In this section, we shall state and prove the main results.
3.1 Exponential Stability Analysis
Consider the following uncertain neutral neural networks:
$$ \begin{aligned} & \dot{x}(t) - C_{\sigma (t)} (t)\dot{x}(t - h(t)) = - A_{\sigma (t)} (t)x(t) + B_{\sigma (t)} (t)f(x(t)) + D_{\sigma (t)} (t)g(x(t - \tau (t))), \\ & x(t) = \varphi (t),\quad \forall t \in [ - \bar{\tau },0], \\ \end{aligned} $$
(7)
Choose a Lyapunov-like-Krasovskii functional candidate:
$$ \begin{aligned} V_{\sigma (t)} (x(t)) & = V_{1\sigma (t)} (x(t)) + V_{2\sigma (t)} (x(t)) + V_{3\sigma (t)} (x(t)) + V_{4\sigma (t)} (x(t)) \\ & \quad + V_{5\sigma (t)} (x(t)) + V_{6\sigma (t)} (x(t)) + V_{7\sigma (t)} (x(t)), \\ \end{aligned} $$
(8)
where
$$ \begin{aligned} V_{1\sigma (t)} (x(t)) & = x^{T} (t)P_{\sigma (t)} x(t), \\ V_{2\sigma (t)} (t) & = \int\limits_{{t - \tau_{1} }}^{t} {e^{\alpha (s - t)} } x^{T} (s)Q_{\sigma (t)} x(s)ds, \\ V_{3\sigma (t)} (x(t)) & = \int\limits_{{t - h_{1} }}^{t} {e^{\alpha (s - t)} } x^{T} (s)R_{\sigma (t)} x(s)ds, \\ V_{4\sigma (t)} (x(t)) & = \int\limits_{t - \tau (t)}^{t} {e^{\alpha (s - t)} } x^{T} (s)S_{\sigma (t)} x(s)ds, \\ V_{5\sigma (t)} (x(t)) & = \int\limits_{t - h(t)}^{t} {e^{\alpha (s - t)} } x^{T} (s)U_{\sigma (t)} x(s)ds, \\ V_{6\sigma (t)} (x(t)) & = \int\limits_{{ - \tau_{1} }}^{0} {\int\limits_{t + \theta }^{t} {e^{\alpha (s - t)} } } \dot{x}^{T} (s)\varLambda_{\sigma (t)} \dot{x}(s)dsd\theta , \\ V_{7\sigma (t)} (x(t)) & = \int\limits_{t - \tau (t)}^{t} {e^{\alpha (s - t)} } g^{T} (x(s))W_{\sigma (t)} g(x(s))ds. \\ \end{aligned} $$
(9)
First of all, we give the following lemma.
Lemma 3
Consider the system (7). For given constants\( \alpha > 0, \)\( \rho_{i} > 0 \), the following inequality holds:
$$ \dot{V}_{i} (x(t)) \le - \alpha V_{i} (x(t)), $$
if there exist matrices \( P_{i} > 0,\;Q_{i} > 0,\;R_{i} > 0,\;S_{i} > 0,\;U_{i} > 0,\;\varLambda_{i} > 0,\;G_{i1} > 0,\;G_{i2} > 0,\;W_{i} > 0,\;X_{i1} > 0 \) and \( X_{i2} > 0 \), such that the following matrices inequality hold for all \( i \in {\mathbb{N}} \):
$$ \varSigma_{i} (t) = \left[ {\begin{array}{*{20}c} {\varXi_{i} (t)} & {\tau_{1} G_{i} } & {\tau_{1} X_{i} } \\ * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } & 0 \\ * & * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } \\ \end{array} } \right] < 0, $$
(10)
where
$$ \varXi_{i} (t) = \left[ {\begin{array}{*{20}c} {\varXi_{11}^{i} (t)} & 0 & 0 & {\varXi_{14}^{i} } & 0 & {\varXi_{16}^{i} (t)} & {\varGamma_{2} } & {\varXi_{18}^{i} (t)} & {\varXi_{19}^{i} (t)} & {\varXi_{1,10}^{i} (t)} \\ * & {\varXi_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & {\varXi_{33}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\varXi_{44}^{i} } & {\varXi_{45}^{i} } & 0 & 0 & {\varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\varXi_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\varXi_{69}^{i} (t)} & {\varXi_{6,10}^{i} (t)} \\ * & * & * & * & * & * & {\varXi_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\varXi_{88}^{i} } & {\varXi_{89}^{i} (t)} & {\varXi_{8,10}^{i} (t)} \\ * & * & * & * & * & * & * & * & {\varXi_{99}^{i} } & {\varXi_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\varXi_{10,10}^{i} (t)} \\ \end{array} } \right], $$
