We construct the Lyapunov Krasovskii functional as:
$$\begin{aligned} V(x_{t},y_{t},t)=\sum \limits _{i=1}^{7} V_{i}(x_{t},y_{t},t), \end{aligned}$$
(11)
where
$$\begin{aligned} V_1(x_{t},y_{t},t)=&x^T(t)\mathcal {P}x(t)+y^T(t)\mathcal {Q}y(t),\\ V_2(x_{t},y_{t},t)=&\delta _1\int _{t-\delta _1}^{t}x^T(s)P_1x(s)ds +\delta _2\int _{t-\delta _2}^{t}y^T(s)P_2y(s)ds,\\ V_3(x_{t},y_{t},t)=&\int _{t-\tau _2(t)}^{t}x^T(s)P_3x(s)ds +\int _{t-\tau _1(t)}^{t}y^T(s)P_4y(s)ds+\int _{t-\tau _2}^{t}x^T(s)P_5x(s)ds\\&+ \int _{t-\tau _1}^{t}y^T(s)P_6y(s)ds+\int _{t-\tau _3}^{t}x^T(s)P_7x(s)ds+\int _{t-\tau _3}^{t}y^T(s)P_8y(s)ds, \\ V_4(x_{t},y_{t},t)=&\tau _2\int _{-\tau _2}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)Q_1\dot{x}(s)dsd\theta + \tau _1\int _{-\tau _1}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)Q_2\dot{y}(s)dsd\theta \\&+ \tau _3\int _{-\tau _3}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)Q_3\dot{x}(s)dsd\theta + \tau _3\int _{-\tau _3}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)Q_4\dot{y}(s)dsd\theta ,\\ V_5(x_{t},y_{t},t)=&\sum \limits _{i=0}^{1}\tau _{(3-i)}\int _{-\tau _{(3-i)}}^{0}\int _{t+\theta }^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)}\\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] dsd\theta \\+&\sum \limits _{i=1}^{2}\tau _{(2i-1)}\int _{-\tau _{(2i-1)}}^{0}\int _{t+\theta }^{t} \left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)}\\ \maltese &{} Z_{(2i)} \end{array}\right] \left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] dsd\theta ,\\ V_6(x_{t},y_{t},t)=&\frac{\tau _2^2}{2!}\int _{-\tau _2}^{0}\int _{\theta }^{0}\int _{t+\vartheta }^{t} \dot{x}^T(s) R_1\dot{x}(s)dsd\vartheta d\theta \\&+\frac{\tau _1^2}{2!}\int _{-\tau _1}^{0}\int _{\theta }^{0} \int _{t+\vartheta }^{t}\dot{y}^T(s) R_2\dot{y}(s)dsd\vartheta d\theta ,\\ V_7(x_{t},y_{t},t)=&\tau _1\int _{-\tau _1}^{0}\int _{t+\theta }^{t}f^T_2(y(s))R_3f_2(y(s))dsd\theta \\&+ \tau _2\int _{-\tau _2}^{0}\int _{t+\theta }^{t}g^T_2(x(s))R_4g_2(x(s))dsd\theta . \end{aligned}$$
We calculate the derivatives
\(\dot{V}_i(x_{t},y_{t},t),\ i=1,2,...,7\) along the trajectories of the system (8) gives,
$$\begin{aligned} \dot{V}_1(x_{t},y_{t},t)=&2x^T(t)\mathcal {P}\dot{x}(t)+2y^T(t)\mathcal {Q}\dot{y}(t),\end{aligned}$$
(12)
$$\begin{aligned} \dot{V}_2(x_{t},y_{t},t)=&\delta _1[x^T(t)P_1x(t)-x^T(t-\delta _1)P_1x(t-\delta _1)] +\delta _2[y^T(t)P_2y(t)-y^T(t-\delta _2)P_2y(t-\delta _2)],\end{aligned}$$
(13)
$$\begin{aligned} \dot{V}_3(x_{t},y_{t},t)\le&x^T(t)[P_3+P_5+P_7]x(t)+y^T(t)[P_4+P_6+P_8]y(t) -(1-\mu _2)x^T(t-\tau _2(t))P_3x(t-\tau _2(t))\nonumber \\&-(1-\mu _1)y^T(t-\tau _1(t))P_4y(t-\tau _1(t))-x^T(t-\tau _2)P_5x(t-\tau _2)-y^T(t-\tau _1) P_6y(t-\tau _1)\nonumber \\&-x^T(t-\tau _3)P_7x(t-\tau _3)-y^T(t-\tau _3)P_8y(t-\tau _3),\end{aligned}$$
(14)
