Exponential and polynomial stabilization of the Kirchhoff string by nonlinear boundary control

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].

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; $y(x,t)$ 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.

for all $t\ge 0$, where L is a positive constant. They established exponential stability. In [15], 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 $b=0$, can be stabilized by the control law in equation (6); see e.g., references [16–18]. Moreover, Shahruz [19], Fung et al. [5], and Li and Hou [20] developed linear boundary control laws for axially moving strings. It is worth mentioning that Kobayashi [21] 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 [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 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.

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 $x\in (0,1)$ and $t\ge 0$. Here $f(x)$ and $g(x)$ in equation (7d) are the initial displacement and velocity of the string, respectively. We assume that $f\in C^1[0,1]$, $g\in C[0,1]$ and that at least one of the functions f or g is not identically zero over $[0,1]$.

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 $r=1$, $L_1=1$, $L_2=2$, and Figure 3 shows a feedback control u satisfying a nonlinear constraint (H) with $r=2$, $L_1=1$, $L_2=2$. Since $u(y_t(1,t))y_t(1,t)\le 0$, for all $t\ge 0$, the boundary control (7c) is the negative feedback of transversal velocity of the string at $x=1$.

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.

Since at least one of the functions f and g is not identically zero over $(0,1)$ we have $E(0)>0$.

In this section we state and prove our main result. For this purpose we establish several lemmas.

Proof See the Appendix. □

Now, we give a property of the energy function E.

for all $t\ge 0$.

for all $t\ge 0$. We obtain equation (14). □

Therefore, the energy E is a decreasing function of time.

During the subsequent stability analysis, we utilize the following inequality.

for all $t\ge 0$.

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 [22], pp.124), which will play an essential role when establishing the stabilization result.

We give a priori estimation for the energy function $E(t)$, which was established in [15]. For the sake of completeness, we give the proof here.

for all $t\ge 0$.

Since $\frac{b}{4}{[{\int }_0^1{y}_x^{2}dx]}^2\ge 0$, we complete the proof of Lemma 3.4. □

where $C_1=\frac{3q+2}{q+1}max\{1,\frac{1}{a}\}$.

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).

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): $r=1$;

Case (II): $r>1$.

where $C_2=C_1+\frac{1}{2}(\frac{1}{L_1}+\frac{L_2}{a})$ and $C_1$ is given in Lemma 3.5.

Now we deal with Case (II). In this case, we choose $q=\frac{r-1}{2}>0$. We first admit the following fact (the proof is given in the Appendix).

where the last inequality follows from Remark 3.1. Finally, by letting $S\to +\mathrm{\infty }$ in (31), (35) and using Lemma 3.3 with $G(t)=E(t)$, 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, $k_1$ and $k_2$ in Theorem 3.1 can be chosen as, respectively, ${\sigma }^{-1}=2max\{1,\frac{1}{a}\}+\frac{1}{2}(\frac{1}{L_1}+\frac{L_2}{a})$, $k_1=eE(0)$, and $k_2=E(0){[2\frac{r+1}{r-1}(C_3{E}^{\frac{r-1}{2}}(0)+C_4)]}^{\frac{2}{r-1}}$ with $C_3=\frac{3r-1}{r+1}max\{1,\frac{1}{a}\}+\frac{a+L_1{L}_2}{a(r+1)L_1}$, $C_4=\frac{1}{r+1}(\frac{1}{L_1}+\frac{L_2^r}{a}){[(1+\frac{1}{a})\frac{r-1}{r+1}]}^{\frac{r-1}{2}}$. This means that the coefficients of the exponential or polynomial decay rate are exactly determined only by the initial tension a, the initial energy $E(0)$ 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.

where $k_1^{\prime }=\sqrt{\frac{2k_1}{a}}$, $k_2^{\prime }=\sqrt{\frac{2k_2}{a}}$ and $k_1$, $k_2$, σ are given in Theorem  3.1.

for all $x\in (0,1)$ and $t\ge 0$. By combining (37) with (30) in Theorem 3.1, we complete the proof of Theorem 3.2. □

Obviously, the feedback control function $u_1$ satisfies the constraint condition (H) with $r=1$, $L_1=1$, $L_2=3$, and $u_2$ satisfies the constraint condition (H) with $r=2$, $L_1=L_2=1$. Then, according to Theorem 3.2, the asymptotic behavior of the transverse vibration of system (38) under control law $u_1$ (or control law $u_2$) possesses exponential decay (or polynomial decay with degree −1, because $r=2$). The string response $y(x,t)$ of closed-loop system (38) with control law $u_1$ and control law $u_2$ are shown in Figure 4 and Figure 5, respectively. The corresponding transversal displacement at $x=0.5$ is shown in Figures 6 and 7, respectively.

From Figures 4 and 5, it can bee seen that, in the case of control law $u_2$, the decay of the transverse vibration relatively slow compared to the case of control law $u_1$. Indeed, from Figure 6 and Figure 7, we know that in the case of control law $u_1$, the transverse vibration has been suppressed exponentially ($|y(0.5,t)|\le K_2{e}^{-\sigma t}$), whereas in the case of control law $u_2$, the transverse vibration has been suppressed polynomially ($|y(0.5,t)|\le K_3{t}^{-1}$). It is coincident with the results of Theorem 3.2.