Appendix 1
To analyze the system stability, we define
$$\begin{aligned} V_2 \left( t \right)= & {} \frac{1}{2}\alpha \rho \int _0^l \left[ \left( {\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}+\left( {\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right. \nonumber \\&\left. +\left( {\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right] \hbox {d}s \nonumber \\&+\,\frac{1}{2}\alpha \int _0^l {\left( {T\left( {s,t} \right) u_s^2 \left( {s,t} \right) +T\left( {s,t} \right) v_s^2 \left( {s,t} \right) } \right) \hbox {d}s} \nonumber \\&+\,\frac{1}{2}\alpha \int _0^l EA\left( s \right) \left( w_s \left( {s,t} \right) +\frac{1}{2}u_s^2 \left( {s,t} \right) \right. \nonumber \\&\left. +\,\frac{1}{2}v_s^2 \left( {s,t} \right) \right) ^{2}\hbox {d}s \end{aligned}$$
(A1)
$$\begin{aligned} V_3 \left( t \right)= & {} \beta \rho \int _0^l s\left( \frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}u_s \left( {s,t} \right) +\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}v_s \left( {s,t} \right) \right. \nonumber \\&\left. +\,\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}w_s \left( {s,t} \right) \right) \hbox {d}s \end{aligned}$$
(A2)
where \(\beta >0\).
A new positive Lyapunov function is considered as
$$\begin{aligned} V\left( t \right) =V_1 \left( t \right) +V_2 \left( t \right) +V_3 \left( t \right) \end{aligned}$$
(A3)
Since \(2w_s^2 \left( {s,t} \right) \le u_s^2 \left( {s,t} \right) \) and \(2w_s^2 \left( {s,t} \right) \le v_s^2 \left( {s,t} \right) \) [41], by using Lemma 2.1 we can obtain
$$\begin{aligned}&-\frac{1}{4\sigma _0 }\int _0^l {u_s^2 \left( {s,t} \right) \hbox {d}s} -\frac{\sigma _0 }{2}\int _0^l {u_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad \le \int _0^l {w_s \left( {s,t} \right) u_s^2 \left( {s,t} \right) \hbox {d}s} \le \frac{1}{4\sigma _0 }\int _0^l {u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad +\,\frac{\sigma _0 }{2}\int _0^l {u_s^4 \left( {s,t} \right) \hbox {d}s} \end{aligned}$$
(A4)
$$\begin{aligned}&-\frac{1}{4\sigma _0 }\int _0^l {v_s^2 \left( {s,t} \right) \hbox {d}s} -\frac{\sigma _0 }{2}\int _0^l {v_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad \le \int _0^l {w_s \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \le \frac{1}{4\sigma _0 }\int _0^l {v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\qquad +\,\frac{\sigma _0 }{2}\int _0^l {v_s^4 \left( {s,t} \right) \hbox {d}s} \end{aligned}$$
(A5)
where \(\sigma _0 >0\).
According to Inequalities (A4) and (A5), \(V_2 \left( t \right) \) can be rewritten as
$$\begin{aligned} V_2 \left( t \right)= & {} \frac{1}{2}\alpha \rho \int _0^l \left[ \left( {\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}+\left( {\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}\right. \nonumber \\&\left. +\left( {\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right] \hbox {d}s \nonumber \\&+\, \frac{1}{2}\alpha \int _0^l {\left( {T\left( {s,t} \right) u_s^2 \left( {s,t} \right) +T\left( {s,t} \right) v_s^2 \left( {s,t} \right) } \right) \hbox {ds}} \nonumber \\&+\,\frac{1}{2}\alpha \int _0^l EA\left( s \right) \left( w_s^2 \left( {s,t} \right) +\frac{1}{4}u_s^4 \left( {s,t} \right) \right. \nonumber \\&+\frac{1}{4}v_s^4 \left( {s,t} \right) +\,w_s \left( {s,t} \right) u_s^2 \left( {s,t} \right) \nonumber \\&\left. +w_s \left( {s,t} \right) v_s^2 \left( {s,t} \right) +\frac{1}{2}u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \right) \hbox {ds} \nonumber \\\le & {} \frac{1}{2}\alpha \rho \int _0^l \left[ \left( {\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}+\left( {\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right. \nonumber \\&\left. +\,\left( {\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right] \hbox {d}s \nonumber \\&+\,\left( {\frac{1}{2}\alpha T_{\max } +\frac{1}{2}\alpha EA_{\max } \frac{1}{4\varepsilon }} \right) \int _0^l {u_s^2 \left( {s,t} \right) \hbox {ds}} \nonumber \\&+\,\left( {\frac{1}{2}\alpha T_{\max } +\frac{1}{2}\alpha EA_{\max } \frac{1}{4\varepsilon }} \right) \int _0^l {v_s^2 \left( {s,t} \right) \hbox {ds}} \nonumber \\&+\,\frac{1}{2}\alpha EA_{\max } \int _0^l {w_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\left( {\frac{1}{8}\alpha EA_{\max } +\frac{1}{4}\alpha EA_{\max } \varepsilon } \right) \int _0^l {u_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\left( {\frac{1}{8}\alpha EA_{\max } +\frac{1}{4}\alpha EA_{\max } \varepsilon } \right) \int _0^l {v_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{1}{4}\alpha EA_{\max } \int _0^l {u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\\le & {} \gamma _1 \theta \left( t \right) \end{aligned}$$
(A6)
where \(\varepsilon >0\) and
$$\begin{aligned} \theta \left( t \right)= & {} \int _0^l \left[ \left( {\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}+\left( {\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right. \nonumber \\&+\left( {\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}\nonumber \\&\left. +\,u_s^2 \left( {s,t} \right) +v_s^2 \left( {s,t} \right) +w_s^2 \left( {s,t} \right) \right. \nonumber \\&\left. {+\,u_s^4 \left( {s,t} \right) +v_s^4 \left( {s,t} \right) +u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) } \right] \hbox {d}s \\ \gamma _1= & {} \frac{1}{2}\alpha \max \left\{ \rho ,T_{\max } +EA_{\max } \frac{1}{4\varepsilon },EA_{\max }, \right. \nonumber \\&\left. \frac{1}{2}EA_{\max } \left( {\frac{1}{2}+\varepsilon } \right) ,\frac{1}{2}EA_{\max } \right\} >0 \end{aligned}$$
Let the positive constant \(\varepsilon \) satisfying \(T_{\min } -EA_{\max } \frac{1}{4\varepsilon }\ge 0\) and \(\frac{1}{2}EA_{\min } -\varepsilon EA_{\max } \ge 0\), similarly, we can obtain
$$\begin{aligned} V_2 \left( t \right) \ge \gamma _2 \theta \left( t \right) \end{aligned}$$
(A7)
where
$$\begin{aligned} \gamma _2= & {} \frac{1}{2}\alpha \min \left\{ \rho ,T_{\min } -\frac{1}{4\varepsilon }EA_{\max }, \right. \\&EA_{\min } ,\frac{1}{2}\left( {\frac{1}{2}EA_{\min } -\varepsilon EA_{\max } } \right) \left. ,\frac{1}{2}EA_{\min } \right\} \\&>0. \end{aligned}$$
Then we focus our attention on \(V_3 \left( t \right) \) in Equation (A2) which satisfies the following inequality
$$\begin{aligned} \left| {V_3 \left( t \right) } \right|\le & {} \frac{1}{2}\beta \rho L\int _0^l \left( \left( {\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2}+u_s^2 \left( {s,t} \right) \right. \nonumber \\&\left. +\,\left( {\frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right. \left. +v_s^2 \left( {s,t} \right) +\left( {\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}} \right) ^{2} \right. \nonumber \\&\left. +\,w_s^2 \left( {s,t} \right) \right) \hbox {d}s \quad \le \beta _1 \theta \left( t \right) \end{aligned}$$
(A8)
where \(\beta _1 =\frac{1}{2}\beta \rho L>0\).
Therefore, from Inequalities (A6–A8), we have
$$\begin{aligned} \left( {\gamma _2 -\beta _1 } \right) \theta \left( t \right) \le V_2 \left( t \right) +V_3 \left( t \right) \le \left( {\beta _1 +\gamma _1 } \right) \theta \left( t \right) \end{aligned}$$
(A9)
Considering that \(\beta _1 \) is a positive weighting constant satisfying \(0<\beta _1 \le \gamma _2 \) and let \(\eta _1 =\gamma _2 -\beta _1 \), \(\eta _2 =\gamma _1 +\beta _1 \), we further have
$$\begin{aligned} \lambda _1 \left( {\theta \left( t \right) +V_1 \left( t \right) } \right) \le V\left( t \right) \le \lambda _2 \left( {\theta \left( t \right) +V_1 \left( t \right) } \right) \end{aligned}$$
(A10)
where \(\lambda _1 =\min \{ {1,\eta _1 } \}>0\), \(\lambda _2 =\max \{ {1,\eta _2 } \}>0\).