$$ \begin{aligned} \varXi_{11}^{i} (t) & = - P_{i} A_{i} (t) - A_{i}^{T} (t)P_{i} + \alpha_{i} P_{i} + Q_{i} + R_{i} + S_{i} + U_{i} + e^{{ - \alpha \tau_{1} }} \left( {G_{i1}^{T} + G_{i1} } \right) - L_{1} - \varGamma_{1} , \\ \varXi_{14}^{i} & = e^{{ - \alpha \tau_{1} }} \left( {G_{i2}^{T} - G_{i1} } \right),\quad \varXi_{16}^{i} (t) = P_{i} B_{i} (t) + L_{2} ,\quad \varXi_{18}^{i} (t) = P_{i} D_{i} (t), \\ \varXi_{19}^{i} (t) & = - \rho_{i} A_{i}^{T} (t)P_{i} ,\quad \varXi_{1,10}^{i} (t) = \rho_{i} A_{i}^{T} (t)P_{i} + P_{i} C_{i} (t),\quad \varXi_{22}^{i} = - (1 - h)e^{{ - \alpha h_{1} }} U_{i} , \\ \varXi_{33}^{i} & = - e^{{ - \alpha h_{1} }} R_{i} ,\;\;\varXi_{44}^{i} = - (1 - \tau )e^{{ - \alpha \tau_{1} }} S_{i} + e^{{ - \alpha \tau_{1} }} \left( { - G_{i2}^{T} - G_{i2} + X_{i1}^{T} + X_{i1} } \right) - \varGamma_{1} , \\ \varXi_{45}^{i} & = e^{{ - \alpha \tau_{1} }} \left( {X_{i2}^{T} - X_{i1} } \right),\quad \varXi_{55}^{i} = - e^{{ - \alpha \tau_{1} }} Q_{i} - e^{{ - \alpha_{i} \tau_{1} }} \left( {X_{i2}^{T} + X_{i2} } \right), \\ \varXi_{69}^{i} (t) & = \rho_{i} B_{i}^{T} (t)P_{i} ,\quad \varXi_{6,10}^{i} (t) = - \rho_{i} B_{i}^{T} (t)P_{i} ,\quad \varXi_{77}^{i} = W_{i} - I, \\ \varXi_{88}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} W_{i} - I,\quad \varXi_{89}^{i} (t) = \rho_{i} D_{i}^{T} (t)P_{i} ,\quad \varXi_{8,10}^{i} (t) = - \rho_{i} D_{i}^{T} (t)P_{i} , \\ \varXi_{99}^{i} & = \tau_{1} \varLambda_{i} -\,2\rho_{i} P_{i} ,\;\;\varXi_{9,10}^{i} = \rho_{i} P_{i} ,\quad \varXi_{10,10}^{i} (t) = -\,2\rho_{i} P_{i} C_{i} (t), \\ G_{i} & = \left[ {\begin{array}{*{20}l} {G_{i1}^{T} } \hfill & 0 \hfill & 0 \hfill & {G_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} , \\ X_{i} & = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {X_{i1}^{T} } \hfill & {X_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} . \\ \end{aligned} $$
Proof
Assume \( \sigma (t_{k} ) = i \), \( \sigma (t_{k}^{ - } ) = j,\;\;i,j \in {\mathbb{N}}. \) When \( t \in [t_{k} ,t_{k + 1} ) \), we have \( \sigma (t) = i \). Along the trajectories of system (7), the time derivative of \( {\text{V}}_{ki} ({\text{x}}({\text{t}})),\;k = 1,2, \ldots ,7, \) can be obtained:
$$ \begin{aligned} \dot{V}_{1i} (x(t)) & = 2x^{T} (t)P_{i} [C_{i} (t)\dot{x}(t - h(t)) - A_{i} (t)x(t) + B_{i} (t)f(x(t)) + D_{i} (t)g(x(t - \tau (t)))], \\ \dot{V}_{2i} (x(t)) & = x^{T} (t)Q_{i} x(t) - e^{{ - \alpha \tau_{1} }} x^{T} (t - \tau_{1} )Q_{i} x(t - \tau_{1} ) - \alpha V_{2i} (x(t)), \\ \dot{V}_{3i} (x(t)) & = x^{T} (t)R_{i} x(t) - e^{{ - \alpha h_{1} }} x^{T} (t - h_{1} )R_{i} x(t - h_{1} ) - \alpha V_{3i} (x(t)), \\ \dot{V}_{4i} (x(t)) & = x^{T} (t)S_{i} x(t) - (1 - \dot{\tau }(t))e^{ - \alpha \tau (t)} x^{T} (t - \tau (t))S_{i} x(t - \tau (t)) - \alpha V_{4i} (x(t)) \\ & \le x^{T} (t)S_{i} x(t) - (1 - \tau )e^{{ - \alpha \tau_{1} }} x^{T} (t - \tau (t))S_{i} x(t - \tau (t)) - \alpha V_{4i} (x(t)), \\ \dot{V}_{5i} (x(t)) & = x^{T} (t)U_{i} x(t) - (1 - \dot{h}(t))e^{ - \alpha h(t)} x^{T} (t - h(t))U_{i} x(t - h(t)) - \alpha V_{5i} (x(t)) \\ & \le x^{T} (t)U_{i} x(t) - (1 - h)e^{{ - \alpha h_{1} }} x^{T} (t - h(t))U_{i} x(t - h(t)) - \alpha V_{5i} (x(t)), \\ \dot{V}_{6i} (x(t)) & = - \alpha V_{6i} (x(t)) + \tau_{1} \dot{x}^{T} (t)\varLambda_{i} \dot{x}(t) - \int\limits_{{t - \tau_{1} }}^{t} {e^{\alpha (s - t)} } \dot{x}^{T} (s)\varLambda_{i} \dot{x}(s)ds \\ & \le - \alpha V_{6i} (x(t)) + \tau_{1} \dot{x}^{T} (t)\varLambda_{i} \dot{x}(t) - e^{{ - \alpha \tau_{1} }} \int\limits_{{t - \tau_{1} }}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds, \\ \end{aligned} $$
$$ \begin{aligned} \dot{V}_{7i} (x(t)) & = - \alpha V_{7i} (x(t)) + g^{T} (x(t))W_{i} g(x(t))\; - (1 - \dot{\tau }(t))e^{ - \alpha \tau (t)}\\ & \times g^{T} (x(t - \tau (t)))W_{i} g(x(t - \tau (t))) \\ & \le - \alpha V_{7i} (x(t)) + g^{T} (x(t))W_{i} g(x(t)) \\ & \quad - (1 - \tau )e^{{ - \alpha \tau_{1} }} g^{T} (x(t - \tau (t)))W_{i} g(x(t - \tau (t))). \\ \end{aligned} $$
(11)
Let
$$ \begin{aligned} \xi (t) & = \left[ {x^{T} (t),\quad x^{T} (t - h(t)),\quad x^{T} (t - h_{1} ),\quad x^{T} (t - \tau (t)),\quad x^{T} (t - \tau_{1} ),\quad f^{T} (x(t)),} \right. \\ & \left. {\quad \quad g^{T} (x(t)), \quad g^{T} (x(t - \tau (t))),\;\;\dot{x}^{T} (t)\;,\;\dot{x}^{T} (t - h(t))} \right]^{T} . \\ \end{aligned} $$
Using the Newton Leibniz formula, it follows
$$ \begin{aligned} 2\xi^{T} (t)G_{i} \left[ {x(t) - x(t - \tau (t)) - \int\limits_{t - \tau (t)}^{t} {\dot{x}} (s)ds} \right] = 0, \hfill \\ 2\xi^{T} (t)X_{i} \left[ {x(t - \tau (t)) - x(t - \tau_{1} ) - \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}(s)} ds} \right] = 0, \hfill \\ \end{aligned} $$
(12)
where
$$ \begin{aligned} G_{i} & = \left[ {\begin{array}{*{20}l} {G_{i1}^{T} } \hfill & 0 \hfill & 0 \hfill & {G_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} , \\ X_{i} & = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {X_{i1}^{T} } \hfill & {X_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} . \\ \end{aligned} $$
According to Lemma 1, one obtains
$$ \begin{aligned} -\,2\xi^{T} (t)G_{i} \int\limits_{t - \tau (t)}^{t} {\dot{x}} (s)ds & = \int\limits_{t - \tau (t)}^{t} { (-\,2)\xi^{T} (t)G_{i} \dot{x}(s)} ds \\ & \le \tau (t)\xi^{T} (t)G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} \xi (t) + \int\limits_{t - \tau (t)}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds \\ & \le \tau_{1} \xi^{T} (t)G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} \xi (t) + \int\limits_{t - \tau (t)}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds, \\ -\,2\xi^{T} (t)X_{i} \int_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}} (s)ds & = \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} { (-\,2)\xi^{T} (t)X_{i} \dot{x}(s)} ds \\ & \le (\tau_{1} - \tau (t))\xi^{T} (t)X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} \xi (t) + \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds \\ & \le \tau_{1} \xi^{T} (t)X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} \xi (t) + \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds. \\ \end{aligned} $$
