses on the Timoshenko beam clamped at the left end and equipped with suitable boundary controls u1 (t) and u2 (t) at the right end: ⎧ ρwtt (x, t) − κ wxx (x, t) − ϕx (x, t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Iρ ϕtt (x, t) − EIϕxx (x, t) − κ wx (x, t) − ϕ(x, t) = 0, w(0, t) = ϕ(0, t) = 0, κ wx (1, t) − ϕ(1, t) = u1 (t), EIϕx (1, t) = u2 (t), (1) ⎪ ⎪ ⎪ ⎪ ⎪ w(x, 0) = w0 (x), wt (x, 0) = w1 (x), ϕ(x, 0) = ϕ0 (x), ϕt (x, 0) = ϕ1 (x), ⎪ ⎪ ⎩ x ∈ (0, 1), t > 0, where w0 (x), w1 (x), ϕ0 (x) and ϕ1 (x) are initial functions, and ρ, Iρ , EI and κ are mass density, moment of mass inertia, rigidity coefficient and shear modulus of elastic beam, respectively. For more precise physical meanings of them, see Timoshenko’s book [19]. 2 2 The abbreviations wt , wtt , wx and wxx represent ∂w , ∂∂tw2 , ∂w and ∂∂xw2 , respectively, and ∂t ∂x thereafter. It is well known that if the feedback control laws are determined by u1 (t) = −α1 wt (1, t)
and
u2 (t) = −α2 ϕt (1, t)
with α1 , α2 > 0,
(2)
then the closed-loop system (1)–(2) is exponentially stable [11]. Especially, when α1 = √ √ √ ρκ = α2 = Iρ EI, it is superstable [3, 4]. When κ/ρ = EI/Iρ , α1 = ρκ and α2 = Iρ EI, the eigenfunctions of the system operator form Riesz basis [22]. These results are based on the ideal situation. However, in engineering, due to the restriction of exter
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