J Dyn Control Syst DOI 10.1007/s10883-015-9293-4

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay Xiu Fang Liu1 · Gen Qi Xu2

Received: 3 September 2014 / Revised: 28 June 2015 © Springer Science+Business Media New York 2015

Abstract In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the 0 outputs of controllers are of the forms α1 u1 (t) + β1 u1 (t − τ ) + −τ g1 (η)u1 (t + η)dη and 0 α2 u2 (t) + β2 u2 (t − τ ) + −τ g2 (η)u2 (t + η)dη respectively, where u1 (t) and u2 (t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially. Keywords Timoshenko beam · Input delay · Exponential stabilization · Boundary control Mathematics Subject Classification (2010) 93C20 · 93D15

1 Introduction In the past decades, the investigation of Timoshenko beam has been a hot topic in mathematics and engineering due to its applications in space science, see [1–3] and the reference therein. Various different control strategy were used to stabilize the system, for instance, This research was supported by the National Science Natural Foundation in China (NSFC–61174080)  Xiu Fang Liu

[email protected] Gen Qi Xu [email protected] 1

School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

2

Department of Mathematics, Tianjin University, Tianjin 300072, China

Xiu Fang Liu and Gen Qi Xu

boundary control [4–6], partial internal control [7], pointwise control [8], and nonlinear control [9]. Besides the single beam, the network system of Timoshenko beam [11–13] and Timoshenko beam with structure memory [14] also were studied. Some nice results have been obtained for the closed-loop system, for example, Riesz basis property [10, 12, 13] and exponential stability [4, 6, 9, 10, 13]. We observe that there is a few work on the control delay problem. This is because the controller delay will destabilize the system even if the system without delay is exponentially stable. Let us briefly survey the studies on the controller problem. The question proposed by Dakto [15], who gave some examples to show that a small time delay in control will destabilize the system. Xu et al. in [16] discussed stabilization of the 1-d wave systems with control delay of the form αu(t) + βu(t − τ ). By a complicated calculation, they proved that the system under negative collocated velocity feedback is exponentially stable if α > β > 0 and unstable if β > α. This result can be explained as the system is exponentially stable as α 1 α 1 1 α+β > 2 and unstable as when α+β < 2 . So it is said to be a 2 -rule. After that, the rule was extended to other complex models, see [17–21]. Different from the early results, Shang et al. in [21] showed that the conditions β > 0 is unnecessary for the Euler-Bernoulli beam. If α > |β|, the stability result still holds true. To remove the restriction on α > |β|, Shang and Xu in [22] designed a dynamic feedback controller. With this dynamic feedback control, the Euler-Bernoulli beam is stabilized exponentially provided that any real α, β satisfy |α|  = |β|. Wang and Xu [23] and Xu and Wang [24] discussed the 1-d wave equation and the Timoshenko beam and proved that such a design of dynamic feedback controller also fits the wave equation and Timoshenko beam. By analyzing the proof of these papers, we see that the restriction condition |α| = |β| mainly comes from the controller zero or the characteristic equation of difference equation: α + βe−τ λ = 0. If the eigenvalues of systems, λn , satisfy α + βe−τ λn = 0, for all n, then the systems are stable. Different from previous control type, we consider the controller with distributed delay of the form:  0 g(η)u(t + η)dη. αu(t) + βu(t − τ ) + −τ

Observe that this is just the output of the system ⎧ ∂z(s,t) ⎨ ∂t = ∂z(s,t) ∂s , s ∈ (−τ, 0), z(0, t) = u(t) ⎩ z(s, 0) = f (s), s ∈ (−τ, 0) 0 and the output v(t) = −τ z(t + s)dα(s), where α(s) is of the form  s g(η)dη α(s) = αH (s) + βH (s + τ ) + 0

where H (s) is the Heaviside function. For such a type of controller delay, Shang and Xu [26] and Liu and Xu [27] studied the Euler-Bernoulli beam and Timoshenko beam, respectively. By using the dynamic feedback controllers, they obtained the exponential stabilization for the Euler-Bernoulli beam and Timoshenko beam, respectively. We observe that, in design of the dynamic feedback controller, the measurement of full state is required, so these results are obtained under the assumption that the full state of the system is known. However, the measurement of

Output-Based Stabilization of Timoshenko Beam

full state is unfeasible in practice. So, in the present paper, we shall consider output-based stabilization problem for the Timoshenko beam with boundary control distributed delay.

2 Preliminaries In this section, we mainly introduce our model under consideration and method used in this paper. Consider the following Timoshenko beam with distributed delay ⎧ ρwtt (x, t) − K(wxx − ϕx )(x, t) = 0, x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ Iρ ϕtt (x, t) − EI ϕxx (x, t) − K(wx − ϕ)(x, t) = 0, x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ w(0, t) = ϕ(0, t) = 0, t > 0, ⎪ 0 ⎪ ⎪ ⎪ K(wx − ϕ)(1, t) = α1 u1 (t) + β1 u1 (t − τ ) + −τ g1 (η)u1 (t + η)dη, ⎪ ⎪  ⎨ 0 EI ϕx (1, t) = α2 u2 (t) + β2 u2 (t − τ ) + −τ g2 (η)u2 (t + η)dη, ⎪ ⎪ w(x, 0) = w0 (x), wt (x, 0) = w1 (x), ⎪ ⎪ ⎪ ϕ(x, 0) = ϕ0 (x), ϕt (x, 0) = ϕ1 (x), ⎪ ⎪ ⎪ ⎪ u1 (θ ) = f1 (θ ), u2 (θ ) = f2 (θ ), θ ∈ (−τ, 0); ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ and y1 (t) = wt (1, t), y2 (t) = ϕt (1, t).

(2.1)

where ui (t)(i = 1, 2) are controller inputs, yi (t)(i = 1, 2) are outputs of the system, τ is the maximal delay time of the controllers, αi , βi ∈ R(i = 1, 2) are the controller parameters, and gi (η) ∈ L2 [−τ, 0], j = 1, 2, are the delay weighted functions of controllers, and fi (θ ), θ ∈ (−τ, 0)(i = 1, 2) are the memory functions of controllers, that are square integrable functions. Equation 2.1 is a general model that includes all models studied earlier time. For instance, if gj ≡ 0, βj = 0 and αj < 0, j = 1, 2, it is a model without delay studied in [4, 10]; if gj ≡ 0, αj < 0 and βj < 0, it is a model with fixed delay studied in [24], wherein the full state is required to be known; if gj = 0, αj , βj ∈ R, and the state is known, it is just the model studied in [27]. The great difference between this paper and [27] is that the state of the system is unknown in Eq. 2.1. Our aim is to stabilize system (2.1) only using the output information. Our idea is to translate (2.1) into a system without delay by a linear transform, and then design the feedback control to stabilize exponentially the system without delay. To this end, at first, we construct the Luenberger observer for system (2.1) and obtain state reconstruction ⎧ ρw tt (x, t) − K( wxx −  ϕx )(x, t) = 0, x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ Iρ  ϕ (x, t) − EI  ϕ (x, t) − K( wx −  ϕ )(x, t) = 0, x ∈ (0, 1), t > 0, ⎪ tt xx ⎪ ⎪ ⎪ ⎪ w  (0, t) =  ϕ (0, t) = 0, t > 0, ⎪ ⎪ ⎨ K( ϕ )(1, t) = −k1 ( w (1, t) − y1 (t)) + α1 u1 (t) + β1 u1 (t − τ ) wx −  0 t + ⎪ −τ g1 (η)u1 (t + η)dη, ⎪ 0 ⎪ ⎪ ⎪ EI  ϕx (1, t) = −k2 ( ϕt (1, t) − y2 (t)) + α2 u2 (t) + β2 u2 (t − τ ) + −τ g2 (η)u2 (t + η)dη, ⎪ ⎪ ⎪ ⎪ (x, 0) = w 0 (x), w t (x, 0) = w 1 (x), ⎪ ⎩w ϕt (x, 0) =  ϕ1 (x),  ϕ (x, 0) =  ϕ0 (x),  (2.2) where ki (i = 1, 2) are positive constants.