So, it follows that
$$ \begin{aligned} \int\limits_{t - \tau (t)}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds & \le \tau_{1} \xi^{T} (t)G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} \xi (t) + 2\xi^{T} (t)G_{i} \int\limits_{t - \tau (t)}^{t} {\dot{x}} (s)ds, \\ \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds & \le \tau_{1} \xi^{T} (t)X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} \xi (t) + 2\xi^{T} (t)X_{i} \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}} (s)ds. \\ \end{aligned} $$
(13)
From (13), we have
$$ \begin{aligned} - e^{{ - \alpha \tau_{1} }} \int\limits_{{t - \tau_{1} }}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds & = - e^{{ - \alpha \tau_{1} }} \left[ {\int\limits_{t - \tau (t)}^{t} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds + \int\limits_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}^{T} } (s)\varLambda_{i} \dot{x}(s)ds} \right] \\ & \le e^{{ - \alpha \tau_{1} }} \left[ {2\xi^{T} (t)G_{i} \int_{t - \tau (t)}^{t} {\dot{x}} (s)ds + \tau_{1} \xi^{T} (t)G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} \xi (t)} \right. \\ & \quad \left. { +\,2\xi^{T} (t)X_{i} \int_{{t - \tau_{1} }}^{t - \tau (t)} {\dot{x}} (s)ds + \tau_{1} \xi^{T} (t)X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} \xi (t)} \right] \\ & = e^{{ - \alpha \tau_{1} }} \left\{ {2\xi^{T} (t)G_{i} [x(t) - x(t - \tau (t))] + 2\xi^{T} (t)X_{i} } \right. \\ & \times \left. { [x(t - \tau (t)) - x(t - \tau_{1} )]} \right. \\ & \quad \left. { +\,\tau_{1} \xi^{T} (t)G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} \xi (t) + \tau_{1} \xi^{T} (t)X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} \xi (t)} \right\}. \\ \end{aligned} $$
(14)
From Assumption 1, we get
$$ \begin{aligned} & (f_{i} (x_{i} (t)) - l_{i}^{ - } x_{i} (t))(l_{i}^{ + } x_{i} (t) - f_{i} (x_{i} (t))) \ge 0, \\ & (g_{i} (x_{i} (t)) - \gamma_{i}^{ - } x_{i} (t))(\gamma_{i}^{ + } x_{i} (t) - g_{i} (x_{i} (t))) \ge 0, \\ & (g_{i} (x_{i} (t - \tau (t))) - \gamma_{i}^{ - } x_{i} (t - \tau (t)))(\gamma_{i}^{ + } x_{i} (t - \tau (t))) - g_{i} (x_{i} (t - \tau (t)))) \ge 0. \\ \end{aligned} $$
So, it follows that
$$ \left[ {\begin{array}{*{20}c} {x(t)} \\ {f(x(t))} \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} { - L_{1} } & {L_{2} } \\ * & { - I} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x(t)} \\ {f(x(t))} \\ \end{array} } \right] \ge 0, $$
(15)
$$ \left[ {\begin{array}{*{20}c} {x(t)} \\ {g(x(t))} \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \varGamma_{1} } & {\varGamma_{2} } \\ * & { - I} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x(t)} \\ {g(x(t))} \\ \end{array} } \right] \ge 0, $$
(16)
$$ \left[ {\begin{array}{*{20}c} {x(t - \tau (t))} \\ {g(x(t - \tau (t)))} \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \varGamma_{1} } & {\varGamma_{2} } \\ * & { - I} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x(t - \tau (t))} \\ {g(x(t - \tau (t)))} \\ \end{array} } \right] \ge 0, $$
(17)
where \( L_{i} ,\varGamma_{i} ,\quad i = 1,2, \) are given by (4).