$$\begin{aligned} \dot{V}_4(x_{t},y_{t},t)=&\dot{x}^T(t)[\tau _2^2Q_1+\tau _3^2Q_3]\dot{x}(t)+ \dot{y}^T(t)[\tau _1^2Q_2+\tau _3^2Q_4]\dot{y}(t)-\tau _2\int _{t-\tau _2}^{t}\dot{x}^T(s)Q_1\dot{x}(s)ds\nonumber \\&-\tau _1\int _{t-\tau _1}^{t}\dot{y}^T(s)Q_2\dot{y}(s)ds-\tau _3\int _{t-\tau _3}^{t}\dot{x}^T(s)Q_3\dot{x}(s)ds -\tau _3\int _{t-\tau _3}^{t}\dot{y}^T(s)Q_4\dot{y}(s)ds,\end{aligned}$$
(15)
$$\begin{aligned} \dot{V}_5(x_{t},y_{t},t)=&\sum \limits _{i=0}^{1}\tau ^2_{(3-i)} \left[ \begin{array} {cc} x(t)\\ \dot{x}(t) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)}\\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(t)\\ \dot{x}(t) \end{array}\right] \nonumber \\ +&\sum \limits _{i=1}^{2}\tau ^2_{(2i-1)} \left[ \begin{array} {cc} y(t)\\ \dot{y}(t) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)}\\ \maltese &{} Z_{(2i)} \end{array}\right] \left[ \begin{array} {cc} y(t)\\ \dot{y}(t) \end{array}\right] \nonumber \\ -&\sum \limits _{i=0}^{1}\tau _{(3-i)}\int _{t-\tau _{(3-i)}}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)}\\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\nonumber \\ -&\sum \limits _{i=1}^{2}\tau _{(2i-1)}\int _{t-\tau _{(2i-1)}}^{t} \left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)}\\ \maltese &{} Z_{(2i)} \end{array}\right] \left[ \begin{array} {cc} y(s)\\ \dot{y}(s), \end{array}\right] ds\end{aligned}$$
(16)
$$\begin{aligned} \dot{V}_6(x_{t},y_{t},t)=&\bigg (\frac{\tau _2^2}{2!}\bigg )^2\dot{x}^T(t)R_1\dot{x}(t) +\bigg (\frac{\tau _1^2}{2!}\bigg )^2\dot{y}^T(t)R_2\dot{y}(t) -\frac{\tau _2^2}{2!}\int _{-\tau _1}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)R_1\dot{x}(s)dsd\theta \nonumber \\&-\frac{\tau _1^2}{2!}\int _{-\tau _1}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)R_2\dot{y}(s)dsd\theta ,\end{aligned}$$
(17)
$$\begin{aligned} \dot{V}_7(x_{t},y_{t},t)=&f_2^T(y(t))\tau _1^2R_3 f_2(y(t))+g_2^T(x(t))\tau _2^2R_4 g_2(x(t))\nonumber \\&-\tau _1\int _{t-\tau _1}^{t}f_2^T(y(s))R_3 f_2(y(s))ds-\tau _2\int _{t-\tau _2}^{t}g_2^T(x(s))R_4 g_2(x(s))ds, \end{aligned}$$
(18)
The integral terms in (
15) can be expressed as following
$$\begin{aligned} -\tau _2\int _{t-\tau _2}^{t}\dot{x}^T(s)Q_1\dot{x}(s)ds=&-\tau _2 \int _{t-\tau _2}^{t-\tau _2(t)}\dot{x}^T(s)Q_1\dot{x}(s)ds- \tau _2\int _{t-\tau _2(t)}^{t}\dot{x}^T(s)Q_1\dot{x}(s)ds,\\ -\tau _1\int _{t-\tau _1}^{t}\dot{y}^T(s)Q_2\dot{y}(s)ds=&- \tau _1\int _{t-\tau _1}^{t-\tau _1(t)}\dot{y}^T(s)Q_2\dot{y}(s)ds-\tau _1\int _{t-\tau _1(t)}^{t}\dot{y}^T(s)Q_2\dot{y}(s)ds,\\ -\tau _3\int _{t-\tau _3}^{t}\dot{x}^T(s)Q_3\dot{x}(s)ds=&- \tau _3\int _{t-\tau _3}^{t-\tau _3(t)}\dot{x}^T(s)Q_3\dot{x}(s)ds-\tau _3\int _{t-\tau _3(t)}^{t}\dot{x}^T(s)Q_3\dot{x}(s)ds,\\ -\tau _3\int _{t-\tau _3}^{t}\dot{y}^T(s)Q_4\dot{y}(s)ds=&- \tau _3\int _{t-\tau _2}^{t-\tau _3(t)}\dot{y}^T(s)Q_4\dot{y}(s)ds-\tau _3\int _{t-\tau _3(t)}^{t}\dot{y}^T(s)Q_4\dot{y}(s)ds. \end{aligned}$$
By applying Lemma 2.2, we obtain
$$\begin{aligned}&-\tau _2\int _{t-\tau _2}^{t-\tau _2(t)}\dot{x}^T(s)Q_1\dot{x}(s)ds\le&\left[ \begin{array} {cc} x(t-\tau _2(t))\\ x(t-\tau _2) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_1 &{} Q_1\\ \maltese &{} -Q_1 \end{array}\right] \left[ \begin{array} {cc} x(t-\tau _2(t))\\ x(t-\tau _2) \end{array}\right] , \end{aligned}$$