Xiu Fang Liu and Gen Qi Xu

Next, we use the reconstructing state to predict partially the state of the system: ⎧ w s (x, s, t) = z(x, s, t), x ∈ (0, 1), s ∈ (0, τ ), ⎪ ⎪ ⎪ s, t), x ∈ (0, 1), s ∈ (0, τ ), ⎪ ϕ (x, s, t) = ψ(x, s ⎪ ⎪ ⎪ ⎪ zs (x, s, t) = Kρ ( wxx (x, s, t) − ϕx (x, s, t)), ⎪ ⎪ ⎪ ⎪ EI K ⎪ (x, s, t) = ϕ (x, s, t) + ( wx (x, s, t) − ϕ (x, s, t)), ψ xx ⎨ s Iρ

Iρ

w (0, s, t) = ϕ (0, s, t) = 0, s ∈ (0, τ ),  −s ⎪ ⎪ ⎪ − ϕ )(1, s, t) = β1 u1 (t + s − τ ) + −τ g1 (η)u1 (t + s + η)dη, s ∈ (0, τ ), K( w ⎪ x ⎪  ⎪ −s ⎪ ⎪ EI ϕx (1, s, t) = β2 u2 (t + s − τ ) + −τ g2 (η)u2 (t + s + η)dη, , s ∈ (0, τ ), t > 0 ⎪ ⎪ ⎪ ⎪ (x, 0, t) = w (x, t), z(x, 0, t) = w t (x, t), x ∈ (0, 1), ⎪w ⎩ t (x, 0, t) =  ϕ (x, 0, t) =  ϕ (x, t), ψ ϕt (x, t), x ∈ (0, 1), t > 0. (2.3) Note that, in this step, we cannot use control information after time t. so we call (2.3) a partial state predictor. Based on Eq. 2.3, we can take its state at the moment s = τ , denote it by τ, t)). (p1 (x, t), p2 (x, t), q1 (x, t), q2 (x, t)) = ( w(x, τ, t), ϕ (x, τ, t), z(x, τ, t), ψ(x, A simple calculation shows that the functions group (p1 (x, t), p2 (x, t), q1 (x, t), q2 (x, t)) satisfy the following equations ⎧ p1,t (x, t) = q1 (x, t) + a1 (x)u1 (t) + a2 (x)u2 (t) + d1 (x)[−k1 ( wt (1, t) − y1 (t))] ⎪ ⎪ ⎪ ⎪ + d2 (x)[−k2 ( ϕt (1, t) − y2 (t))] x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ p2,t (x, t) = q2 (x, t) + a3 (x)u1 (t) + a4 (x)u2 (t) + d3 (x)[−k1 ( wt (1, t) − y1 (t))] ⎪ ⎪ ⎪ ⎪ + d (x)[−k ( ϕ (1, t) − y (t))], x ∈ (0, 1), t > 0, t 4 2 2 ⎪ ⎪ ⎪ ⎪ q1,t (x, t) = Kρ (p1,xx − p2,x )(x, t) + b1 (x)u1 (t) + b2 (x)u2 (t) ⎪ ⎪ ⎪ ⎪ ⎪ + e1 (x)[−k1 ( wt (1, t) − y1 (t))] + e2 (x)[−k2 ( ϕt (1, t) − y2 (t))], ⎪ ⎪ ⎪ EI K ⎪ q (x, t) = p (x, t) + (p − p )(x, t) + b (x)u 2,t 2,xx 1,x 2 3 1 (t) + b4 (x)u2 (t) ⎪ Iρ Iρ ⎨ wt (1, t) − y1 (t))] + e4 (x)[−k2 ( ϕt (1, t) − y2 (t))], + e3 (x)[−k1 ( ⎪ ⎪ ⎪ p1 (0, t) = p2 (0, t) = q1 (0, t) = q2 (0, t) = 0, t > 0, ⎪ ⎪ ⎪ K(p1,x − p2 )(1, t) = β1 u1 (t), t > 0, ⎪ ⎪ ⎪ ⎪ EIp2,x (1, t) = β2 u2 (t), t > 0, ⎪ ⎪ 0 0 ⎪ ⎪ w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a1 (x, s)f1 (s)ds − −τ a2 (x, s)f2 (s)ds, p1 (x, 0) = E1 ( ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ p2 (x, 0) = E2 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a3 (x, s)f1 (s)ds − −τ a4 (x, s)f2 (s)ds, ⎪ ⎪   ⎪ 0 0 ⎪ ⎪ q1 (x, 0) = E3 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b1 (x, s)f1 (s)ds + −τ b2 (x, s)f2 (s)ds, ⎪ ⎪ 0 0 ⎩ q2 (x, 0) = E4 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b3 (x, s)f1 (s)ds + −τ b4 (x, s)f2 (s)ds,

(2.4)

where the coefficient functions ai (x, r), bi (x, r), ai (x), bi (x), di (x), ei (x)(i = 1, 2, 3, 4) are scalar functions and Ei (i = 1, 2, 3, 4) are bounded linear operators on [H 1 [0, 1] × L2 [0, 1]]2 , they will be determined later. By now, we have translated the delayed system (2.1) into a system without delay (2.4). To obtain the control signals, we design the feedback controls for Eq. 2.4 as u1 (t) = −U1 (p1 , p2 , q1 , q2 ),

u2 (t) = −U2 (p1 , p2 , q1 , q2 )

(2.5)

Output-Based Stabilization of Timoshenko Beam

where U1 (p1 , p2 , q1 , q2 )  1  1 = β1 q1 (1, t) + K(p1,x (x, t) − p2 (x, t))(a1 (x) − a3 (x))dx + ρq1 (x, t)b1 (x)dx 0

 1  1 + EIp2,x (x, t)a3 (x)dx + Iρ q2 (x, t)b3 (x)dx; 0

0

0

and U2 (p1 , p2 , q1 , q2 )  1  1  = β2 q2 (1, t) + K(p1,x (x, t) − p2 (x, t))(a2 (x) − a4 (x))dx + ρq1 (x, t)b2 (x)dx 0

 1  1 + EIp2,x (x, t)a4 (x)dx + Iρ q2 (x, t)b4 (x)dx. 0

0

0

Thus, the closed-loop system associated with Eq. 2.4 is ⎧ p1,t (x, t) = q1 (x, t) − a1 (x)U1 (p1 , p2 , q1 , q2 ) − a2 (x)U2 (p1 , p2 , q1 , q2 ) ⎪ ⎪ ⎪ ⎪ + d1 (x)[−k1 ( wt (1, t) − y1 (t))] ⎪ ⎪ ⎪ ⎪ (x)[−k ( + d 2 2 ϕt (1, t) − y2 (t))], x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ p (x, t) = q (x, t) − a 2,t 2 3 (x)U1 (p1 , p2 , q1 , q2 ) − a4 (x)U2 (p1 , p2 , q1 , q2 ) ⎪ ⎪ ⎪ ⎪ + d (x)[−k ( wt (1, t) − y1 (t))] 3 1 ⎪ ⎪ ⎪ ⎪ + d (x)[−k ( ϕt (1, t) − y2 (t))], x ∈ (0, 1), t > 0, 4 2 ⎪ ⎪ ⎪ ⎪ q1,t (x, t) = K (p1,xx − p2,x )(x, t) − b1 (x)U1 (p1 , p2 , q1 , q2 )−b2 (x)U2 (p1 , p2 , q1 , q2 ) ⎪ ρ ⎪ ⎪ ⎪ ⎪ + e1 (x)[−k1 ( wt (1, t) − y1 (t))] + e2 (x)[−k2 ( ϕt (1, t) − y2 (t))] ⎪ ⎪ ⎪ K ⎨ q2,t (x, t) = EI p (x, t) + (p − p )(x, t) − b (x)U 2,xx 1,x 2 3 1 (p1 , p2 , q1 , q2 ) I I ρ