Differentiating Eq. (A1) leads to
$$\begin{aligned} {\dot{V}}_2 \left( t \right)= & {} \alpha \rho \int _0^l \left( \frac{\hbox {D}v\left( {s,t} \right) }{\hbox {D}t}\frac{\hbox {D}^{2}v\left( {s,t} \right) }{\hbox {D}t^{2}} \right. \nonumber \\&+\,\frac{\hbox {D}u\left( {s,t} \right) }{\hbox {D}t}\frac{\hbox {D}^{2}u\left( {s,t} \right) }{\hbox {D}t^{2}}\nonumber \\&\left. +\frac{\hbox {D}w\left( {s,t} \right) }{\hbox {D}t}\frac{\hbox {D}^{2}w\left( {s,t} \right) }{\hbox {D}t^{2}} \right) \hbox {d}s \nonumber \\&+\, \frac{1}{2}\alpha \int _0^l \left( {\dot{T}} \left( {s,t} \right) u_s^2 \left( {s,t} \right) \right. \nonumber \\&\left. \hbox {+\,2}T\left( {s,t} \right) u_s \left( {s,t} \right) {\dot{u}}_s \left( {s,t} \right) +{\dot{T}}\left( {s,t} \right) v_s^2 \left( {s,t} \right) \right. \nonumber \\&\left. {+ \,2T\left( {s,t} \right) v_s \left( {s,t} \right) {\dot{v}}_s \left( {s,t} \right) } \right) \hbox {d}s \nonumber \\&+\, \alpha \int _0^l {EA\left( s \right) \left( {w_s \left( {s,t} \right) +\frac{1}{2}u_s^2 \left( {s,t} \right) +\frac{1}{2}v_s^2 \left( {s,t} \right) } \right) } \nonumber \\&\quad \left( {{\dot{w}}_s \left( {s,t} \right) +u_s \left( {s,t} \right) {\dot{u}}_s \left( {s,t} \right) +v_s \left( {s,t} \right) {\dot{v}}_s \left( {s,t} \right) } \right) \hbox {d}s \nonumber \\&+\,\frac{1}{2}\alpha {\dot{l}}EA\left( l \right) \left( {w_s \left( {l,t} \right) +\frac{1}{2}u_s^2 \left( {l,t} \right) +\frac{1}{2}v_s^2 \left( {l,t}\right) } \right) ^{2} \nonumber \\&-\,\frac{1}{2}\alpha {\dot{l}}EA\left( 0 \right) \left( {w_s \left( {0,t} \right) +\frac{1}{2}u_s^2 \left( {0,t} \right) +\frac{1}{2}v_s^2 \left( {0,t} \right) } \right) ^{2} \nonumber \\&+\,\frac{1}{2}\alpha {\dot{l}}\left( {T\left( {l,t} \right) u_s^2 \left( {l,t} \right) +T\left( {l,t} \right) v_s^2 \left( {l,t} \right) } \right) \nonumber \\&-\,\frac{1}{2}\alpha {\dot{l}}\left( {T\left( {0,t} \right) u_s^2 \left( {0,t} \right) +T\left( {0,t} \right) v_s^2 \left( {0,t} \right) } \right) \nonumber \\ \end{aligned}$$
(A11)
Then, substituting Equations (6)-(8) into Equation (A11), we have the following equation
$$\begin{aligned} \dot{V}_2 \left( t \right)= & {} \alpha \left( {\dot{u}\left( {l,t} \right) +\dot{l}u_s \left( {l,t} \right) } \right) T\left( {l,t} \right) u_s \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{u}\left( {l,t} \right) +\dot{l}u_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) u_s^3 \left( {l,t} \right) \nonumber \\&+\,\frac{\alpha }{2}\int _0^l {\dot{l}u_s^2 \left( {s,t} \right) T_s \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\alpha \int _0^l {\dot{l}\frac{1}{4}EA_s \left( s \right) u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\alpha \int _0^l {\dot{l}\frac{1}{8}EA_s \left( s \right) u_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{\alpha }{2}\int _0^l {\dot{l}EA_s \left( s \right) w_s \left( {s,t} \right) u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\alpha \left( {\dot{u}\left( {l,t} \right) +\dot{l}u_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) u_s \left( {l,t} \right) v_s^2 \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{u}\left( {l,t} \right) +\dot{l}u_s \left( {l,t} \right) } \right) EA\left( l \right) w_s \left( {l,t} \right) u_s \left( {l,t} \right) \nonumber \\&-\,\alpha \left( {\dot{u}\left( {0,t} \right) +\dot{l}u_s \left( {0,t} \right) } \right) T\left( {0,t} \right) u_s \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{u}\left( {0,t} \right) +\dot{l}u_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) u_s^3 \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{u}\left( {0,t} \right) +\dot{l}u_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) u_s \left( {0,t} \right) v_s^2 \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{u}\left( {0,t} \right) +\dot{l}u_s \left( {0,t} \right) } \right) EA\left( 0 \right) w_s \left( {0,t} \right) u_s \left( {0,t} \right) \nonumber \\&+\,\alpha \left( {\dot{v}\left( {l,t} \right) +\dot{l}v_s \left( {l,t} \right) } \right) T \left( {l,t} \right) v_s \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{v}\left( {l,t} \right) +\dot{l}v_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) v_s^3 \left( {l,t} \right) \nonumber \\&+\,\frac{\alpha }{2}\int _0^l {\dot{l}v_s^2 \left( {s,t} \right) T_s \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\alpha \int _0^l {\dot{l}\frac{1}{8}EA_s \left( s \right) v_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\alpha \left( {\dot{v}\left( {l,t} \right) +\dot{l}v_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) u_s^2 \left( {l,t} \right) v_s \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{v}\left( {l,t} \right) +\dot{l}v_s \left( {l,t} \right) } \right) EA\left( l \right) w_s \left( {l,t} \right) v_s \left( {l,t} \right) \nonumber \\&-\,\alpha \left( {\dot{v}\left( {0,t} \right) +\dot{l}v_s \left( {0,t} \right) } \right) T\left( {0,t} \right) v_s \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{v}\left( {0,t} \right) +\dot{l}v_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) v_s^3 \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{v}\left( {0,t} \right) +\dot{l}v_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) u_s^2 \left( {0,t} \right) v_s \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{v}\left( {0,t} \right) +\dot{l}v_s \left( {0,t} \right) } \right) EA\left( 0 \right) w_s \left( {0,t} \right) v_s \left( {0,t} \right) \nonumber \\&+\,\alpha \left( {\dot{w}\left( {l,t} \right) +\dot{l}w_s \left( {l,t} \right) } \right) EA\left( l \right) w_s \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{w}\left( {l,t} \right) +\dot{l}w_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) u_s^2 \left( {l,t} \right) \nonumber \\&+\,\alpha \left( {\dot{w}\left( {l,t} \right) +\dot{l}w_s \left( {l,t} \right) } \right) \frac{1}{2}EA\left( l \right) v_s^2 \left( {l,t} \right) \nonumber \\&-\,\alpha \left( {\dot{w}\left( {0,t} \right) +\dot{l}w_s \left( {0,t} \right) } \right) EA\left( 0 \right) w_s \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{w}\left( {0,t} \right) +\dot{l}w_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) u_s^2 \left( {0,t} \right) \nonumber \\&-\,\alpha \left( {\dot{w}\left( {0,t} \right) +\dot{l}w_s \left( {0,t} \right) } \right) \frac{1}{2}EA\left( 0 \right) v_s^2 \left( {0,t} \right) \nonumber \\&+\,\frac{1}{2}\alpha \int _0^l {\dot{l}EA_s \left( s \right) w_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{\alpha }{2}\int _0^l {\dot{l}EA_s \left( s \right) w_s \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\, \frac{1}{2}\alpha \int _0^l {\left( {\dot{T}\left( {s,t} \right) u_s^2 \left( {s,t} \right) +\dot{T}\left( {s,t} \right) v_s^2 \left( {s,t} \right) } \right) \hbox {d}s}\nonumber \\ \end{aligned}$$
(A12)
Then, differentiating Equation (A2) and substituting Eqs. (6–8), we arrive at
$$\begin{aligned} {\dot{V}}_3 \left( t \right)= & {} \frac{1}{2}\beta lT\left( {l,t} \right) u_s^2 \left( {l,t} \right) \nonumber \\&-\frac{\beta }{2}\int _0^l {T\left( {s,t} \right) u_s^2 \left( {s,t} \right) } \hbox {ds} \nonumber \\&+\,\frac{\beta }{2}\int _0^l {sT_s \left( {s,t} \right) u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta l\frac{3}{8}EA\left( l \right) u_s^4 \left( {l,t} \right) \nonumber \\&-\,\beta \int _0^l {\frac{3}{8}EA\left( s \right) u_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{1}{8}\beta \int _0^l {sEA_s \left( s \right) u_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta lEA\left( l \right) w_s \left( {l,t} \right) u_s^2 \left( {l,t} \right) \nonumber \\&-\,\beta \int _0^l {EA\left( s \right) w_s \left( {s,t} \right) u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta l\frac{3}{4}EA\left( l \right) u_s^2 \left( {l,t} \right) v_s^2 \left( {l,t} \right) \nonumber \\&-\,\beta \int _0^l {\frac{3}{4}EA\left( s \right) u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta \rho