From (7), one obtains
$$ \begin{aligned} & 2\rho_{i} [\dot{x}^{T} (t) - \dot{x}^{T} (t - h(t))]P_{i} [ - \dot{x}(t) + C_{i} (t)\dot{x}(t - h(t)) - A_{i} (t)x(t) \\ & \quad \left. { +\,B_{i} (t)f(x(t)) + D_{i} (t)g(x(t - \tau (t)))} \right] = 0, \\ \end{aligned} $$
(18)
From (11), (14)–(18), we have
$$ \dot{V}_{i} (x(t)) + \alpha V_{i} (x(t)) \le \xi^{T} (t)\left[ {\varXi_{i} (t) + \tau_{1} e^{{ - \alpha \tau_{1} }} \left( {G_{i} \varLambda_{i}^{ - 1} G_{i}^{T} + X_{i} \varLambda_{i}^{ - 1} X_{i}^{T} } \right)} \right]\xi (t), $$
(19)
where
$$ \begin{array}{*{20}c} {\varXi_{i} (t) = \left[ {\begin{array}{*{20}c} {\varXi_{11}^{i} (t)} & 0 & 0 & {\varXi_{14}^{i} } & 0 & {\varXi_{16}^{i} (t)} & {\varGamma_{2} } & {\varXi_{18}^{i} (t)} & {\varXi_{19}^{i} (t)} & {\varXi_{1,10}^{i} (t)} \\ * & {\varXi_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & {\varXi_{33}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\varXi_{44}^{i} } & {\varXi_{45}^{i} } & 0 & 0 & {\varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\varXi_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\varXi_{69}^{i} (t)} & {\varXi_{6,10}^{i} (t)} \\ * & * & * & * & * & * & {\varXi_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\varXi_{88}^{i} } & {\varXi_{89}^{i} (t)} & {\varXi_{8,10}^{i} (t)} \\ * & * & * & * & * & * & * & * & {\varXi_{99}^{i} } & {\varXi_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\varXi_{10,10}^{i} (t)} \\ \end{array} } \right]} \\ \end{array} $$
$$ \begin{aligned} \varXi_{11}^{i} (t) & = - P_{i} A_{i} (t) - A_{i}^{T} (t)P_{i} + \alpha P_{i} + Q_{i} + R_{i} + S_{i} + U_{i} + e^{{ - \alpha \tau_{1} }} (G_{i1}^{T} + G_{i1} ) - L_{1} - \varGamma_{1} , \\ \varXi_{14}^{i} & = e^{{ - \alpha \tau_{1} }} (G_{i2}^{T} - G_{i1} ),\;\;\varXi_{16}^{i} (t) = P_{i} B_{i} (t) + L_{2} ,\;\;\varXi_{18}^{i} (t) = P_{i} D_{i} (t), \\ \varXi_{19}^{i} (t) & = - \rho_{i} A_{i}^{T} (t)P_{i} ,\;\;\varXi_{1,10}^{i} (t) = \rho_{i} A_{i}^{T} (t)P_{i} + P_{i} C_{i} (t),\;\;\varXi_{22}^{i} = - (1 - h)e^{{ - \alpha h_{1} }} U_{i} , \\ \varXi_{33}^{i} & = - e^{{ - \alpha h_{1} }} R_{i} ,\;\;\varXi_{44}^{i} = - (1 - \tau )e^{{ - \alpha \tau_{1} }} S_{i} + e^{{ - \alpha \tau_{1} }} ( - G_{i2}^{T} - G_{i2} + X_{i1}^{T} + X_{i1} ) - \varGamma_{1} , \\ \varXi_{45}^{i} & = e^{{ - \alpha \tau_{1} }} (X_{i2}^{T} - X_{i1} ),\;\;\varXi_{55}^{i} = - e^{{ - \alpha \tau_{1} }} Q_{i} - e^{{ - \alpha \tau_{1} }} (X_{i2}^{T} + X_{i2} ), \\ \varXi_{69}^{i} (t) & = \rho_{i} B_{i}^{T} (t)P_{i} ,\;\;\varXi_{6,10}^{i} (t) = - \rho_{i} B_{i}^{T} (t)P_{i} ,\;\;\varXi_{77}^{i} = W_{i} - I, \\ \end{aligned} $$
$$ \begin{aligned} \varXi_{88}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} W_{i} - I,\quad \varXi_{89}^{i} (t) = \rho_{i} D_{i}^{T} (t)P_{i} ,\quad \varXi_{8,10}^{i} (t) = - \rho_{i} D_{i}^{T} (t)P_{i} , \\ \varXi_{99}^{i} & = \tau_{1} \varLambda_{i} -\,2\rho_{i} P_{i} ,\quad \varXi_{9,10}^{i} = \rho_{i} P_{i} ,\quad \varXi_{10,10}^{i} (t) = -\,2\rho_{i} P_{i} C_{i} (t). \\ \end{aligned} $$
From (19), using Schur complement and (10), we can get
$$ \dot{V}_{i} (x(t)) \le - \alpha V_{i} (x(t)). $$
This completes the proof.□
Theorem 1