ρ

wt (1, t) − y1 (t))] − b4 (x)U2 (p1 , p2 , q1 , q2 ) + e3 (x)[−k1 ( ⎪ ⎪ ⎪ ⎪ ϕt (1, t) − y2 (t))], + e4 (x)[−k2 ( ⎪ ⎪ ⎪ ⎪ p1 (0, t) = p2 (0, t) = q1 (0, t) = q2 (0, t) = 0, t > 0, ⎪ ⎪ ⎪ ⎪ K(p 1,x − p2 )(1, t) = −β1 U1 (p1 , p2 , q1 , q2 ), t > 0, ⎪ ⎪ ⎪ ⎪ EIp (1, t) = −β2 U2 (p1 , p2 , q1 , q2 ), t > 0, 2,xx ⎪ ⎪ 0 0 ⎪ ⎪ (x, 0) = E1 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a1 (x, s)f1 (s)ds − −τ a2 (x, s)f1 (s)ds, p ⎪ 1 ⎪ 0 0 ⎪ ⎪ ⎪ w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a3 (x, s)f1 (s)ds − −τ a4 (x, s)f1 (s)ds, ⎪ ⎪ p2 (x, 0) = E2 (   ⎪ 0 0 ⎪ ⎪ q1 (x, 0) = E3 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b1 (x, s)f1 (s)ds + −τ b2 (x, s)f1 (s)ds, ⎪ ⎪ 0 0 ⎩ q2 (x, 0) = E4 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b3 (x, s)f1 (s)ds + −τ b4 (x, s)f1 (s)ds,

(2.6) In the present paper, we shall prove that under certain conditions the closed-loop system (2.6) is exponentially stable. Furthermore, we prove that the system (2.1) with the control signals (2.5) also is exponentially stable. The rest of this paper is organized as follows. In Section 2, we shall show that Eq. 2.2 is an exponential-type observer of Eq. 2.1. In Section 3, we use Eq. 2.3 to determine functions ai (x), bi (x), di (x), ei (x), ai (x, s), bi (x, s) (i = 1, 2, 3, 4) appearing in Eq. 2.4. In Section 4, we shall prove that the system (2.6) is exponentially stable. In Section 5, we shall prove that the system (2.1) with control (2.5) also is exponentially stable by estimating the error between Eqs. 2.1 and 2.6. In Section 6, we conclude the paper.

Xiu Fang Liu and Gen Qi Xu

3 Analysis of the Systems (2.1) and (2.2) In this section, we prove that the system (2.2) is an exponential-type observer of Eq. 2.1. To this end, we consider the error between Eqs. 2.1 and 2.2 1 (x, t) = w (x, t) − w(x, t),

2 (x, t) =  ϕ (x, t) − ϕ(x, t)

Obviously, 1 (x, t), 2 (x, t) satisfy the following equation: ⎧ ρ 1,tt (x, t) = K( 1,xx − 2,x )(x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ I ⎪ ρ 2,tt (x, t) = EI 2,xx (x, t) + K( 1,x − 2 )(x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎨ 1 (0, t) = 2 (0, t) = 0, t > 0, K( 1,x − 2 )(1, t) = −k1 1,t (1, t), ⎪ ⎪ ⎪ EI 2,x (1, t) = −k2 2,t (1, t), ⎪ ⎪ ⎪ (x, 0) = w 0 (x) − w0 (x), 1,t (x, 0) = w 1 (x) − w1 (x), ⎪ ⎪ ⎩ 1 ϕ0 (x) − ϕ0 (x), 2,t (x, 0) =  ϕ1 (x) − ϕ1 (x), 2 (x, 0) = 

(3.1)

For the error system (3.1), we have the following result [4]. Theorem 3.1 The energy function E(t) of the system (3.1) decays exponentially to zero, that is, there exist two constants ω1 > 0 and M1 > 0 such that: E(t) ≤ M1 e−ω1 t E(0),

∀t ≥ 0,

where E(t) is defined by 1 1 ||( 1 , 2 )||2V 1 (0,1)×V 1 (0,1) + ||( 1,t , 2,t )||2L2 (0,1)×L2 (0,1) ρ 2 2 K EI Iρ  1  1 1 1 = K| 1,x (x, t) − 2 (x, t)|2 dx + EI | 2,x (x, t)|2 dx 2 0 2 0   1 1 1 1 + ρ| 1,t (x, t)|2 dx + + Iρ | 2,t (x, t)|2 dx. 2 0 2 0

E(t) =

Theorem 3.1 asserts that the system (2.2) is an exponentially-type state observer of Eq. 2.1. For the use later, we give a corollary of Theorem 3.1. Corollary 3.1 Let ( 1 (x, t), 2 (x, t)) be the solution of the system (3.1). Then there exist constants ω1 and M1 > 0 such that, for any 0 < t1 < t2 , the following inequality holds  t2  t2 k1 | 1,t (1, t)|2 dt + k2 | 2,t (1, t)|2 dt ≤ M1 E(0)(e−ω1 t1 + e−ω1 t2 ). t1

t1

Proof A simple calculation gives dE(t) = −k1 | 1,t (1, t)|2 − k2 | 2,t (1, t)|2 . dt So,

 E(t1 ) − E(t2 ) = k1

t2

t1

 | 1,t (1, t)|2 dt + k2

t2

t1

| 2,t (1, t)|2 dt.

Output-Based Stabilization of Timoshenko Beam

By Theorem 3.1, we have  t2  2 1,t (1, t)dt + k2 k1 t1

t2

t1

2 2,t (1, t)dt ≤ M1 E(0)(e−ω1 t1 + e−ω1 t2 ).

4 Representation of the Systems (2.4) In this section, we shall determine all coefficient functions ai (x), bi (x), di (x), ei (x), ai (x, s), bi (x, s) (i = 1, 2, 3, 4) in Eq. 2.4 by using Eqs. 2.2 and 2.3. We begin with introducing a useful lemma. Lemma 4.1 [24] Let the differential operator L in L2ρ (0, 1) × L2Iρ (0, 1) be defined as

L(w, ϕ) = (− with domain

K  EI  K ϕ (x) − (w  (x) − ϕ(x)))T , (w (x) − ϕ  (x)), − ρ Iρ Iρ

⎫ ⎬

w(0) = ϕ(0) = 0

D(L) = (w(x), ϕ(x)) ∈ H 2 (0, 1) × H 2 (0, 1) K(w (1) − ϕ(1)) = 0 . ⎭ ⎩ EI ϕ  (1) = 0

(4.1)

⎧ ⎨

(4.2)

Then L is a positive define operator with compact resolvent in L2ρ (0, 1) × L2Iρ (0, 1), its eigenvalues are 0 < μ1 < μ2 < · · · < μn < · · · (4.3) T and the eigenfunctions n (x) = (wn (x), ϕn (x)) corresponding to μn are real functions and form a normalized orthogonal basis for L2ρ (0, 1) × L2Iρ (0, 1). For the sake of convenience, we write the Eq. 2.1 into the vector form    K  − Kρ ∂x w(x, t) wtt (x, t) ρ ∂xx =0 − K K EI ϕ(x, t) ϕtt (x, t) Iρ ∂x Iρ ∂xx − Iρ   w(0, t) and boundary conditions are = 0, and ϕ(0, t)    w(x, t) K∂x −K ϕ(x, t) x=1 0 EI ∂x       β1 0 α1 0 u1 (t) u1 (t − τ ) + = 0 α2 u2 (t) 0 β2 u2 (t − τ )    0 u1 (t + η) g1 (η) 0 + dη. 0 g2 (η) u2 (t + η) −τ The initial datum are     w0 (x) w(x, 0) , = ϕ0 (x) ϕ(x, 0)



wt (x, 0) ϕt (x, 0)



 =

w1 (x) ϕ1 (x)

 .