(19)
$$\begin{aligned}&-\tau _2\int _{t-\tau _2(t)}^{t}\dot{x}^T(s)Q_1\dot{x}(s)ds\le&\left[ \begin{array} {cc} x(t)\\ x(t-\tau _2(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_1 &{} Q_1\\ \maltese &{} -Q_1 \end{array}\right] \left[ \begin{array} {cc} x(t)\\ x(t-\tau _2(t)) \end{array}\right] ,\end{aligned}$$
(20)
$$\begin{aligned}&-\tau _1\int _{t-\tau _1}^{t-\tau _1(t)}\dot{y}^T(s)Q_2\dot{y}(s)ds\le&\left[ \begin{array} {cc} y(t-\tau _1(t))\\ y(t-\tau _1) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_2 &{} Q_2\\ \maltese &{} -Q_2 \end{array}\right] \left[ \begin{array} {cc} y(t-\tau _1(t))\\ y(t-\tau _1) \end{array}\right] ,\end{aligned}$$
(21)
$$\begin{aligned}&-\tau _1\int _{t-\tau _1(t)}^{t}\dot{y}^T(s)Q_2\dot{y}(s)ds\le&\left[ \begin{array} {cc} y(t)\\ y(t-\tau _1(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_2 &{} Q_2\\ \maltese &{} -Q_2 \end{array}\right] \left[ \begin{array} {cc} y(t)\\ y(t-\tau _1(t)) \end{array}\right] ,\end{aligned}$$
(22)
$$\begin{aligned}&-\tau _3\int _{t-\tau _3}^{t-\tau _3(t)}\dot{x}^T(s)Q_3\dot{x}(s)ds\le&\left[ \begin{array} {cc} x(t-\tau _3(t))\\ x(t-\tau _3) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_3 &{} Q_3\\ \maltese &{} -Q_3 \end{array}\right] \left[ \begin{array} {cc} x(t-\tau _3(t))\\ x(t-\tau _3) \end{array}\right] ,\end{aligned}$$
(23)
$$\begin{aligned}&-\tau _3\int _{t-\tau _3(t)}^{t}\dot{x}^T(s)Q_3\dot{x}(s)ds\le&\left[ \begin{array} {cc} x(t)\\ x(t-\tau _3(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_3 &{} Q_3\\ \maltese &{} -Q_3 \end{array}\right] \left[ \begin{array} {cc} x(t)\\ x(t-\tau _3(t)) \end{array}\right] ,\end{aligned}$$
(24)
$$\begin{aligned}&-\tau _3\int _{t-\tau _3}^{t-\tau _3(t)}\dot{y}^T(s)Q_4\dot{y}(s)ds\le&\left[ \begin{array} {cc} y(t-\tau _3(t))\\ y(t-\tau _3) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_4 &{} Q_4\\ \maltese &{} -Q_4 \end{array}\right] \left[ \begin{array} {cc} y(t-\tau _3(t))\\ y(t-\tau _3) \end{array}\right] ,\end{aligned}$$
(25)
$$\begin{aligned}&-\tau _3\int _{t-\tau _3(t)}^{t}\dot{y}^T(s)Q_4\dot{y}(s)ds\le&\left[ \begin{array} {cc} y(t)\\ y(t-\tau _3(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -Q_4 &{} Q_4\\ \maltese &{} -Q_4 \end{array}\right] \left[ \begin{array} {cc} y(t)\\ y(t-\tau _3(t)) \end{array}\right] . \end{aligned}$$
(26)
By applying the Lemmas
2.1 and
2.3 in
\(\dot{V}_5(x_{t},y_{t},t)\), we obtain the following results