Under Assumption 1, for given constants\( \alpha > 0,\,\mu \ge 1 \), \( \rho_{i} > 0,\varepsilon_{i} > 0,\;i \in {\mathbb{N}} \), the system (7) is exponentially stable for any switching signal with the average dwell time satisfying\( T_{a} > {{\left( {ln\mu } \right)} \mathord{\left/ {\vphantom {{\left( {ln\mu } \right)} \alpha }} \right. \kern-0pt} \alpha } \), if there exist symmetric and positive definite matrices\( Y_{i} \), \( \bar{Q}_{i} \), \( \bar{R}_{i} \), \( \bar{S}_{i} \), \( \bar{U}_{i} \), \( \bar{\varLambda }_{i} \), \( W_{i} \)and any matrices\( \bar{G}_{i1} \), \( \bar{G}_{i2} \), \( \bar{X}_{i1} \)and\( \bar{X}_{i2} \)such that the following LMIs hold for all\( i,j \in {\mathbb{N}}\):
$$ \begin{array}{*{20}c} {\bar{\varSigma }_{i} = \left[ {\begin{array}{*{20}c} {\bar{\varXi }_{i} } & {\tau_{1} \bar{G}_{i} } & {\tau_{1} \bar{X}_{i} } & {(\bar{\varTheta }_{2}^{i} )^{T} } \\ * & { - \tau_{1} e^{{\alpha \tau_{1} }} \bar{\varLambda }_{i} } & 0 & 0 \\ * & * & { - \tau_{1} e^{{\alpha \tau_{1} }} \bar{\varLambda }_{i} } & 0 \\ * & * & * & { - \varepsilon_{i} I} \\ \end{array} } \right] < 0} \\ \end{array} , $$
(20)
$$ Y_{i} \le \mu Y_{j} ,\quad \bar{Q}_{i} \le \mu \bar{Q}_{j} ,\quad \bar{R}_{i} \le \mu \bar{R}_{j} ,\quad \bar{S}_{i} \le \mu \bar{S}_{j} ,\quad \bar{U}_{i} \le \mu \bar{U}_{j} ,\quad \bar{\varLambda }_{i} \le \mu \bar{\varLambda }_{j} ,\quad W_{i} \le \mu W_{j} , $$
(21)
where
$$ \begin{array}{*{20}c} {\bar{\varXi }_{i} = \left[ {\begin{array}{*{20}c} {\bar{\varXi }_{11}^{i} } & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{14}^{i} } & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{16}^{i} } & {Y_{i} \varGamma_{2} } & {D_{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{19}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{1,10}^{i} } \\ * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & { - e^{{ - \alpha h_{1} }} \bar{R}_{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\bar{\varXi }_{44}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{45}^{i} } & 0 & 0 & {Y_{i} \varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{69}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{6,10}^{i} } \\ * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{88}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{89}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{8,10}^{i} } \\ * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{99}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{10,10}^{i} } \\ \end{array} } \right]} \\ \end{array} $$
$$ \begin{aligned} \bar{\varXi }_{11}^{i} & = - A_{i} Y_{i} - Y_{i} A_{i}^{T} + \alpha Y_{i} + \bar{Q}_{i} + \bar{R}_{i} + \bar{S}_{i} + \bar{U}_{i} + e^{{ - \alpha \tau_{1} }} (\bar{G}_{i1}^{T} + \bar{G}_{i1} ) - L_{1} Y_{i} - Y_{i} L_{1} \\ & \quad - \varGamma_{1} Y_{i} - Y_{i} \varGamma_{1} + 2I + \varepsilon_{i} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{14}^{i} & = e^{{ - \alpha \tau_{1} }} (\bar{G}_{i2}^{T} - \bar{G}_{i1} ),\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{16}^{i} = B_{i} + Y_{i} L_{2} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{19}^{i} = - \rho_{i} Y_{i} A_{i}^{T} + \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{1,10}^{i} & = \rho_{i} Y_{i} A_{i}^{T} + C_{i} Y_{i} - \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{22}^{i} = - (1 - h)e^{{ - \alpha h_{1} }} \bar{U}_{i} , \\ \bar{\varXi }_{44}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} \bar{S}_{i} + e^{{ - \alpha \tau_{1} }} ( - \bar{G}_{i2}^{T} - \bar{G}_{i2} + \bar{X}_{i1}^{T} + \bar{X}_{i1} ) - \varGamma_{1} Y_{i} - Y_{i} \varGamma_{1} + I, \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{45}^{i} & = e^{{ - \alpha \tau_{1} }} (\bar{X}_{i2}^{T} - \bar{X}_{i1} ),\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{55}^{i} = - e^{{ - \alpha \tau_{1} }} \bar{Q}_{i} - e^{{ - \alpha \tau_{1} }} (\bar{X}_{i2}^{T} + \bar{X}_{i2} ), \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{69}^{i} & = \rho_{i} B_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{6,10}^{i} = - \rho_{i} B_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{77}^{i} = W_{i} - I, \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{88}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} W_{i} - I,\;\quad \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{89}^{i} = \rho_{i} D_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{8,10}^{i} = - \rho_{i} D_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{99}^{i} & = \tau_{1} \bar{\varLambda }_{i} -\,2\rho_{i} Y_{i} + \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{9,10}^{i} = \rho_{i} Y_{i} - \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{10,10}^{i} & = -\,2\rho_{i} C_{i} Y_{i} + \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} , \\ \end{aligned} $$
$$ \begin{aligned} \bar{G}_{i} & = \left[ {\begin{array}{*{20}l} {\bar{G}_{i1}^{T} } \hfill & 0 \hfill & 0 \hfill & {\bar{G}_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} , \\ \bar{X}_{i} & = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {\bar{X}_{i1}^{T} } \hfill & {\bar{X}_{i2}^{T} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]^{T} , \\ \bar{\varTheta }_{2}^{i} & = \left[ {\begin{array}{*{20}l} { - M_{1}^{i} Y_{i} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {M_{2}^{i} } \hfill & 0 \hfill & {M_{4}^{i} } \hfill & 0 \hfill & {M_{3}^{i} Y_{i} } \hfill \\ \end{array} } \right]. \\ \end{aligned} $$
Proof
Note that \( \varSigma_{i} (t) < 0 \) is not standard LMIs due to the existence of parameter uncertainties, which will be further dealt with via the following approach. \( \varSigma_{i} (t) \) can be written as
$$ \varSigma_{i} (t) = \hat{\varSigma }_{i} + \Delta \hat{\varSigma }_{i} (t), $$
(22)
where
$$ \hat{\varSigma }_{i} = \left[ {\begin{array}{*{20}c} {\hat{\varXi }_{i} } & {\tau_{1} G_{i} } & {\tau_{1} X_{i} } \\ * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } & 0 \\ * & * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } \\ \end{array} } \right],\quad \Delta \hat{\varSigma }_{i} (t) = \left[ {\begin{array}{*{20}c} {\Delta \hat{\varXi }_{i} (t)} & 0 & 0 \\ * & 0 & 0 \\ * & * & 0 \\ \end{array} } \right], $$