l{\dot{l}}\left( {{\dot{u}}\left( {l,t} \right) +{\dot{l}}u_s \left( {l,t} \right) } \right) u_s \left( {l,t} \right) \nonumber \\&+\,\beta \int _0^l {s\frac{1}{4}EA_s \left( s \right) u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&-\,\frac{1}{2}\beta \rho l{\dot{l}}^{2}u_s^2 \left( {l,t} \right) +\frac{1}{2}\beta \rho \int _0^l {{\dot{l}}^{2}u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\hbox {+}\,\frac{1}{2}\beta \rho l{\dot{u}}^{2}\left( {l,t} \right) -\frac{1}{2}\beta \rho \int _0^l {{\dot{u}}^{2}\left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{1}{2}\beta lT\left( {l,t} \right) v_s^2 \left( {l,t} \right) \nonumber \\&-\frac{\beta }{2}\int _0^l {T\left( {s,t} \right) v_s^2 \left( {s,t} \right) } \hbox {ds} \nonumber \\&+\,\frac{\beta }{2}\int _0^l {sT_s \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s}\nonumber \\&+\beta l\frac{3}{8}EA\left( l \right) v_s^4 \left( {l,t} \right) \nonumber \\&-\,\beta \int _0^l {\frac{3}{8}EA\left( s \right) v_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\frac{1}{8}\beta \int _0^l {sEA_s \left( s \right) v_s^4 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta \rho l{\dot{l}}\left( {{\dot{v}}\left( {l,t} \right) +{\dot{l}}v_s \left( {l,t} \right) } \right) v_s \left( {l,t} \right) \nonumber \\&+\,\beta lEA\left( l \right) w_s \left( {l,t} \right) v_s^2 \left( {l,t} \right) \nonumber \\&-\,\beta \int _0^l {EA\left( s \right) w_s \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&-\,\frac{1}{2}\beta \rho l{\dot{l}}^{2}v_s^2 \left( {l,t} \right) \nonumber \\&+\,\frac{1}{2}\beta \rho \int _0^l {{\dot{l}}^{2}v_s^2 \left( {s,t} \right) \hbox {d}s} \hbox {+}\frac{1}{2}\beta \rho l{\dot{v}}^{2}\left( {l,t} \right) \nonumber \\&-\,\frac{1}{2}\beta \rho \int _0^l {{\dot{v}}^{2}\left( {s,t} \right) \hbox {d}s} +\frac{1}{2}\beta lEA\left( l \right) w_s^2 \left( {l,t} \right) \nonumber \\&-\,\frac{1}{2}\beta \int _0^l {EA\left( s \right) w_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta \rho l{\dot{l}}\left( {{\dot{w}}\left( {l,t} \right) +{\dot{l}}w_s \left( {l,t} \right) } \right) w_s \left( {l,t} \right) \nonumber \\&+\,\frac{1}{2}\beta \int _0^l {sEA_s \left( s \right) w_s^2 \left( {s,t} \right) \hbox {d}s}\nonumber \\&-\,\frac{1}{2}\beta \rho l{\dot{l}}^{2}w_s^2 \left( {l,t} \right) \nonumber \\&+\,\frac{1}{2}\beta \rho \int _0^l {{\dot{l}}^{2}w_s^2 \left( {s,t} \right) \hbox {d}s} +\frac{1}{2}\beta \rho l{\dot{w}}^{2}\left( {l,t} \right) \nonumber \\&-\,\frac{1}{2}\beta \rho \int _0^l {{\dot{w}}^{2}\left( {s,t} \right) \hbox {d}s} \nonumber \\&+\,\beta \int _0^l {s\frac{1}{2}EA_s \left( s \right) u_s^2 \left( {s,t} \right) w_s \left( {s,t} \right) \hbox {d}s}\nonumber \\&+\,\beta \int _0^l {s\frac{1}{2}EA_s \left( s \right) v_s^2 \left( {s,t} \right) w_s \left( {s,t} \right) \hbox {d}s}\nonumber \\ \end{aligned}$$
(A13)
By applying Inequalities (A4) and (A5), substituting Eqs. (53), (A12) and (A13) into the derivative of Eq. (A3) leads to
$$\begin{aligned}&{\dot{V}}\left( t \right) \le -\zeta _1 \int _0^l {u_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad -\zeta _2 \int _0^l {u_s^2 \left( {s,t} \right) v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad -\,\zeta _3 \int _0^l {u_s^4 \left( {s,t} \right) \hbox {d}s} -\zeta _4 \int _0^l {v_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad -\,\zeta _5 \int _0^l {v_s^4 \left( {s,t} \right) \hbox {d}s} -\zeta _6 \int _0^l {w_s^2 \left( {s,t} \right) \hbox {d}s} \nonumber \\&\quad -\,\frac{1}{4}\beta \rho \int _0^l {\left( {{\dot{u}}\left( {s,t} \right) +{\dot{l}}u_s \left( {s,t} \right) } \right) ^{2}\hbox {d}s} \nonumber \\&\quad -\,\frac{1}{4}\beta \rho \int _0^l {\left( {{\dot{v}}\left( {s,t} \right) +{\dot{l}}v_s \left( {s,t} \right) } \right) ^{2}\hbox {d}s} \nonumber \\&\quad -\,\frac{1}{4}\beta \rho \int _0^l {\left( {{\dot{w}}\left( {s,t} \right) +{\dot{l}}w_s \left( {s,t} \right) } \right) ^{2}\hbox {d}s} -\zeta _7 u_s^2 \left( {l,t} \right) \nonumber \\&\quad -\,\zeta _8 u_s^4 \left( {l,t} \right) -\zeta _9 u_s^2 \left( {l,t} \right) v_s^2 \left( {l,t} \right) +\alpha \bar{{d}}_U \varepsilon _1 k_U \nonumber \\&\quad +\,\alpha \bar{{d}}_V \varepsilon _2 k_V +\alpha \bar{{d}}_W \varepsilon _3 k_W \nonumber \\&\quad -\,\frac{\alpha c_4 }{2}\left( {{\dot{u}}\left( {l,t} \right) +{\dot{l}}{{u}}_s \left( {l,t} \right) +u_s \left( {l,t} \right) } \right) ^{2}\nonumber \\&\quad -\,\zeta _{10} \left( {{\dot{u}}\left( {l,t} \right) +{\dot{l}}u_s \left( {l,t} \right) } \right) ^{2}\nonumber \\&\quad -\,\zeta _{11} v_s^2 \left( {l,t} \right) -\zeta _{12} v_s^4 \left( {l,t} \right) \nonumber \\&\quad -\,\frac{\alpha c_5 }{2}\left( {{\dot{v}}\left( {l,t} \right) +{\dot{l}}v_s \left( {l,t} \right) +v_s \left( {l,t} \right) } \right) ^{2}\nonumber \\&\quad -\,\zeta _{13} \left( {{\dot{v}}\left( {l,t} \right) +{\dot{l}}v_s \left( {l,t} \right) } \right) ^{2}-\zeta _{14} w_s^2 \left( {l,t} \right) \nonumber \\&\quad -\,\frac{\alpha c_6 }{2}\left( {{\dot{w}}\left( {l,t} \right) +{\dot{l}}w_s \left( {l,t} \right) +w_s \left( {l,t} \right) } \right) ^{2}\nonumber \\&\quad -\,\zeta _{15} \left( {{\dot{w}}\left( {l,t} \right) +{\dot{l}}w_s \left( {l,t} \right) } \right) ^{2} \nonumber \\&\quad \hbox {+}\,\alpha \left( {\frac{\partial h_1 \left( {\tau _1 \left( t \right) } \right) }{\partial \tau _1 \left( t \right) }{{{\mathcal {N}}}}\left( {\varsigma _1 \left( t \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_1 \left( t \right) }{k_1 }\nonumber \\&\quad -\alpha c_7 \bar{{u}}_2^2 \left( t \right) +\,\alpha \left( {\frac{\partial h_2 \left( {\tau _2 \left( t \right) } \right) }{\partial \tau _2 \left( t \right) }{{{\mathcal {N}}}}\left( {\varsigma _2 \left( t \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_2 \left( t \right) }{k_2 } \nonumber \\&\quad -\,\alpha c_8 \bar{{v}}_2^2 \left( t \right) +\alpha \left( {\frac{\partial h_3 \left( {\tau _3 \left( t \right) } \right) }{\partial \tau _3 \left( t \right) }{{{\mathcal {N}}}}\left( {\varsigma _3 \left( t \right) } \right) -1} \right) \nonumber \\&\quad \times \frac{{\dot{\varsigma }}_3 \left( t \right) }{k_3 }-\alpha c_9 \bar{{w}}_2^2 \left( t \right) \end{aligned}$$
(A14)
where \(\zeta _i \), \(i=1,2,3\ldots 15\), are
$$\begin{aligned} \zeta _1= & {} \frac{\beta }{2}T_{\min } -\frac{\alpha }{2}T_{s\max } {\dot{l}}_{\max } -\frac{\alpha }{2}{\dot{l}}_{\max } EA_{s\max } \frac{1}{4\delta _1 }\nonumber \\&-\,\frac{1}{2}\alpha {\dot{T}}_{\max } -\frac{\beta }{2}LT_{s\max } -\beta EA_{\max } \frac{1}{4\delta _3 }\nonumber \\&-\,\beta \rho {\dot{l}}_{\max }^2 -\frac{1}{2}\beta LEA_{s\max } \frac{1}{4\delta _5 }, \\ \zeta _2= & {} \beta \frac{3}{4}EA_{\min } -\alpha \frac{1}{4}EA_{s\max } {\dot{l}}_{\max } -L\frac{1}{4}EA_{s\max } \beta , \\ \zeta _3= & {} \beta \frac{3}{8}EA_{\min } -\alpha \frac{1}{8}{\dot{l}}_{\max } EA_{s\max } -\frac{\alpha }{4}{\dot{l}}_{\max } EA_{s\max } \delta _1 \nonumber \\&-\,\frac{1}{8}\beta LEA_{s\max } -\frac{1}{2}\beta EA_{\max } \delta _3 -\frac{1}{4}\beta LEA_{s\max } \delta _5 , \\ \zeta _4= & {} \frac{\beta }{2}T_{\min } -\frac{\alpha }{2}{\dot{l}}_{\max } T_{s\max } -\frac{\alpha }{2}{\dot{l}}_{\max } EA_{s\max } \frac{1}{4\delta _2 }\nonumber \\&-\,\frac{1}{2}\alpha {\dot{T}}_{\max } -\frac{\beta }{2}LT_{s\max } -\beta EA_{\max } \frac{1}{4\delta _4 }-\beta \rho {\dot{l}}_{\max }^2\nonumber \\&-\,\frac{1}{2}\beta LEA_{s\max } \frac{1}{4\delta _6 }, \\ \zeta _5= & {} \beta \frac{3}{8}EA_{\min } -\alpha \frac{1}{8}EA_{s\max } {\dot{l}}_{\max } -\frac{\alpha }{4}{\dot{l}}_{\max } EA_{s\max } \delta _2\nonumber \\&-\,\frac{1}{8}\beta LEA_{s\max } -\frac{1}{2}\beta EA_{\max } \delta _4 -\frac{1}{4}\beta LEA_{s\max } \delta _6 , \\ \zeta _6= & {} \frac{1}{2}\beta EA_{\min } -\frac{1}{2}\alpha {\dot{l}}_{\max } EA_{s\max }\nonumber \\&-\,\frac{1}{2}\beta LEA_{s\max } -\beta \rho {\dot{l}}_{\max }^2 , \\ \zeta _7= & {} \alpha T_{\min } -\frac{3}{2}\alpha EA_{\max } \frac{1}{4\sigma _1 }-\frac{\alpha c_4 }{2}\nonumber \\&-\,\frac{1}{2}\beta LT_{\max } -\beta LEA_{\max } \frac{1}{4\sigma _3 }, \\ \zeta _8= & {} \frac{1}{2}\alpha EA_{\min } -\frac{3}{4}\alpha EA_{\max } \sigma _1 -\beta L\frac{3}{8}EA_{\max } \nonumber \\&-\,\frac{1}{2}\beta LEA_{\max } \sigma _3 , \quad \zeta _9 =\alpha EA_{\min } \nonumber \\&-\,\beta L\frac{3}{4}EA_{\max } , \zeta _{10} =\frac{\alpha c_4 }{2}-\frac{1}{2}\beta \rho L, \quad \zeta _{11} \nonumber \\= & {} \alpha T_{\min } -\frac{3}{2}\alpha EA_{\max } \frac{1}{4\sigma _2 }-\frac{\alpha c_5 }{2}\nonumber \\&-\,\frac{1}{2}\beta LT_{\max } -\beta LEA_{\max } \frac{1}{4\sigma _4 }, \\ \zeta _{12}= & {} \frac{1}{2}\alpha EA_{\min } -\frac{3}{4}\alpha EA_{\max } \sigma _2 -\beta L\frac{3}{8}EA_{\max } \nonumber \\&-\,\frac{1}{2}\beta LEA_{\max } \sigma _4 , \quad \zeta _{13} =\frac{\alpha c_5 }{2}-\frac{1}{2}\beta \rho L \\ \zeta _{14}= & {} \alpha EA_{\min } -\frac{\alpha c_6 }{2}-\frac{1}{2}\beta LEA_{\max } , \quad \zeta _{15} =\frac{\alpha c_6 }{2}\nonumber \\&-\,\frac{1}{2}\beta \rho L \end{aligned}$$
As a result, if the constants \(\delta _1 \), \(\delta _2 \), \(\delta _3 \), \(\delta _4 \), \(\delta _5 \), \(\delta _6 \), \(\sigma _1 \), \(\sigma _2 \), \(\sigma _3 \), \(\sigma _4 \), \(c_4 \), \(c_5 \), \(c_6 \), \(\alpha \), \(\beta \) are selected appropriately to satisfy \(\zeta _i >0\), \(i=1,2,3\ldots 15\), we have
$$\begin{aligned} {\dot{V}}\left( t \right)\le & {} -\lambda V\left( t \right) +\alpha \bar{{d}}_U \varepsilon _1 k_U \nonumber \\&+\,\alpha \bar{{d}}_V \varepsilon _2 k_V +\alpha \bar{{d}}_W \varepsilon _3 k_W \nonumber \\&+\,\alpha \left( {\xi _1 \left( t \right) {{{\mathcal {N}}}}\left( {\varsigma _1 \left( t \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_1 \left( t \right) }{k_1 } \nonumber \\&+\,\alpha \left( {\xi _2 \left( t \right) {{{\mathcal {N}}}}\left( {\varsigma _2 \left( t \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_2 \left( t \right) }{k_2 }\nonumber \\&+\,\alpha \left( {\xi _3 \left( t \right) {{{\mathcal {N}}}}\left( {\varsigma _3 \left( t \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_3 \left( t \right) }{k_3 } \end{aligned}$$
(A15)
where
$$\begin{aligned}&0<\xi _1 \left( t \right) =\frac{\partial h_1 \left( {\tau _1 \left( t \right) } \right) }{\partial \tau _1 \left( t \right) }=\frac{4}{\left( {e^{\frac{\tau _1 \left( t \right) }{\tau _{UM} }}\hbox {+}e^{-\frac{\tau _1 \left( t \right) }{\tau _{UM} }}} \right) ^{2}}\le 1, \\&\quad 0<\xi _2 \left( t \right) =\frac{\partial h_2 \left( {\tau _2 \left( t \right) } \right) }{\partial \tau _2 \left( t \right) }=\frac{4}{\left( {e^{\frac{\tau _2 \left( t \right) }{\tau _{VM} }}\hbox {+}e^{-\frac{\tau _2 \left( t \right) }{\tau _{VM} }}} \right) ^{2}} \\&\quad \le 1, 0<\xi _3 \left( t \right) =\frac{\partial h_3 \left( {\tau _3 \left( t \right) } \right) }{\partial \tau _3 \left( t \right) }=\frac{4}{\left( {e^{\frac{\tau _3 \left( t \right) }{\tau _{WM} }}\hbox {+}e^{-\frac{\tau _3 \left( t \right) }{\tau _{WM} }}} \right) ^{2}} \\&\quad \le 1, \quad \lambda _3 =\min \left\{ \frac{1}{4}\beta \rho ,\zeta _1 ,\zeta _2 ,\zeta _3 ,\zeta _4 ,\zeta _5 ,\zeta _6 , \right. \\&\left. \quad \frac{c_4 }{m},\frac{c_5 }{m},\frac{c_6 }{m},2c_7 ,2c_8 ,2c_9 \right\} , \quad \lambda =\frac{\lambda _3 }{\lambda _2 }. \end{aligned}$$
Multiplying Inequality (A15) by \(e^{\lambda t}\) and by integration, we obtain
$$\begin{aligned} V\left( t \right) \le V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}\left( t \right) \end{aligned}$$
(A16)
where