$$\begin{aligned}&-\sum \limits _{i=0}^{1}\tau _{(3-i)}\int _{t-\tau _{(3-i)}}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)}\\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\quad =-\sum \limits _{i=0}^{1}\tau _{(3-i)}\Bigg [\int _{t-\tau _{(3-i)}(t)}^{t} +\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)}\Bigg ] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds,\\&\quad =-\sum \limits _{i=0}^{1}\tau _{(3-i)}\Biggl [-\tau _{(3-i)}\int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\qquad -\tau _{(3-i)}\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\Biggl ],\\&\quad =-\sum \limits _{i=0}^{1}\tau _{(3-i)}\Biggl [\frac{\tau _{(3-i)}}{\tau _{(3-i)}(t)}\times \tau _{(3-i)}(t)\int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\qquad -\frac{\tau _{(3-i)}(\tau _{(3-i)}-\tau _{(3-i)}(t))}{(\tau _{(3-i)}-\tau _{(3-i)}(t))}\times \int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\Biggl ],\\&\quad \le -\sum \limits _{i=0}^{1}\tau _{(3-i)}\Biggl [\frac{\tau _{(3-i)}}{\tau _{(3-i)}(t)}\int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T ds \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\qquad -\frac{\tau _{(3-i)}}{(\tau _1-\tau _{(3-i)}(t))}\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T ds \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\Biggl ],\\&\qquad \le -\sum \limits _{i=0}^{1}\tau _{(3-i)}\Biggl [\int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T ds \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\qquad -2\int _{t-\tau _{(3-i)}(t)}^{t} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T ds \left[ \begin{array} {cc} M_{(3-2i)} &{} N_{(3-2i)}\\ H_{(3-2i)} &{} T_{(3-2i)} \end{array}\right] \int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\&\qquad -\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ^T ds \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)} \left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\Biggl ], \end{aligned}$$
$$\begin{aligned} =-\sum \limits _{i=0}^{1}\tau _{(3-i)}\left[ \begin{array}{c} \begin{array}{c} \int _{t-\tau _{(3-i)}(t)}^{t}\left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\ \hline -\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)}\left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds \end{array} \\ \end{array} \right] ^T&\left[ \begin{array}{c|c} \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] &{} \left[ \begin{array}{cc} M_{(3-2i)} &{} N_{(3-2i)} \\ H_{(3-2i)} &{} T_{(3-2i)} \end{array}\right] \\ \hline \maltese &{} \left[ \begin{array} {cc} X_{(3-2i)} &{} Y_{(3-2i)} \\ \maltese &{} Z_{(3-2i)} \end{array}\right] \end{array} \right] \nonumber \\ {}&\times \left[ \begin{array}{c} \begin{array}{c} \int _{t-\tau _{(3-i)}(t)}^{t}\left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds\\ \hline -\int _{t-\tau _{(3-i)}}^{t-\tau _{(3-i)}(t)}\left[ \begin{array} {cc} x(s)\\ \dot{x}(s) \end{array}\right] ds \end{array} \\ \end{array} \right] . \end{aligned}$$
(27)
We can estimate the three terms in inequality (
16) in a similar way to (
27) as
