Abstract
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 polynomialtype 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.
1 Introduction
Stabilization and vibration controllability of string or beam systems arising from different engineering backgrounds has attracted attention of many researchers [14]. In particular, boundary feedback stabilization of string and beam systems has become an important research area [57]. 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
which was proposed by Kirchhoff [10]. 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;
In this paper, we consider Kirchhoff string (1) with the following boundary conditions (see Figure 1):
for all
for all
Figure 1. Schematic of the nonlinear Kirchhoff string with boundary control.
Shahruz and Krishna [13] 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
In the literature mentioned above, such as [13,16] and [20], 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 polynomialtype 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 infinitedimensional 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
To obtain a precise stabilization result, we make the following hypothesis on the
continuous control feedback
(H) There exist constants
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 polynomialtype constraint.
For example, Figure 2 illustrates a feedback control u satisfying a linear sector constraint (H) with
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 define the natural energy function of time for system (7a)(7d) and denote it by
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
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
Proof See the Appendix. □
Now, we give a property of the energy function E.
Proposition 3.1The timederivative 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
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
for all
Proof Applying the CauchySchwarz inequality, we get
for all
for all
Together with (19) and (20), we get equation (18). Hence we complete the proof of Lemma 3.2. □
Now, we present a Gronwalltype lemma (see Komornik [22], pp.124), which will play an essential role when establishing the stabilization result.
Lemma 3.3Let
Then the following estimation is true, for all
We give a priori estimation for the energy function
Lemma 3.4The energy functionEin equation (10), along the solution of system (7a)(7d), satisfies
for all
Proof We multiply equation (7a) by
using equations (13) and (12) in Lemma 3.1. It follows from (10) that
Since
Hence, substituting equation (23) into equation (22) and using equation (24), one has
Since
Lemma 3.5For any constant
where
Proof According to inequality (21) in Lemma 3.4, we have, for all
Moreover, using integration by parts, we get
Hence, inequality (26) becomes
where
Firstly, we estimate
where the last inequality follows from the fact
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
Remark 3.2 It is worth to mention that Theorem 2.4 in [13] 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 [13].
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
Hence, from inequality (25) and equation (14), we deduce that, for all
where
Now we deal with Case (II). In this case, we choose
Claim 1For any
Now, inserting inequalities (32) and (33) into (25), we obtain, for all
where
Recalling
where the last inequality follows from Remark 3.1. Finally, by letting
Remark 3.3 According to the proof of Theorem 3.1, it is easy to see that the constants σ,
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
where
Proof According to the fact that
for all
4 Numerical results
In this section we consider a computational example for the closedloop 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 dynamic responses of the controlled Kirchhoff string were simulated under two feedback control laws:
and
Obviously, the feedback control function
Figure 4. String response of closedloop system (38) under control law
Figure 5. String response of closedloop system (38) under control law
Figure 6. Transverse displacement of the string at
Figure 7. Transverse displacement of the string at
From Figures 4 and 5, it can bee seen that, in the case of control law
Appendix
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
and
for all
A.2 Proof of Claim 1
Since
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
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
Inserting (47) into (46), we get inequality (33) in Claim 1.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
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).
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