$$\begin{aligned} \bar{{\upsilon }}\left( t \right)= & {} \alpha e^{-\lambda t}\int _0^t e^{\lambda \tau }\left( \left( {\xi _1 \left( \tau \right) {{{\mathcal {N}}}}\left( {\varsigma _1 \left( \tau \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_1 \left( \tau \right) }{k_1 } \right. \\&\left. +\,\left( {\xi _2 \left( \tau \right) {{{\mathcal {N}}}}\left( {\varsigma _2 \left( \tau \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_2 \left( \tau \right) }{k_2 } \right. \\&\left. +\,\left( {\xi _3 \left( \tau \right) {{{\mathcal {N}}}}\left( {\varsigma _3 \left( \tau \right) } \right) -1} \right) \frac{{\dot{\varsigma }}_3 \left( \tau \right) }{k_3 }+\bar{{d}}_U \varepsilon _1 k_U \right. \\&\left. +\,\bar{{d}}_V \varepsilon _2 k_V +\bar{{d}}_W \varepsilon _3 k_W \right) \hbox {d}\tau \end{aligned}$$
Applying Lemma 2.3, we can come to the conclusion that V(t), \(\varsigma _i ( t )\) and \(\int _0^t {( {\xi _i ( \tau ){{{\mathcal {N}}}}( {\varsigma _i ( \tau )} )-1} ){\dot{\varsigma }}_i ( \tau )\hbox {d}\tau } \) are bounded on \([ {0,t} ), \quad i=1,2,3\), which indicates that \(\bar{{u}}_1 ( t )\), \(\bar{{v}}_1 ( t )\), \(\bar{{w}}_1 ( t )\), \(\bar{{u}}_2 ( t )\), \(\bar{{v}}_2 ( t )\), \(\bar{{w}}_2 ( t )\), \(u_s ( {s,t} )\), \(v_s ( {s,t} )\), \(w_s ( {s,t})\), \(\frac{\hbox {D}u ( {s,t} )}{\hbox {D}t}\), \(\frac{\hbox {D}v ( {s,t} )}{\hbox {D}t}\), \(\frac{\hbox {D}w ( {s,t} )}{\hbox {D}t}\) and \(\bar{{\upsilon }}( t )\) are all bounded on [ 0, t ). The range of \(\bar{{\upsilon }}( t )\) is described as \(| {\bar{{\upsilon }}( t )} |\le \bar{{\upsilon }}_{\max } \), where \(\bar{{\upsilon }}_{\max } \) is a positive constant.
By Lemma 2.2 and Inequality (A10), we can obtain that
$$\begin{aligned}&\frac{1}{L}u^2 \left( {s,t} \right) \le \int _0^l {u_s^2 \left( {s,t} \right) \hbox {d}s} \le \frac{V\left( t \right) }{\lambda _1 } \nonumber \\&\quad \le \frac{V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max } }{\lambda _1 }, \quad \forall s\in \left[ {0,L} \right] \end{aligned}$$
(A17)
$$\begin{aligned}&\frac{1}{L}v^2 \left( {s,t} \right) \le \int _0^l {v_s^2 \left( {s,t} \right) \hbox {d}s} \le \frac{V\left( t \right) }{\lambda _1 } \nonumber \\&\quad \le \frac{V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max } }{\lambda _1 }, \quad \forall s\in \left[ {0,L} \right] \end{aligned}$$
(A18)
$$\begin{aligned}&\frac{1}{L}w^2 \left( {s,t} \right) \le \int _0^l {w_s^2 \left( {s,t} \right) \hbox {d}s} \le \frac{V\left( t \right) }{\lambda _1 } \nonumber \\&\quad \le \frac{V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max } }{\lambda _1 }, \quad \forall s\in \left[ {0,L} \right] \end{aligned}$$
(A19)
which implies
$$\begin{aligned}&\left| {u\left( {s,t} \right) } \right| \le \sqrt{\frac{L}{\lambda _1 }\left( {V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max }} \right) }, \quad \forall s\in \left[ {0,L} \right] \nonumber \\ \end{aligned}$$
(A20)
$$\begin{aligned}&\left| {v\left( {s,t} \right) } \right| \le \sqrt{\frac{L}{\lambda _1 }\left( {V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max } } \right) }, \quad \forall s\in \left[ {0,L} \right] \nonumber \\ \end{aligned}$$
(A21)
$$\begin{aligned}&\left| {w\left( {s,t} \right) } \right| \le \sqrt{\frac{L}{\lambda _1 }\left( {V\left( 0 \right) e^{-\lambda t}+\bar{{\upsilon }}_{\max } } \right) } , \quad \forall s\in \left[ {0,L} \right] \nonumber \\ \end{aligned}$$
(A22)
and
$$\begin{aligned}&\mathop {\lim }\limits _{t\rightarrow \infty } \left| {u\left( {s,t} \right) } \right| \le C, \quad \mathop {\lim }\limits _{t\rightarrow \infty } \left| {v\left( {s,t} \right) } \right| \nonumber \\&\quad \le C, \quad \mathop {\lim }\limits _{t\rightarrow \infty } \left| {w\left( {s,t} \right) } \right| \le C \end{aligned}$$
(A23)
where \(C=\sqrt{\frac{L\bar{{\upsilon }}_{\max } }{\lambda _1 }}\).
And from Eq. (25–27), we know that the bounds of control inputs can be written as
$$\begin{aligned}&\left| {\tau _U \left( t \right) } \right| \le \tau _{UM} , \quad \left| {\tau _V \left( t \right) } \right| \nonumber \\&\quad \le \tau _{VM} , \quad \left| {\tau _W \left( t \right) } \right| \le \tau _{WM} \end{aligned}$$
(A24)
From the above proof, we can conclude that the vibrations of the flexible string can finally converge to the compact sets with the proposed constrained boundary control inputs.