$$\begin{aligned}&-\sum \limits _{i=1}^{2}\tau _{(2i-1)}\int _{t-\tau _{(2i-1)}}^{t} \left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ^T \left[ \begin{array} {cc} X_{2i} &{} Y_{2i}\\ \maltese &{} Z_{2i} \end{array}\right] \left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ds\nonumber \\ {}&\le -\sum \limits _{i=1}^{2}\tau _{(2i-1)} \left[ \begin{array}{c} \begin{array}{c} \int _{t-\tau _{(2i-1)}(t)}^{t}\left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ds\\ \hline \int _{t-\tau _{(2i-1)}}^{t-\tau _{(2i-1)}(t)}\left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ds \end{array} \\ \end{array} \right] ^T \left[ \begin{array}{c|c} \left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)} \\ \maltese &{} Z_{(2i)} \end{array}\right] &{} \left[ \begin{array}{cc} M_{(2i)} &{} N_{(2i)} \\ H_{(2i)} &{} T_{(2i)} \end{array}\right] \\ \hline \maltese &{} \left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)} \\ \maltese &{} Z_{(2i)} \end{array}\right] \end{array} \right] \nonumber \\ {}&\hspace{2cm}\times \left[ \begin{array}{c} \begin{array}{c} \int _{t-\tau _{(2i-1)}(t)}^{t}\left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ds\\ \hline \int _{t-\tau _{(2i-1)}}^{t-\tau _{(2i-1)}(t)}\left[ \begin{array} {cc} y(s)\\ \dot{y}(s) \end{array}\right] ds \end{array} \\ \end{array} \right] . \end{aligned}$$
(28)
The upper bound of the reciprocally convex combination in
\(\dot{V}_6(x_{t},y_{t},t)\) can be obtained as
$$\begin{aligned}&-\frac{\tau _2^2}{2!}\int _{-\tau _2}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)R_1\dot{x}(s)dsd\theta \nonumber \\&\quad \le \left[ \begin{array} {cc} \int _{-\tau _2(t)}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)ds\\ \int _{-\tau _2}^{-\tau _2(t)}\int _{t+\theta }^{t}\dot{x}^T(s)ds \end{array}\right] ^T \left[ \begin{array} {cc} -R_1 &{} -{\tilde{R}}_1\\ \maltese &{} -R_1 \end{array}\right] \left[ \begin{array} {cc} \int _{-\tau _2(t)}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)ds\\ \int _{-\tau _2}^{-\tau _2(t)}\int _{t+\theta }^{t}\dot{x}^T(s)ds \end{array}\right] ,\end{aligned}$$
(29)
$$\begin{aligned}&-\frac{\tau _1^2}{2!}\int _{-\tau _1}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)R_2\dot{y}(s)dsd\theta \nonumber \\&\quad \le \left[ \begin{array} {cc} \int _{-\tau _1(t)}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)ds\\ \int _{-\tau _1}^{-\tau _1(t)}\int _{t+\theta }^{t}\dot{y}^T(s)ds \end{array}\right] ^T \left[ \begin{array} {cc} -R_2 &{} -{\tilde{R}}_2\\ \maltese &{} -R_2 \end{array}\right] \left[ \begin{array} {cc} \int _{-\tau _1(t)}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)ds\\ \int _{-\tau _1}^{-\tau _1(t)}\int _{t+\theta }^{t}\dot{y}^T(s)ds \end{array}\right] . \end{aligned}$$
(30)
According to Lemma 2.1 the following inequality’s are hold:
$$\begin{aligned}&-\tau _1\int _{t-\tau _1}^{t}f_2^T(y(s))R_3 f_2(y(s))ds\le - \Bigg (\int _{t-\tau _1}^{t}f_2(y(s))ds\Bigg )^T R_3\Bigg (\int _{t-\tau _1}^{t}f_2(y(s))ds\Bigg ),\end{aligned}$$
(31)
$$\begin{aligned}&-\tau _2\int _{t-\tau _2}^{t}g_2^T(x(s))R_4 g_2(x(s))ds\le - \Bigg (\int _{t-\tau _2}^{t}g_2(x(s))ds\Bigg )^T R_4\Bigg (\int _{t-\tau _2}^{t}g_2(x(s))ds\Bigg ). \end{aligned}$$
(32)
On the other hand, for any matrices
\(S_1\) and
\(S_2\) with appropriate dimensions the following equations holds:
$$\begin{aligned}&0=[x(t)+\dot{x}(t)] S_1[-\dot{x}(t)+\dot{x}(t)],\nonumber \\&0=[x(t)+\dot{x}(t)] S_1[-\dot{x}(t)-Ax(t-\delta _1)+W_1f_1(y(t-\tau _1(t))) +W_2\int _{t-\tau _1}^{t}f_2(y(s))ds+\mathcal {K}x(t-\tau _3(t))],\end{aligned}$$
(33)
$$\begin{aligned}&0=[y(t)+\dot{y}(t)]S_2[-\dot{y}(t)+\dot{y}(t)],\nonumber \\&0=[y(t)+\dot{y}(t)]S_2[-\dot{y}(t)-By(t-\delta _2)+V_1g_1(x(t-\tau _2(t))) +V_2\int _{t-\tau _2}^{t}g_2(x(s))ds+\mathcal {M}y(t-\tau _3(t))]. \end{aligned}$$
(34)
Furthermore, based on Assumption (A), we have
$$\begin{aligned}&\Big [f_{ki}(y_i(t))-F_{ki}^{-}y_i(t)\Big ]^T\Big [f_{ki}(y_i(t))-F_{ki}^{+}y_i(t)\Big ]\le 0,\ i=1,2,...,n,\\&\Big [g_{kj}(x_j(t))-G_{kj}^{-}x_j(t)\Big ]^T\Big [g_{kj}(x_i(t))-G_{kj}^{+}x_j(t)\Big ]\le 0,\ j=1,2,...,n, \end{aligned}$$
where
\(k=1,2,\)
which is equivalent to
$$\begin{aligned}&\left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] ^T \left[ \begin{array} {cc} F_{ki}^{-}F_{ki}^{+}y_iy_i^T &{} -\frac{F_{ki}^{-}+F_{ki}^{+}}{2}y_iy_i^T\\ -\frac{F_{ki}^{-}+F_{ki}^{+}}{2}y_iy_i^T &{} y_iy_i^T \end{array}\right] \left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] \le 0,\\&\left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] ^T \left[ \begin{array} {cc} G_{kj}^{-}G_{kj}^{+}x_jx_j^T &{} -\frac{G_{kj}^{-}+G_{kj}^{+}}{2}x_jx_j^T\\ -\frac{G_{kj}^{-}+G_{kj}^{+}}{2}x_jx_j^T &{} x_jx_j^T \end{array}\right] \left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] \le 0. \end{aligned}$$
where
\(x_i\) and
\(y_j\) denotes the units column vector having element 1 on its
\(i^{th}\) row,
\(j^{th}\) row, and zeros elsewhere. Let
\(U_k=diag\{u^k_{11},u^k_{12},...,u^k_{1n}\}>0,\) \(J_k=diag\{{\overline{j}}^k_{11},{\overline{j}}^k_{12},...,{\overline{j}}^k_{1n}\}>0,\). Here
\(x_i\) and
\(y_j\) stand for the units column vector, which has zeros on the other rows and element 1 on the
\(i^{th}\) row,
\(j^{th}\) rows. It is simple to observe that if
\(U_k=diag\{u^k_{11},u^k_{12},...,u^k_{1n}\}>0,\) \(J_k=diag\{{\overline{j}}^k_{11},{\overline{j}}^k_{12},...,{\overline{j}}^k_{1n}\}>0,\)$$\begin{aligned}&\sum \limits _{i=1}^{n}u^k_{1i} \left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] ^T \left[ \begin{array} {cc} F_{ki}^{-}F_{ki}^{+}y_iy_i^T &{} -\frac{F_{ki}^{-}+F_{ki}^{+}}{2}y_iy_i^T\\ -\frac{F_{ki}^{-}+F_{ki}^{+}}{2}y_iy_i^T &{} y_iy_i^T \end{array}\right] \left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] \le 0,\\&\sum \limits _{j=1}^{n}{\overline{j}}^k_{1j} \left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] ^T \left[ \begin{array} {cc} G_{kj}^{-}G_{kj}^{+}x_jx_j^T &{} -\frac{G_{kj}^{-}+G_{kj}^{+}}{2}x_jx_j^T\\ -\frac{G_{kj}^{-}+G_{kj}^{+}}{2}x_jx_j^T &{} x_jx_j^T \end{array}\right] \left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] \le 0, \end{aligned}$$
That is,
$$\begin{aligned}&\left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -F_{k1}U_{k} &{} F_{k2}U_{k}\\ F_{k2}U_{k} &{} -U_{k} \end{array}\right] \left[ \begin{array} {cc} y(t)\\ f_{k}(y(t)) \end{array}\right] \ge 0,\end{aligned}$$
(35)