Set W (x, t) = (w(x, t), ϕ(x, t))T and U (t) = (u1 (t), u2 (t))T . Define 2 × 2 matrices       α1 0 β1 0 g1 (η) 0 1 = , 2 = , 3 (η) = 0 α2 0 β2 0 g2 (η)

Xiu Fang Liu and Gen Qi Xu

and define an operator B from R2 to H −1 (0, 1) × H −1 (0, 1), where Vω1 (0, 1) = {f ∈ H 1 (0, 1) f (0) = 0} and H −1 (0, 1) = (Vω1 (0, 1))∗ is the dual space, and   δ(x − 1) 0 , B= 0 δ(x − 1) we know reference [25] for δ− function and define an operator N from H 2 (0, 1) × H 2 (0, 1) to R2 by   K(w  (1) − ϕ(1)) N W = EI ϕ  (1) where W (x) = (w(x), ϕ(x))T . With these notations, we can rewrite Eq. 2.1 into ⎧ 0 ⎪ W (x, t) + LW (x, t) = B(1 U (t) + 2 U (t − τ ) + −τ 3 (η)U (t + η)dη), t > 0 ⎪ ⎨ tt W (0, t) = 0, N W (1, t) = 0, ⎪ W (x, 0) = W0 (x), ⎪ ⎩ Wt (x, 0) = W1 (x), (4.4) Similarly, we can write the Eq. 2.2 into the vector form ⎧  (x, t) = B[−K(W t (1, t) − Y (t)) + 1 U (t) + 2 U (t − τ ) Wtt (x, t) + LW ⎪ ⎪ 0 ⎪ ⎪ + −τ 3 (η)U (t + η)dη], ⎨  (1, t) = 0,  (0, t) = 0, N W (4.5) W ⎪ ⎪  (x, 0) = W 0 (x), ⎪W ⎪ ⎩ t (x, 0) = W 1 (x), W where  (x, t) = W



w (x, t)  ϕ (x, t)



 ,

Y (t) =

y1 (t) y2 (t)



 K=

,

k1 0 0 k2

 .

And (2.3) is of the form ⎧ (x, s, t), s ∈ (0, τ ) Ws (x, s, t) = V ⎪ ⎪  ⎪ (x, s, t) = B2 U (t + s − τ ) + B −s 3 (η)U (t + s + η)dη, s (x, s, t) + LW ⎪V ⎨ −τ (1, s, t) = 0, (0, s, t) = 0, N W (4.6) W ⎪ ⎪  ⎪ W (x, 0, t) = W (x, t), ⎪ ⎩ t (x, t) V (x, 0, t) = W (x, s, t) = ( (x, s, t) = ( s, t))T . z(x, s, t), ψ(x, where W w(x, s, t), ϕ (x, s, t))T , V 2 We define two families of the bounded linear operators on Lρ (0, 1) × L2Iρ (0, 1) by Cos(t L)F =

∞  n=1

Sin(t L)F =

cos

√ μn t (F, n )L2 ×L2 n , ρ

Iρ

√ ∞  sin μn t (F, n )L2 ×L2 n √ ρ Iρ μn n=1

where μn and n are given as Lemma 4.1. Clearly, the following equalities hold, for any t ∈ R,  t d (Cos(t L)) = −LSin(t L). Sin(t L) = Cos(t L)dt, dt 0

Output-Based Stabilization of Timoshenko Beam

It is easy to see that the vector-valued function  t W (x, t) = Cos(t L)W0 + Sin(t L)W1 + Sin((t − s)L)B[1 U (s) + 2 U (s − τ )]ds  t  + Sin((t − s)L)B

0 0

−τ

0

3 (η)U (s + η)dηds

is differentiable with respect to t and



t

Wt (x, t) = −LSin(t L)W0 + Cos(t L)W1 +

Cos((t − s)L)B[1 U (s)

0



t

+2 U (s − τ )]ds +

(4.7)



Cos((t − s)L)B

0

−τ

0

3 (η)U (s + η)dηds. (4.8)

Furthermore, W (x, t) satisfies Eq. 4.4. About the regularity of U we refer to [28]. Similarly, the vector-valued functions  t 0 + Sin(t L)W 1 +  (x, t) = Cos(t L)W Sin((t − s)L)B[1 U (s) + 2 U (s − τ )]ds W  t  + Sin((t − s)L)B

0 0

−τ

0

3 (η)U (s + η)dηds

 t t (1, s) − Y (s))]ds + Sin((t − s)L)B[−K(W

(4.9)

0

and 0 + Cos(t L)W 1 + t (x, t) = −LSin(t L)W W  +2 U (s − τ )]ds +



t

Cos((t − s)L)B[1 U (s)x

0 t

Cos((t − s)L)B

0



0

−τ

3 (η)U (s + η)dηds

 t t (1, s) − Y (s))]ds + Cos((t − s)L)B[−K(W 0

satisfy Eq. 4.5. The vector-valued functions  (·, t) + Sin(s L)W t (·, t) (x, s, t) = Cos(s L)W W  s + Sin((s − r)L)B2 U (t + r − τ )dr 0  s  −r + Sin((s − r)L)B 3 (η)U (t + r + η)dηdr 0

−τ

and  (·, t) + Cos(s L)W t (·, t) (x, s, t) = −LSin(s L)W V  s + Cos((s − r)L)B2 U (t + r − τ )dr 0  s  −r + Cos((s − r)L)B 3 (η)U (t + r + η)dηdr 0

satisfy Eq. 4.6. Set

(x, τ, t), P (x, t) = W

−τ

(x, τ, t). Q(x, t) = V

(4.10)

Xiu Fang Liu and Gen Qi Xu

Then we have       (x, t) W Cos(τ L) Sin(τ L) P (x, t) = t (x, t) −LSin(τ L) Cos(τ L) Q(x, t) W   τ Sin((τ − r)L) + B2 U (t + r − τ )dr Cos((τ − r)L) 0   −r  τ Sin((τ − r)L) B 3 (η)U (t + r + η)dηdr + Cos((τ − r)L) 0 −τ and      t (x, t) W Cos(τ L) Sin(τ L) Pt (x, t) = tt (x, t) Qt (x, t) −LSin(τ L) Cos(τ L) W     τ 0 Cos((τ − r)L) + B2 U (t + r − τ )dr + B2 U (t) −LSin((τ − r)L) 0     0  Sin(τ L) Sin(τ L) − B2 U (t − τ ) − B 3 (η)U (t + η)dη Cos(τ L) Cos(τ L) −τ   −r  τ Cos((τ − r)L) + B 3 (η)U (t + r + η)dηdr − LSin((τ − r)L) 0 −τ   0 Sin((τ + η)L) + B3 (η)U (t)dη. Cos((τ + η)L) −τ Since 

t (x, t) W tt (x, t) W



 = +

0 I −L 0 



 (x, t) W t (x, t) W



0  t (1, t) − Y (t)) + 1 U (t) + 2 U (t − τ ) + 0 3 (η)U (t + η)dη] B[−K(W −τ

 ,

so it holds that  Pt (x, t)  Qt (x, t)   0 I P (x, t) = − Q(x, t) L 0   t (1, t) − Y (t)) + 1 U (t)] + 0 Sin((τ + η)L)B3 (η)U (t)dη Sin(τ L)B[−K(W −τ 0 + . t (1, t) − Y (t)) + 1 U (t)] + Cos(τ L)B[−K(W −τ Cos((τ + η)L)B3 (η)U (t)dη + B2 U (t) 

Thus, we get equations ⎧ t (1, t) − Y (t)) + 1 U (t)] P (x, t) = Q(x, t) + Sin(τ L)B[−K(W ⎪ ⎪ 0 ⎪ t ⎪ ⎪ + −τ Sin((τ + η)L)B3 (η)U (t)dη, ⎨ t (1, t) − Y (t)) + 1 U (t)] Qt (x, t) = −LP (x, t) + Cos(τ L)B[−K(W 0 ⎪ ⎪ ⎪ + −τ Cos((τ + η)L)B3 (η)U (t)dη, ⎪ ⎪ ⎩ P (0, t) = Q(0, t) = 0, N P (·, t) = 2 U (t) and initial conditions

