This paper addresses the stabilization problem of the nonlinear Kirchhoff string using nonlinear boundary control. Nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a polynomial-type constraint. Employing the multiplier method, we establish explicit exponential and polynomial stability for the Kirchhoff string. The theoretical results are assured by numerical results of the asymptotic behavior for the system.
Stabilization and vibration controllability of string or beam systems arising from different engineering backgrounds has attracted attention of many researchers [1-4]. In particular, boundary feedback stabilization of string and beam systems has become an important research area [5-7]. This is because, in a practice system, vibration is more easily controlled through a boundary point than using point sensors or actuators away from the boundaries [8,9].
There are several nonlinear mathematical models that describe the transversal vibration of stretched strings. One such model is presented in the following equation:
for all and , where , are two constants. Obviously, the above equation is a simple prototype of the classical equation
which was proposed by Kirchhoff . Here l is the length of the string; E is Young’s modulus of the material; ρ is density; h is the area of the cross section; is the transversal displacement of the point x of the string at time t. This model has been studied by researchers from the physical and mathematical points of view; see, e.g., references [11-13] and the references therein.
In this paper, we consider Kirchhoff string (1) with the following boundary conditions (see Figure 1):
for all . denotes the tension in the string at time t. The boundary condition in equation (2) implies that the string is fixed at . The boundary condition in equation (3) represents the balance of the transversal component of the tension in the string and the control input u which is applied transversally at . Because the tension in the string represented by equation (1) is not constant and is given by
for all (see ), the boundary condition in equation (3) can be written as
Figure 1. Schematic of the nonlinear Kirchhoff string with boundary control.
Shahruz and Krishna  investigated the stabilization of Kirchhoff string (1) with a linear negative velocity control, which means the boundary control u has a linear negative velocity feedback form
for all , where L is a positive constant. They established exponential stability. In , the absolute stability of the Kirchhoff string (1) with linear sector boundary control was considered. It is well known that linear strings represented by equation (1), for which , can be stabilized by the control law in equation (6); see e.g., references [16-18]. Moreover, Shahruz , Fung et al., and Li and Hou  developed linear boundary control laws for axially moving strings. It is worth mentioning that Kobayashi  designed a linear parallel compensator based on boundary displacement observer and proved the string (1) can be stabilized by parallel compensator control.
In the literature mentioned above, such as [13,16] and , the exponential stabilization result for various string systems by linear boundary control mainly relies on the Lyapunov direct method. In this work, we investigate the stabilization of string (1) with a more general and ‘flexible’ boundary control (see hypothesis (H) in Section 2). The feedback function u is not required to satisfy a strict control law such as (6), but just satisfies some appropriate polynomial-type constraint. In this general boundary control case, it seems that the Lyapunov direct method is no more applicable. So, we need to use a more meticulous method to deal with the stabilization problem. Applying the multiplier method, we establish not only exponential stability result but also polynomial stability result for Kirchhoff string (1).
The remainder of this technical paper is arranged as follows. Section 2 describes the model of the Kirchhoff nonlinear string and introduces the control assumption. The problem of exponential and polynomial stability is addressed in Section 3. Finally, a numerical example is demonstrated where the nonlinear distributed parameter infinite-dimensional equation is solved by applying the finite element method in Section 4.
2 Problem formulation
Consider the nonlinear Kirchhoff string model as shown in Figure 1. For the sake of easy reading and later referring, the governing equation, the boundary conditions and the initial functions are put together as
for all and . Here and in equation (7d) are the initial displacement and velocity of the string, respectively. We assume that , and that at least one of the functions f or g is not identically zero over .
To obtain a precise stabilization result, we make the following hypothesis on the continuous control feedback :
(H) There exist constants and such that
Remark 2.1 Obviously, condition (H) is equivalent to, for all ,
Obviously, hypothesis (H) is a ‘flexible’ and ‘robust’ condition, which allows the feedback function u to vary in an appropriate geometric region given by a polynomial-type constraint. For example, Figure 2 illustrates a feedback control u satisfying a linear sector constraint (H) with , , , and Figure 3 shows a feedback control u satisfying a nonlinear constraint (H) with , , . Since , for all , the boundary control (7c) is the negative feedback of transversal velocity of the string at .
For the existence and uniqueness of the solution of the general Kirchhoff equation, we refer to [11,12] and references therein. In this work, we study the stabilization of the string in (7a) by this negative feedback boundary control u, which provides a dissipative effect.