$$\begin{aligned}&\left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] ^T \left[ \begin{array} {cc} -G_{k1}J_{k} &{} G_{k2}J_{k}\\ G_{k2}J_{k} &{} -J_{k} \end{array}\right] \left[ \begin{array} {cc} x(t)\\ g_{k}(x(t)) \end{array}\right] \ge 0. \end{aligned}$$
(36)
Similar to above, for
\(U_{k+2}=diag\{u^{k+2}_{11},u^{k+2}_{12},...,u^{k+2}_{1n}\}>0\),
\(J_{k+2}=diag\{{\overline{j}}^{k+2}_{11},{\overline{j}}^{k+2}_{12},...,{\overline{j}}^{k+2}_{1n}\}>0,\) we can obtain the following inequalities:
$$\begin{aligned}&\left[ \begin{array} {cc} y(t-\tau _1(t))\\ f_{k}(y(t-\tau _1(t))) \end{array}\right] ^T \left[ \begin{array} {cc} -F_{k1}U_{k+2} &{} F_{k2}U_{k+2}\\ F_{k2}U_{k+2} &{} -U_{k+2} \end{array}\right] \left[ \begin{array} {cc} y(t-\tau _1(t))\\ f_{k}(y(t-\tau _1(t))) \end{array}\right] \ge 0,\end{aligned}$$
(37)
$$\begin{aligned}&\left[ \begin{array} {cc} x(t-\tau _2(t))\\ g_{k}(x(t-\tau _2(t))) \end{array}\right] ^T \left[ \begin{array} {cc} -G_{k1}J_{k+2} &{} G_{k2}J_{k+2}\\ G_{k2}J_{k+2} &{} -J_{k+2} \end{array}\right] \left[ \begin{array} {cc} x(t-\tau _2(t))\\ g_{k}(x(t-\tau _2(t))) \end{array}\right] \ge 0. \end{aligned}$$
(38)
Now, combining (
12)–(
38), we have
$$\begin{aligned} \dot{V}(x_{t},y_{t},t)\le -\zeta ^T(t)\Omega \zeta (t), \end{aligned}$$
(39)
where
$$\begin{aligned} \zeta ^T(t)=&[\zeta _1^T(t)\ \zeta _2^T(t)],\\ \zeta _1^T(t)=&\bigg [x^T(t)\quad \dot{x}^T(t)\\ {}&\quad x^T(t-\delta _1)\quad x^T(t-\tau _2(t))\quad x^T(t-\tau _2)\quad x^T(t-\tau _3(t))\quad x^T(t-\tau _3)\quad g^T_1(x(t)\\ {}&g^T_1(x(t-\tau _2(t)) \quad g^T_2(x(t)\quad g^T_2(x(t-\tau _2(t))\\ {}&\quad \int _{t-\tau _2}^{t}g^T_2(x(s)ds\quad \int _{t-\tau _2(t)}^{t}x^T(s)ds\quad \int _{t-\tau _2(t)}^{t}\dot{x}^T(s)ds\\&\int _{t-\tau _2}^{t-\tau _2(t)} x^T(s)ds\quad \int _{t-\tau _2}^{t-\tau _2(t)} \dot{x}^T(s)ds\quad \int _{-\tau _2(t)}^{0}\int _{t+\theta }^{t}\dot{x}^T(s)ds \\ {}&\quad \int _{-\tau _2}^{-\tau _2(t)}\int _{t+\theta }^{t}\dot{x}^T(s)ds\quad \int _{t-\tau _3(t)}^{t}x^T(s)ds\\&\int _{t-\tau _3(t)}^{t}\dot{x}^T(s)ds\quad \int _{t-\tau _3}^{t-\tau _3(t)} x^T(s)ds\quad \int _{t-\tau _3}^{t-\tau _3(t)}\dot{x}^T(s)ds\bigg ],\\ \zeta _2^T(t)=&\bigg [y^T(t)\quad \dot{y}^T(t)\\ {}&\quad y^T(t-\delta _2)\ y^T(t-\tau _1(t))\ y^T(t-\tau _1)\quad y^T(t-\tau _3(t))\quad y^T(t-\tau _3) \quad f^T_1(y(t)\\ {}&f^T_1(y(t-\tau _1(t))\\ {}&\quad f^T_2(y(t)\quad f^T_2(y(t-\tau _1(t))\ \int _{t-\tau _1}^{t}f^T_2(y(s)ds\quad \int _{t-\tau _1(t)}^{t}y^T(s)ds\quad \int _{t-\tau _1(t)}^{t}\dot{y}^T(s)ds\\ {}&\int _{t-\tau _1}^{t-\tau _1(t)} y^T(s)ds\quad \int _{t-\tau _1}^{t-\tau _1(t)}\dot{y}^T(s)ds\ \int _{-\tau _1(t)}^{0}\int _{t+\theta }^{t}\dot{y}^T(s)ds \ \int _{-\tau _1}^{-\tau _1(t)}\int _{t+\theta }^{t}\dot{y}^T(s)ds\\&\int _{t-\tau _3(t)}^{t}y^T(s)ds \int _{t-\tau _3(t)}^{t}\dot{y}^T(s)ds\\ {}&\quad \int _{t-\tau _3}^{t-\tau _3(t)} y^T(s)ds\ \int _{t-\tau _3}^{t-\tau _3(t)}\dot{y}^T(s)ds\bigg ], \end{aligned}$$
and
\(\Omega \) is defined as in (
10).