 0 + Sin(τ L)W 1 − 0 Sin(s L)B2 f (s)ds P (x, 0) = Cos(τ L)W −τ 0 s + −τ −τ Sin((τ − s + η)L)B3 (η)f (s)dηds

(4.11)

Output-Based Stabilization of Timoshenko Beam

and

 0 + Cos(τ L)W 1 + 0 Cos(s L)B2 f (s)ds Q(x, 0) = −LSin(τ L)W −τ 0 s + −τ −τ Cos((τ − s + η)L)B3 (η)f (s)dηds

where f (s) = (f1 (s), f2 (s))T . Since all entries of B are meaningful as linear functional on H 1 (0, 1), for any Z = (z1 , z2 ) ∈ R2 and n (x) ∈ H 1 (0, 1) × H 1 (0, 1), it holds that 1 1 (BZ, n )L2 (0,1)×L2 (0,1) = z1 0 ρδ(x − 1)wn (x)dx + z2 0 Iρ δ(x − 1)ϕn (x)dx ρ

Iρ

= ρz1 wn (1) + Iρ z2 ϕn (1) = [z1 , z2 ][ρwn (1), Iρ ϕn (1)]T

Therefore, we have the following result. Theorem 4.1 Let {μn ; n ∈ N} be the list of all eigenvalues of L. Then the functions appearing in Eq. 2.4 are given as follows ⎧   √ √ ∞ 0  sin τ μn sin(τ +η) μn ⎪ √ √ ⎪ a (x) = ρ w (1)w (x) α + g (η) dη n n 1 1 1 ⎪ −τ μn μn ⎪ ⎪ n=1 ⎪   √ √ ⎪ ∞   ⎪ sin τ μ sin(τ +η) μn 0 ⎪ ⎪ √ ϕn (1)wn (x) α2 √μn n + −τ g2 (η) dη ⎨ a2 (x) = Iρ μn n=1 (4.12)   √ √ ∞   sin(τ +η) μn sin τ μn ⎪ 0 ⎪ √ √ a (x) = ρ w (1)ϕ (x) α + g (η) dη ⎪ n n 3 1 1 −τ ⎪ μn μn ⎪ n=1 ⎪ ⎪   √ √ ∞ ⎪   ⎪ sin τ μ sin(τ +η) μn 0 ⎪ √ ⎩ a4 (x) = Iρ ϕn (1)ϕn (x) (α2 √μn n + −τ g2 (η) dη μn n=1

⎧   ∞ 0  √ √ ⎪ ⎪ b1 (x) = ρ wn (1)wn (x) α1 cos τ μn + −τ g1 (η) cos(τ + η) μn dη ⎪ ⎪ ⎪ n=1 ⎪   ⎪ ∞ 0  ⎪ √ √ ⎪ ⎪ b (x) = I ϕ (1)w (x) α cos τ μ + g (η) cos(τ + η) μ dη ρ n n n n 2 2 2 ⎨ −τ n=1   ∞   √ √ ⎪ 0 ⎪ b (x) = ρ w (1)ϕ (x) α cos τ μ + g (η) cos(τ + η) μ dη ⎪ n n n n 3 1 1 −τ ⎪ ⎪ n=1 ⎪ ⎪   ∞ ⎪   √ √ ⎪ 0 ⎪ ⎩ b4 (x) = Iρ ϕn (1)ϕn (x) α2 cos τ μn + −τ g2 (η) cos(τ + η) μn dη n=1

(4.13)

⎧ √ ∞  sin τ μ ⎪ ⎪ d1 (x) = ρ wn (1)wn (x) √μn n ⎪ ⎪ ⎪ n=1 ⎪ √ ⎪ ∞  ⎪ sin τ μ ⎪ ⎪ ϕn (1)wn (x) √μn n ⎨ d2 (x) = Iρ n=1

√ ∞  sin τ μn ⎪ ⎪ √ d (x) = ρ w (1)ϕ (x) ⎪ n n 3 ⎪ μn ⎪ n=1 ⎪ ⎪ √ ∞ ⎪  ⎪ sin τ μ ⎪ ⎩ d4 (x) = Iρ ϕn (1)ϕn (x) √μ n n=1

(4.14)

n

⎧ ∞  √ ⎪ ⎪ e1 (x) = ρ wn (1)wn (x) cos τ μn ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞  ⎪ √ ⎪ ⎪ e (x) = I ϕn (1)wn (x) cos τ μn ρ ⎨ 2 n=1

∞  √ ⎪ ⎪ e3 (x) = ρ wn (1)ϕn (x) cos τ μn ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪  √ ⎪ ⎪ ⎩ e4 (x) = Iρ ϕn (1)ϕn (x) cos τ μn n=1

(4.15)

Xiu Fang Liu and Gen Qi Xu

⎧   √ √ ∞ s  sin s μ sin(τ −s+η) μn ⎪ √ ⎪ a1 (x, s) = ρ wn (1)wn (x) β1 √μn n − −τ g1 (η) dη ⎪ μ n ⎪ ⎪ n=1 ⎪   √ √ ⎪ ∞ s  ⎪ sin s μ sin(τ −s+η) μn ⎪ ⎪ √ a (x, s) = I ϕn (1)wn (x) β2 √μn n − −τ g2 (η) dη ρ ⎨ 2 μn n=1   √ √ ∞ s  sin s μ sin(τ −s+η) μn ⎪ ⎪ √ a (x, s) = ρ wn (1)ϕn (x) β1 √μn n − −τ g1 (η) dη ⎪ 3 ⎪ μ n ⎪ n=1 ⎪ ⎪   √ √ ∞ ⎪ s  ⎪ sin(τ −s+η) μn sin s μ ⎪ √ ⎩ a4 (x, s) = Iρ ϕn (1)ϕn (x) β2 √μn n − −τ g2 (η) dη μn

(4.16)

n=1

⎧ ∞ s    √ √ ⎪ ⎪ wn (1)wn (x) β1 cos s μn + −τ g1 (η) cos(τ − s + η) μn dη ⎪ b1 (x, s) = ρ ⎪ ⎪ n=1 ⎪ ⎪ ∞ s    ⎪ √ √ ⎪ ⎪ ϕn (1)wn (x) β2 cos s μn + −τ g2 (η) cos(τ − s + η) μn dη ⎨ b2 (x, s) = Iρ n=1

∞ s    √ √ ⎪ ⎪ wn (1)ϕn (x) β1 cos s μn + −τ g1 (η) cos(τ − s + η) μn dη b3 (x, s) = ρ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪ s    √ √ ⎪ ⎪ ⎩ b4 (x, s) = Iρ ϕn (1)ϕn (x) β2 cos s μn + −τ g2 (η) cos(τ − s + η) μn dη n=1

(4.17) and the linear operators are ⎧ ∞  √ ⎪ 0 , n ) + ⎪ E1 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) = [cos τ μn (W ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞  ⎪ √ ⎪ 0 , n ) + ⎪ w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) = [cos τ μn (W ⎨ E2 ( n=1

√ sin τ μn  √ μn (W1 , n )]wn (x) √ sin τ μn  √ μn (W1 , n )]ϕn (x)

∞  √ √ ⎪ ⎪ 0 , n ) + cos τ √μn (W 1 , n )]wn (x) E3 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) = [− μn sin τ μn (W ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ∞ ⎪  √ √ ⎪ ⎪ 0 , n ) + cos τ √μn (W 1 , n )]ϕn (x) ⎩ E4 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) = [− μn sin τ μn (W n=1

(4.18)