Remark 2.2 According to boundary condition (7b) at , we easily get
We define the natural energy function of time for system (7a)-(7d) and denote it by . The scalar-valued function E is defined as
Especially, from the initial displacement and velocity condition (7d), we obtain the initial energy as
Since at least one of the functions f and g is not identically zero over we have .
3 Stabilization by boundary control
In this section we state and prove our main result. For this purpose we establish several lemmas.
Lemma 3.1Let be the solution for system (7a)-(7d). Then
Proof See the Appendix. □
Now, we give a property of the energy function E.
Proposition 3.1The time-derivative of the energy functionEin equation (10), along the solution of system (7a)-(7d) satisfies
for all .
Proof Differentiating the energy function (10) with respect to t, we get
According to equation (11) and boundary control (7c), we get
Substituting equation (16) into equation (15) and observing (7a), we obtain
for all . We obtain equation (14). □
Remark 3.1 From Proposition 3.1, we obtain the energy identity for system (7a)-(7d),
Therefore, the energy E is a decreasing function of time.
During the subsequent stability analysis, we utilize the following inequality.
Lemma 3.2Let be the solution for system (7a)-(7d). Then
for all .
Proof Applying the Cauchy-Schwarz inequality, we get
for all . On the other hand, the definition of energy function (10) implies
for all . It follows from the above inequality that
Together with (19) and (20), we get equation (18). Hence we complete the proof of Lemma 3.2. □
Now, we present a Gronwall-type lemma (see Komornik , pp.124), which will play an essential role when establishing the stabilization result.
Lemma 3.3Let be a non-increasing function. Assume that there exists a constant such that
Then the following estimation is true, for all ,
We give a priori estimation for the energy function , which was established in . For the sake of completeness, we give the proof here.
Lemma 3.4The energy functionEin equation (10), along the solution of system (7a)-(7d), satisfies
for all .
Proof We multiply equation (7a) by and do integration over , with respect to x. We obtain
using equations (13) and (12) in Lemma 3.1. It follows from (10) that
Since , according to boundary control (7c), we have
Hence, substituting equation (23) into equation (22) and using equation (24), one has
Since , we complete the proof of Lemma 3.4. □
Lemma 3.5For any constant , the energy functionEalong the solution of system (7a)-(7d) satisfies the following estimation, for all ,
Proof According to inequality (21) in Lemma 3.4, we have, for all ,
Moreover, using integration by parts, we get
Hence, inequality (26) becomes
Firstly, we estimate and ,
where the last inequality follows from the fact is a decreasing function. On the other hand,
Finally, inserting the two inequalities, (28) and (29), in (27), we get inequality (25). This completes the proof of Lemma 3.5. □
We now state the main stabilization result for system (7a)-(7d).
Theorem 3.1Assume that assumption (H) holds. Then there exist three constants such that, for all ,
Remark 3.2 It is worth to mention that Theorem 2.4 in  can be viewed as a special cases of Theorem 3.1. Indeed, in the linear control case (6), the exponential stability in Theorem 3.1 coincides with the result in .
Proof of Theorem 3.1 We distinguish two cases related to the parameter r to establish the energy decay rate.
Case (I): ;
Case (II): .
In Case (I), we choose . According to hypothesis (H), we know that
Hence, from inequality (25) and equation (14), we deduce that, for all ,
where and is given in Lemma 3.5.
Now we deal with Case (II). In this case, we choose . We first admit the following fact (the proof is given in the Appendix).
Claim 1For any , we have the following estimates, for all ,
Now, inserting inequalities (32) and (33) into (25), we obtain, for all ,
where , and is given in Lemma 3.5. Now we choose . Then it is obvious that . Hence, inequality (34) becomes
Recalling , we get . Hence, the above inequality is rewritten as
where the last inequality follows from Remark 3.1. Finally, by letting in (31), (35) and using Lemma 3.3 with , we complete the proof of Theorem 3.1. □
Remark 3.3 According to the proof of Theorem 3.1, it is easy to see that the constants σ, and in Theorem 3.1 can be chosen as, respectively, , , and with , . This means that the coefficients of the exponential or polynomial decay rate are exactly determined only by the initial tension a, the initial energy and the feedback control u. However, in the polynomial decay case, the order of decay rate is determined only by the feedback control u.
Finally, it is shown that the boundary control u stabilizes the nonlinear Kirchhoff string.
Theorem 3.2Assume that assumption (H) holds. Then there exist two constants such that for all and ,
where , and , , σare given in Theorem 3.1.