Thus, the equilibrium point of (
8) is globally asymptotically stable. It may be inferred from the inequality (
39) that,
$$\begin{aligned} V(x_{t},y_{t},t)+\int _{0}^{t}\zeta ^T(s)\Omega \zeta (s)ds\le V(0)<\infty , \end{aligned}$$
(40)
where
$$\begin{aligned} V(0)\le \big [&\lambda _{max}(\mathcal {P})+\delta _1\lambda _{max}(P_1)+\tau _2\lambda _{max}(P_3)+\tau _2\lambda _{max}(P_5) +\tau _3\lambda _{max}(P_7)+\frac{\tau _2^2}{2!}\lambda _{max}(Q_1)\\ +&\frac{\tau _3^2}{2!}\lambda _{max}(Q_3)+\sum \limits _{i=0}^{1}\frac{\tau _{(3-i)}^2}{2!}\lambda _{max}\left[ \begin{array} {cc} X_{(2i+1)} &{} Y_{(2i+1)}\\ \maltese &{} Z_{(2i+1)} \end{array}\right] + \frac{\tau _2^3}{3!}\lambda _{max}(R_1)+\frac{\tau _2^2}{2!}\lambda _{max}(R_3)\big ] \mathbf \Psi _{x_t}^2\\+&\big [\lambda _{max}(\mathcal {Q})+\delta _2\lambda _{max}(P_2)+\tau _1\lambda _{max}(P_4) +\tau _1\lambda _{max}(P_6)+\tau _3\lambda _{max}(P_8)+\frac{\tau _1^2}{2!}\lambda _{max}(Q_2)\\ +&\frac{\tau _3^2}{2!}\lambda _{max}(Q_4)+\sum \limits _{i=1}^{2}\frac{\tau _{(2i-1)}^2}{2!}\lambda _{max}\left[ \begin{array} {cc} X_{(2i)} &{} Y_{(2i)}\\ \maltese &{} Z_{(2i)} \end{array}\right] +\frac{\tau _1^3}{3!}\lambda _{max}(R_2)+ \frac{\tau _1^2}{2!}\lambda _{max}(R_4)\big ]\mathbf \Psi _{y_t}^2, \end{aligned}$$
$$\begin{aligned} =\varSigma _1\mathbf \Psi _{x_t}^2+\varSigma _2\mathbf \Psi _{y_t}^2, \end{aligned}$$
(41)
where
$$\begin{aligned}&\mathbf \Psi _{x_t}=max\{\sup \limits _{\theta \in [-\upsilon _{x_t},0]}\Vert \psi _{x_t} (\theta )\Vert ,\ \sup \limits _{\theta \in [-\upsilon _2,0]}\Vert {\dot{\psi }}_{x_t}(\theta )\Vert \},\\ {}&\mathbf \Psi _{y_t} =max\{\sup \limits _{\theta \in [-\upsilon _{y_t},0]}\Vert \psi _{y_t} (\theta )\Vert ,\ \sup \limits _{\theta \in [-\upsilon _2,0]}\Vert {\dot{\psi }}_{y_t}(\theta )\Vert \}. \end{aligned}$$
On the other hand, by the definition of
\(V(x_{t},y_{t},t)\), we get
$$\begin{aligned} V(x_{t},y_{t},t)\ge&x^T(t)\mathcal {P}x(t)+y^T(t)\mathcal {Q}y(t),\nonumber \\ \ge&\lambda _{min}\mathcal {P}x^T(t)x(t)+\lambda _{min}\mathcal {Q}y^T(t)y(t),\nonumber \\ =&\lambda _{min}\mathcal {P}\Vert x(t)\Vert ^2+\lambda _{min}\mathcal {Q}\Vert y(t)\Vert ^2,\nonumber \\ =&\min \{\lambda _{min}\mathcal {P},\lambda _{min}\mathcal {Q}\}\big (\Vert x(t)\Vert ^2+\Vert y(t)\Vert ^2\big ), \end{aligned}$$
(42)
Then, combining (41) and (42), we obtain
$$\begin{aligned} \Vert x(t)\Vert ^2+\Vert y(t)\Vert ^2\le \frac{V(0)}{\min \{\lambda _{min}\mathcal {P},\lambda _{min}\mathcal {Q}\}}. \end{aligned}$$
(43)
It is clear that the system (
8) is globally asymptotic stable for
\(\Omega <0\), by Lyapunov stability theory. This completes the proof.
\(\square \)