5 The Stability of the System (2.6) In this section, we shall prove the system (2.6) is exponentially stable. To this aim, we first consider the following system, which is the system (2.4) without the disturbance terms. ⎧ p1,t (x, t) = q1 (x, t) + a1 (x)u1 (t) + a2 (x)u2 (t), ⎪ ⎪ ⎪ p (x, t) = q (x, t) + a (x)u (t) + a (x)u (t), ⎪ 2,t 2 3 1 4 2 ⎪ ⎪ ⎪ K ⎪ q (x, t) = (p − p )(x, t) + b (x)u ⎪ 1,t 1,xx 2,x 1 1 (t) + b2 (x)u2 (t), ρ ⎪ ⎪ EI K ⎪ ⎪ (x, t) = p (x, t) + (p − p )(x, t) + b3 (x)u1 (t) + b4 (x)u2 (t), q 2,t 2,xx 1,x 2 ⎪ Iρ Iρ ⎪ ⎪ ⎪ ⎪ (0, t) = p (0, t) = q (0, t) = q (0, t) = 0, t > 0, p 2 1 2 ⎨ 1 K(p1,x − p2 )(1, t) = β1 u1 (t), t > 0, (5.1) ⎪ ⎪ EIp2,x (1, t) = β2 u2 (t), t > 0, ⎪ ⎪ 0 0 ⎪ ⎪ w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a1 (x, s)f1 (s)ds − −τ a2 (x, s)f2 (s)ds, p1 (x, 0) = E1 ( ⎪ ⎪  0 ⎪ 0 ⎪ ⎪ p2 (x, 0) = E2 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a3 (x, s)f1 (s)ds − −τ a4 (x, s)f2 (s)ds, ⎪ ⎪   ⎪ 0 0 ⎪ ⎪ q1 (x, 0) = E3 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b1 (x, s)f1 (s)ds + −τ b2 (x, s)f2 (s)ds, ⎪ ⎪ 0 0 ⎩ q2 (x, 0) = E4 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b3 (x, s)f1 (s)ds + −τ b4 (x, s)f2 (s)ds,

Output-Based Stabilization of Timoshenko Beam

The energy functional of this system is defined by 1 1 ||(p1 , p2 )||2V 1 (0,1)×V 1 (0,1) + ||(q1 , q2 )||2L2 (0,1)×L2 (0,1) ρ 2 2 K EI Iρ  1  1 1 1 = K|p1,x (x, t) − p2 (x, t)|2 dx + EI |p2,x (x, t)|2 dx 2 0 2 0   1 1 1 1 2 + ρ|q1 (x, t)| dx + Iρ |q2 (x, t)|2 dx. 2 0 2 0

E(t) =

A direct calculation gives   1 dE(t) = u1 (t) β1 q1 (1, t) + K(p1,x − p2 )[a1 (x) − a3 (x)]dx dt 0   1  1  1  + EIp2,x (x, t)a3 (x)dx + ρq1 (x, t)b1 (x)dx + Iρ q2 (x, t)b3 (x)dx 0



0



1

+u2 (t) β2 q2 (1, t) + 0

0

K(p1,x − p2 )[a2 (x) − a4 (x)]dx

  1  1  1 ρq1 (x, t)b2 (x)dx + Iρ q2 (x, t)b4 (x)dx + EIp2,x (x, t)a4 (x)dx + 0

0

0

Set U1 (p1 , p2 , q1 , q2 )  1  1 = β1 q1 (1, t) + K(p1,x (x, t) − p2 (x, t))(a1 (x) − a3 (x))dx + ρq1 (x, t)b1 (x)dx 0

 1  1 + EIp2,x (x, t)a3 (x)dx + Iρ q2 (x, t)b3 (x)dx; 0

0

0

U2 (p1 , p2 , q1 , q2 )  1  1 = β2 q2 (1, t) + K(p1,x (x, t) − p2 (x, t))(a2 (x) − a4 (x))dx + ρq1 (x, t)b2 (x)dx 0

 1  1 + EIp2,x (x, t)a4 (x)dx + Iρ q2 (x, t)b4 (x)dx. 0

0

0

We take the feedback control law as u1 (t) = −U1 (p1 , p2 , q1 , q2 ),

u2 (t) = −U2 (p1 , p2 , q1 , q2 ).

Xiu Fang Liu and Gen Qi Xu

With this control law, we get the closed loop system: ⎧ p 1,t (x, t) = q 1 (x, t) − a1 (x)U1 (p 1 , p 2 , q 1 , q 2 ) − a2 (x)U2 (p 1 , p 2 , q 1 , q 2 ), ⎪ ⎪ ⎪ p (x, t) = q (x, t) − a (x)U (p , p , q , q ) − a (x)U (p , p , q , q ), ⎪ 3 1 1 4 2 1 ⎪ 2,t 2 2 1 2 2 1 2 ⎪ ⎪ K ⎪ q (x, t) = (p − p )(x, t) − b (x)U (p , p , q , q ) ⎪ 1 1 1,t 1,xx 2,x 1 2 1 2 ρ ⎪ ⎪ ⎪ ⎪ − b2 (x)U2 (p 1 , p 2 , q 1 , q 2 ), ⎪ ⎪ K ⎪ ⎪ q 2,t (x, t) = EI ⎪ Iρ p 2,xx (x, t) + Iρ (p 1,x − p 2 )(x, t) − b3 (x)U1 (p 1 , p 2 , q 1 , q 2 ) ⎪ ⎪ ⎪ ⎪ − b4 (x)U2 (p 1 , p 2 , q 1 , q 2 ), ⎨ p 1 (0, t) = p 2 (0, t) = q 1 (0, t) = q 2 (0, t) = 0, t > 0, (5.2) ⎪ ⎪ ⎪ K(p 1,x − p2 )(1, t) = β1 U1 (p 1 , p 2 , q 1 , q 2 ), t > 0, ⎪ ⎪ ⎪ EI p 2,x (1, t) = β2 U2 (p 1 , p2 , q 1 , q 2 ), t > 0, ⎪ ⎪ 0 0 ⎪ ⎪ p 1 (x, 0) = E1 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a1 (x, s)f1 (s)ds − −τ a2 (x, s)f2 (s)ds, ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) − −τ a3 (x, s)f1 (s)ds − −τ a4 (x, s)f2 (s)ds, ⎪ ⎪ p 2 (x, 0) = E2 (   ⎪ 0 0 ⎪ ⎪ q 1 (x, 0) = E3 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b1 (x, s)f1 (s)ds + −τ b2 (x, s)f2 (s)ds, ⎪ ⎪ 0 0 ⎩ q 2 (x, 0) = E4 ( w0 ,  ϕ0 , w 1 ,  ϕ1 )(x) + −τ b3 (x, s)f1 (s)ds + −τ b4 (x, s)f2 (s)ds, Now, we let (p1 (x, t), p2 (x, t), q1 (x, t), q2 (x, t)) be the solution of (2.6). We set i (x, t) = pi (x, t) − p i (x, t), p

 qi (x, t) = qi (x, t) − q i (x, t), i = 1, 2.