Proof According to the fact that , for all , we get
for all and . By combining (37) with (30) in Theorem 3.1, we complete the proof of Theorem 3.2. □
4 Numerical results
In this section we consider a computational example for the closed-loop system (7a)-(7d). To illustrate the control performance of the boundary control law satisfying condition (H), numerical simulations by using the finite element method (FEM) are performed. We use Lagrange ‘hat’ basis with FEM equidistant meshes. The system parameters used in the simulations are , . The initial conditions are and . That is we consider the following Kirchhoff system:
The dynamic responses of the controlled Kirchhoff string were simulated under two feedback control laws:
Obviously, the feedback control function satisfies the constraint condition (H) with , , , and satisfies the constraint condition (H) with , . Then, according to Theorem 3.2, the asymptotic behavior of the transverse vibration of system (38) under control law (or control law ) possesses exponential decay (or polynomial decay with degree −1, because ). The string response of closed-loop system (38) with control law and control law are shown in Figure 4 and Figure 5, respectively. The corresponding transversal displacement at is shown in Figures 6 and 7, respectively.
Figure 4. String response of closed-loop system (38) under control law .
Figure 5. String response of closed-loop system (38) under control law .
Figure 6. Transverse displacement of the string at under control law .
Figure 7. Transverse displacement of the string at under control law .
From Figures 4 and 5, it can bee seen that, in the case of control law , the decay of the transverse vibration relatively slow compared to the case of control law . Indeed, from Figure 6 and Figure 7, we know that in the case of control law , the transverse vibration has been suppressed exponentially ( ), whereas in the case of control law , the transverse vibration has been suppressed polynomially ( ). It is coincident with the results of Theorem 3.2.
A.1 Proof of Lemma 3.1
Integrating by parts and (9) we get, for all ,
So we obtain equation (11). Next, integrating by parts we compute
for all . That is, equation (12) holds. Finally, we write
for all . Substituting equation (40) into equation (39), we obtain equation (13). Thus the proof of Lemma 3.1 is complete.
A.2 Proof of Claim 1
Since , hypothesis (H) implies
Hence, it is true that, for all ,
Then, from (41) and Proposition 3.1 we have
From Young’s inequality, we have, for any ,
where and . Applying the above inequality with , , and , we obtain, for any ,
By inserting (45) into (43), we deduce inequality (32) in Claim 1. Similarly, from (42) and Proposition 3.1 we have
Applying again inequality (44) with , , and , we obtain, for any
Inserting (47) into (46), we get inequality (33) in Claim 1.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
This work is supported by the National Natural Science Foundation of China (Grant 11226128) and the Natural Science Foundation of Heilongjiang Province of China (Grant F201113).
Chen, G, Delfour, MC, Krall, AM, Payre, G: Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim.. 25, 526–546 (1987). Publisher Full Text
Lagnese, J: Decay of solutions of wave equations in a bounded domain with boundary dissipation. J. Differ. Equ.. 50, 163–182 (1983). Publisher Full Text
Shahruz, SM: Bounded-input bounded-output stability of a damped nonlinear string. IEEE Trans. Autom. Control. 41, 1179–1182 (1996). Publisher Full Text
Guo, BZ: Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim.. 39, 1736–1747 (2001). Publisher Full Text
Fung, RF, Wu, JW, Wu, SL: Exponential stabilization of an axially moving string by linear boundary feedback. Automatica. 35, 177–181 (1999). Publisher Full Text
Morgül, Ö: Stabilization disturbance rejection for the beam equation. IEEE Trans. Autom. Control. 46(12), 1913–1918 (2001). Publisher Full Text
Arosio, A, Panizzi, S: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc.. 348(1), 305–330 (1996). Publisher Full Text
Ono, K: Global existence, decay, and blow up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differ. Equ.. 137, 273–301 (1997). Publisher Full Text
Shahruz, SM, Krishna, LG: Boundary control of a non-linear string. J. Sound Vib.. 195, 169–174 (1996). Publisher Full Text
Oplinger, DW: Frequency response of a nonlinear stretched string. J. Acoust. Soc. Am.. 32, 1529–1538 (1960). Publisher Full Text
Chen, G: A note on boundary stabilization of the wave equation. SIAM J. Control Optim.. 19, 106–113 (1981). Publisher Full Text
Lagnese, JE: Note on boundary stabilization of wave equations. SIAM J. Control Optim.. 26, 1250–1256 (1988). Publisher Full Text
Li, TC, Hou, ZC: Exponential stabilization of an axially moving string with geometrical nonlinearity by linear boundary feedback. J. Sound Vib.. 296, 861–870 (2006). Publisher Full Text
Kobayashi, T: Boundary position feedback control of Kirchhoff’s non-linear strings. Math. Methods Appl. Sci.. 27, 79–89 (2004). Publisher Full Text