Substituting them into Eq. 2.6 lead to ⎧ 1,t (x, t) =  2 ,  2 ,  p q1 (x, t) − a1 (x)U1 ( p1 , p q1 ,  q2 ) − a2 (x)U2 ( p1 , p q1 ,  q2 ) ⎪ ⎪ ⎪ ⎪ − k d (x) (1, t) − k d (x) (1, t), ⎪ 1 1 1,t 2 2 2,t ⎪ ⎪ ⎪ 2,t (x, t) =  2 ,  2 ,  p q2 (x, t) − a3 (x)U1 ( p1 , p q1 ,  q2 ) − a4 (x)U2 ( p1 , p q1 ,  q2 ) ⎪ ⎪ ⎪ ⎪ d (x) (1, t) − k d (x) (1, t), − k ⎪ 1 3 1,t 2 4 2,t ⎪ ⎪ ⎪ ⎪ 2,x )(x, t) − b1 (x)U1 ( 2 ,  2 ,  q1,t (x, t) = Kρ ( p1,xx − p p1 , p q1 ,  q2 ) − b2 (x)U2 ( p1 , p q1 ,  q2 ) ⎪ ⎪ ⎨ − k1 e1 (x) 1,t (1, t) − k2 e2 (x) 2,t (1, t),  2,xx (x, t) + IKρ ( 2 )(x, t) − b3 (x)U1 ( 2 ,  q2,t (x, t) = EI p1,x − p p1 , p q1 ,  q2 ) ⎪ ⎪ Iρ p ⎪ ⎪ ⎪ 2 ,  p1 , p q1 ,  q2 ) − k1 e3 (x) 1,t (1, t) − k2 e4 (x) 2,t (1, t), − b4 (x)U2 ( ⎪ ⎪ ⎪ ⎪ 1 (0, t) = p 2 (0, t) =  p q1 (0, t) =  q2 (0, t) = 0, t > 0, ⎪ ⎪ ⎪ ⎪ 2 )(1, t) = β1 U1 ( 2 ,  p1 , p q1 ,  q2 ), t > 0, K( p1,x − p ⎪ ⎪ ⎪ ⎪     EI p (1, t) = β U ( p , p , q , q ), t > 0, ⎪ 2,x 2 2 1 2 1 2 ⎪ ⎩ 2 (x, 0) = 0,  1 (x, 0) = 0, p q1 (x, 0) = 0,  q2 (x, 0) = 0, p (5.3) 1 (0, 1) × L2 [0, 1] × L2 [0, 1] with energy We take the energy space H = VK1 (0, 1) × VEI ρ Iρ norm. In space H, the system (5.3) can be rewritten as abstract evolution equation: 

= AY (t), t > 0, Y (0) = Y0 dY (t) dt

(5.4)

Let S(t) be the C0 semigroup generated by A. According to the result in [27], the system (5.2) under certain conditions is exponentially (or asymptotically) stable, that is, there is a positive constant ω2 such that ||S(t)|| ≤ Me−ω2 t , ∀t ≥ 0 (or

lim ||S(t)Y0 || = 0).

t→∞

Output-Based Stabilization of Timoshenko Beam

Then, the solution of Eq. 5.2 is given by (P (t), Q(t)) = S(t)(P (0), Q(0)), and the solution of the system (5.3) is (t), Q(t))  (0), Q(0))  (P = S(t)(P +



t

 S(t − s)D(−K t (1, s))ds,

0

 = ( (0), Q(0))  (t) = ( 2 (x, t))T , Q(t) q1 (x, t),  q2 (x, t))T , (P = (0, 0), where P p1 (x, t), p       k1 0  = d1 (x) d2 (x) , t (1, s) = 1,t (1, s) . , D K= 0 k2 d3 (x) d4 (x) 2,t (1, s) Let 0 < t < t. Then  t (t), Q(t))   (P = S(t − s)D(−K t (1, s))ds 0

 t  t   (1, s))ds − S(t − s)D(K = − S(t − s)D(K t t (1, s))ds 0



= −S(t − t)

t t

 S(t − s)D(K t (1, s))ds −

0



t

 S(t − s)D(K t (1, s))ds.

t

From the above, we can get the following Theorem. Theorem 5.1 If the system (5.2) is exponentially stable, so is the system (5.3); if the system (5.2) is asymptotically stable, then the system (5.3) is also asymptotically stable. Proof We estimate the norm of the solution of Eq. 5.3 as  2  t   2    ( P (t), Q(t))   ≤ 2 S(t − t) S(t − s)D(K  t (1, s))ds    0  t 2     +2   S(t − s)D(K t (1, s))ds  t

2   t  2     ≤ 2 S(t − t)  S(t − s)D(K t (1, s))ds    0  t 2     +2   S(t − s)D(K t (1, s))ds  . t

 is admissible for S(t) (see [29, Proposition 5.5.2, pp.165]), we have inequalities Since D     t  t t 2 2 2 2 2 

1,t (1, s) ds + k

2,t (1, s) ds ,

S(t − s)D(K t (1, s))ds ≤ M2 k 0

1

0

2

0

   t  t  t 2 2 2 2 2 

S(t − s)D(K t (1, s))ds ≤ M3 k1 1,t (1, s) ds + k2 2,t (1, s) ds , t

where Mi > 0, i = 2, 3 are two constants.

t

t

Xiu Fang Liu and Gen Qi Xu

According to Corollary 3.1, there exists constants Mi > 0, i = 4, 5 such that  2  t      S(t − s)D(K t (1, s))ds  ≤ M4 E(0),  0   t 2      S(t − s)D(K  ≤ M5 E(0) e−ω1 t + e−ω1 t .  (1, s))ds t   t

Thus, we get 2 (t), Q(t))  ≤ 2M4 M 2 e−2ω2 (t−t) E(0) + 2M5 E(0)(e−ω1 t + e−ω1 t ).

(P

Taking t = 2t , we obtain 2 (t), Q(t))  ≤ 2M4 M 2 E(0)e−ω2 t + 4M5 E(0)e−

(P

ω1 2 t

≤ M6 e−ωt E(0) where ω = min{ ω21 , ω2 } and M6 = max{2M4 M 2 , 4M5 }. So (5.3) also is exponentially stable. If the system (5.2) is asymptotically stable, i.e., limt→∞ S(t)(P , Q) = 0, from the previous estimation we see (x, t), Q(x,  t)) 2 = 0. lim (P

t→∞

So the system (5.3) is asymptotically stable. The desired results follow.

6 The Stability of the System (2.1) with Control (2.5) In this section, we shall prove the system (2.1) with control (2.5) has the same stability as the closed-loop system (2.6). Herein, we mainly estimate the error:

P (·, t) − W (·, t + τ ) 2V 1 (0,1)×V 1 K

EI (0,1)

+ Q(·, t) − Wt (·, t + τ ) 2L2 (0,1)×L2 ρ

According to the calculation in Section 4, we have  (x, t + τ ) P (x, t) − W  τ = − Sin((τ − r)L)B1 U (t + r)dr 0

 τ  0 − Sin((τ − r)L)B 3 (η)U (t + r + η)dηdr 0 −τ  τ t (1, r + t) − Y (r + t))]dr − Sin((τ − r)L)B[−K(W 0

Iρ (0,1)

.

Output-Based Stabilization of Timoshenko Beam

and t (x, t + τ ) Q(x, t) − W  τ = − Cos((τ − r)L)B1 U (t + r)dr 0

 τ  0 − Cos((τ − r)L)B 3 (η)U (t + r + η)dηdr 0 −τ  τ t (1, r + t) − Y (r + t))]dr. − Cos((τ − r)L)B[−K(W 0

We have the following inequality t (·, t + τ ) 2 2 + Q(·, t) − W L (0,1)×L2

 (·, t + τ ) 2 1

P (·, t) − W V (0,1)×V 1 ≤ 4(α1 ρ)2

∞ 



sin

+4ρ

2



+4Iρ2 +4ρ 2 +4Iρ2

0

 |ϕn (1)| |

+4ρ

2

+4Iρ2 +4ρ 2 +4Iρ2

n=1 ∞  n=1 ∞  n=1 ∞  n=1



√ sin μn (τ − r)



g1 (η)u1 (r + t + η)dη dr|2

0

−τ



g2 (η)u2 (r + t + η)dη dr|2

√ μn (τ − r)(−k1 ( wt (1, r + t) − y1 (r + t))dr|2

τ

|wn (1)|2 |



0

−τ



0

sin 0



τ

|ϕn (1)|2 |

sin

√ μn (τ − r)(−k2 ( ϕt (1, r + t) − y2 (r + t))dr|2

0 ∞ 



n=1 ∞  2

cos

√ μn (τ − r)u1 (r + t)dr|2

cos

√ μn (τ − r)u2 (r + t)dr|2

τ

|wn (1)| | 2

0

 |ϕn (1)|2 |

τ

0

n=1 ∞ 

τ

2

2

+4(α2 Iρ )

√ μn (τ − r)u2 (r + t)dr|2

√ sin μn (τ − r)

τ

|wn (1)| |

n=1

+4(α1 ρ)

sin 0

 2

n=1 ∞  n=1 ∞ 

√ μn (τ − r)u1 (r + t)dr|2

τ

|ϕn (1)|2 |

n=1 ∞ 

ρ

0

∞  n=1

∞ 

τ

|wn (1)|2 |

n=1

+4(α2 Iρ )2

EI (0,1)

K



τ

|wn (1)| | 2

√ cos μn (τ − r)

0



τ

|ϕn (1)| | 2

√ cos μn (τ − r)

0



τ

|wn (1)|2 |

cos





0

−τ 0

−τ

 g1 (η)u1 (r + t + η)dη dr|2 

g2 (η)u2 (r + t + η)dη dr|2

√ μn (τ − r)(−k1 ( wt (1, r + t) − y1 (r + t))dr|2

0

 |ϕn (1)|2 |

τ

cos 0

√ μn (τ − r)(−k2 ( ϕt (1, r + t) − y2 (r + t))dr|2

Iρ (0,1)

Xiu Fang Liu and Gen Qi Xu

√ √ Since {cos μn t, sin μn t; n ∈ N} is a Bessel sequence for L2 [0, τ ](see[30]), there i (i = 1, 2, · · · , 6) such that exists positive constants M 2  (·, t + τ ) 2 1 

P (·, t) − W 1 (0,1) + Q(·, t) − Wt (·, t + τ ) L2 (0,1)×L2 (0,1) VK (0,1)×VEI ρ Iρ  τ  τ 2 2 2 2   ≤ M |u1 (r + t)| dr + M |u2 (r + t)| dr 1 2 0 0  τ

 0  τ

 0



2 2 2   g1 (η)u1 (t + r + η)dη| dr + M4 g2 (η)u2 (t + r + η)dη|2 dr +M3



0 −τ 0 −τ  τ  τ 2 2 2 +M k | w (1, r + t) − y (r + t)| dr + M k2 | ϕt (1, r + t) − y2 (r + t)|2 dr t 1 1 5 6 0 0  τ  τ 2 2 2  ≤ M |u (r + t)| dr + M |u2 (r + t)|2 dr 1 1 2 0

2 +M 3 2 +M 4



0

−τ 0

|g1 (η)| dη 2

 τ 0



|g2 (η)| dη

0

0

−τ 0

|u1 (t + r + η)|2 dηdr

 τ

|u2 (t + r + η)|2 dηdr −τ 0 −τ  τ  τ 2 2  2 +M k | w (1, r + t) − y (r + t)| dr + M k2 | ϕt (1, r + t) − y2 (r + t)|2 dr t 1 1 5 6 2

0

0

Let (P (t), Q(t)) be the solution to Eq. 5.2, E(t) be its energy functional. Then we have E(t) =

1

(P (t), Q(t)) 2H 2

and dE(t) = −U12 (P , Q)(t) − U22 (P , Q)(t) = −||U (t)||2 dt where U (t) = (U1 (t), U2 (t))T . The above equality implies U (t) ∈ L2 (R+ ) that will be used later. From above, we get  τ  t+τ

U (t + r) 2R2 dr = (U12 (P , Q)(r) + U22 (P , Q)(r))dr = E(t) − E(t + τ ). 0

t

 τ

0

−τ

0

and



U (t + r 

τ

0

+ η) 2R2 dηdr

=

τ

(E(t + r − τ ) − E(t + r))dr

0

U (t − r) 2R2 dr = E(t − τ ) − E(t).

According to Corollary 3.1, we have  τ  τ | wt (1, r + t) − y1 (r + t)|2 dr + k2 | ϕt (1, r + t) − y2 (r + t)|2 dr k1 0

= E(t) − E(t + τ ) ≤ M1 E(0)e−ω1 t .

0

Output-Based Stabilization of Timoshenko Beam

We further estimate t (·, t + τ ) 2 2 + Q(·, t) − W L (0,1)×L2

 (·, t + τ ) 2 1

P (·, t) − W V (0,1)×V 1

EI (0,1)

K

ρ

Iρ (0,1)

 2 }[E(t) − E(t + τ )] + max {M  2 }[E(t) − E(t + τ )] ≤ max {M i i i=1,2



2 } + max {M j j =3,4

i=5,6

0

−τ



τ

(|g1 (η)| + |g2 (η)| )dη 2

2

[E(t + r − τ ) − E(t + r)]dr.

0

Obviously, if the system (5.2) is exponential stable, i.e., E(t) ≤ ME(0)e−ω2 t , then  (·, t + τ ) 2 1

P (·, t) − W ≤

1 (0,1) VK (0,1)×VEI

 −ω2 (t−τ ) Me

t (·, t + τ ) 2 + Q(·, t) − W 2

Lρ (0,1)×L2Iρ (0,1)

 is an appropriate positive constant. where ω = min{ ω21 , ω2 }, and M Since

P (·, t) − W ·, t + τ ) 2V 1 (0,1)×V 1

EI (0,1)

K

 (·, t ≤ 2 P (·, t) − W

+ τ ) 2V 1 (0,1)×V 1 (0,1) K EI

−W (·, t + τ ) 2V 1 (0,1)×V 1 K

+ Q(·, t) − Wt (·, t + τ ) 2L2 (0,1)×L2

EI (0,1)

Iρ (0,1)

ρ

 (·, t + τ ) + 2 W

t (·, t + τ ) 2 2 + 2 Q(·, t) − W L (0,1)×L2 ρ

Iρ (0,1)

t (·, t + τ ) − Wt (·, t + τ ) 2 2 +2 W L (0,1)×L2 ρ

 ≤ 2Me

−ω(t−τ )

+ 2ME(0)e

Iρ (0,1)

−ω1 t

so (W (x, t), Wt (x, t)) also decays exponentially. τ When the system (5.2) is strongly stable, it holds that limt→∞ 0 U (t + s) 2 ds = 0, we also have lim (W (x, t), Wt (x, t) = 0. t→∞

Summarizing discussion, we arrive at the following result. Theorem 6.1 If the system (2.6) is exponential stability, then the system (2.1) with controls (2.5) is also exponential stability; if the system (2.6) is asymptotically stable, then the system (2.1) with controls (2.5) is also asymptotically stable. Finally, as an application of the result in [27], we give a complete description for stability of Eq. 2.1 with controls (2.5). Theorem 6.2 Suppose that Kρ = EI Iρ . Let μn , n ∈ N be the eigenvalues of L, which associates with the free system (the system (2.1) without controls). Set  0  0 √ √ g1 (η)e−i μn (τ +η) dη, ξn(2) = g2 (η)e−i μn (τ +η) dη ξn(1) = −τ

−τ

Then the following assertions are true: 1) when



√

β1

−iτ μn (1)

+ ξn > 0, inf + α1 e n ρ



√

β2

−iτ μn (2)

+ ξn > 0, inf + α2 e n Iρ

the system (2.1) with controls (2.5) is exponentially stable;

Xiu Fang Liu and Gen Qi Xu

2) If for all n ∈ N,



√

β1

+ α1 e−iτ μn + ξ (1) > 0, n

ρ but



√

β1

−iτ μn (1)

+ ξn = 0, inf + α1 e n ρ



√

β2

+ α2 e−iτ μn + ξ (2) > 0 n

I ρ



√

β2

−iτ μn (2)

+ ξn = 0, inf + α2 e n Iρ

then the system (2.1) with controls (2.5) is asymptotically stable.

7 Conclusion In this paper, based on the output of the system, we obtained the exponential stabilization of Timoshenko beam with distributed delay by the dynamic feedback controller. The dynamic feedback controller is composed of the Luenberger observer, partial state predictor, and static feedback controller, in which the Luenberger observer and partial state predictor are used to translate the delayed system into a system without delay, the static feedback controller is used to obtain right control signals from the system without delay. By feeding back the obtained signals to the original system, we realize the exponential stabilization for the system. This result is an improvement of [27]. Such a control strategy of Luenberger observer, partial state predictor plus state feedback can be applied to other models.

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