$$ \begin{array}{*{20}c} {\hat{\varXi }_{i} = \left[ {\begin{array}{*{20}c} {\hat{\varXi }_{11}^{i} } & 0 & 0 & {\varXi_{14}^{i} } & 0 & {\hat{\varXi }_{16}^{i} } & {\varGamma_{2} } & {P_{i} D_{i} } & {\hat{\varXi }_{19}^{i} } & {\hat{\varXi }_{1,10}^{i} } \\ * & {\varXi_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & { - e^{{ - \alpha h_{1} }} R_{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\varXi_{44}^{i} } & {\varXi_{45}^{i} } & 0 & 0 & {\varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\varXi_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\hat{\varXi }_{69}^{i} } & {\hat{\varXi }_{6,10}^{i} } \\ * & * & * & * & * & * & {\varXi_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\varXi_{88}^{i} } & {\hat{\varXi }_{89}^{i} } & {\hat{\varXi }_{8,10}^{i} } \\ * & * & * & * & * & * & * & * & {\varXi_{99}^{i} } & {\varXi_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\hat{\varXi }_{10,10}^{i} } \\ \end{array} } \right]} \\ \end{array} , $$
$$ \begin{array}{*{20}c} {\Delta \hat{\varXi }_{i} (t) = \left[ {\begin{array}{*{20}c} {\Delta \hat{\varXi }_{11}^{i} (t)} & 0 & 0 & 0 & 0 & {\Delta \hat{\varXi }_{16}^{i} (t)} & 0 & {\Delta \hat{\varXi }_{18}^{i} (t)} & {\Delta \hat{\varXi }_{19}^{i} (t)} & {\Delta \hat{\varXi }_{1,10}^{i} (t)} \\ * & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & 0 & 0 & 0 & {\Delta \hat{\varXi }_{69}^{i} (t)} & {\Delta \hat{\varXi }_{6,10}^{i} (t)} \\ * & * & * & * & * & * & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & 0 & {\Delta \hat{\varXi }_{89}^{i} (t)} & {\Delta \hat{\varXi }_{8,10}^{i} (t)} \\ * & * & * & * & * & * & * & * & 0 & 0 \\ * & * & * & * & * & * & * & * & * & {\Delta \hat{\varXi }_{10,10}^{i} (t)} \\ \end{array} } \right]} \\ \end{array} , $$
$$ \begin{array}{*{20}c} \begin{aligned} \hat{\varXi }_{11}^{i} & = - P_{i} A_{i} - A_{i}^{T} P_{i} + \alpha P_{i} + Q_{i} + R_{i} + S_{i} + U_{i} + e^{{ - \alpha \tau_{1} }} (G_{i1}^{T} + G_{i1} ) - L_{1} - \varGamma_{1} , \\ \hat{\varXi }_{16}^{i} & = P_{i} B_{i} + L_{2} ,\;\quad \hat{\varXi }_{19}^{i} = - \rho_{i} A_{i}^{T} P_{i} ,\quad \hat{\varXi }_{1,10}^{i} = \rho_{i} A_{i}^{T} P_{i} + P_{i} C_{i} , \\ \;\hat{\varXi }_{69}^{i} & = \rho_{i} B_{i}^{T} P_{i} ,\quad \hat{\varXi }_{6,10}^{i} = - \rho_{i} B_{i}^{T} P_{i} ,\quad \hat{\varXi }_{89}^{i} = \rho_{i} D_{i}^{T} P_{i} , \\ \hat{\varXi }_{8,10}^{i} & = - \rho_{i} D_{i}^{T} P_{i} ,\;\;\hat{\varXi }_{10,10}^{i} = -\,2\rho_{i} P_{i} C_{i} , \\ \Delta \hat{\varXi }_{11}^{i} (t) & = - P_{i} \Delta A_{i} (t) - \Delta A_{i}^{T} (t)P_{i} ,\quad \Delta \hat{\varXi }_{16}^{i} (t) = P_{i} \Delta B_{i} (t),\quad \\ \Delta \hat{\varXi }_{18}^{i} (t) & = P_{i} \Delta D_{i} (t),\quad \Delta \hat{\varXi }_{19}^{i} (t) = - \rho_{i} \Delta A_{i}^{T} (t)P_{i} , \\ \Delta \hat{\varXi }_{1,10}^{i} (t) & = \rho_{i} \Delta A_{i}^{T} (t)P_{i} + P_{i} \Delta C_{i} (t),\quad \Delta \hat{\varXi }_{69}^{i} (t) = \rho_{i} \Delta B_{i}^{T} (t)P_{i} , \\ \Delta \hat{\varXi }_{6,10}^{i} (t) & = - \rho_{i} \Delta B_{i}^{T} (t)P_{i} ,\quad \Delta \hat{\varXi }_{89}^{i} (t) = \rho_{i} \Delta D_{i}^{T} (t)P_{i} , \\ \Delta \hat{\varXi }_{8,10}^{i} (t) & = - \rho_{i} \Delta D_{i}^{T} (t)P_{i} ,\quad \Delta \hat{\varXi }_{10,10}^{i} (t) = -\,2\rho_{i} P_{i} \Delta C_{i} (t). \\ \end{aligned} \\ \end{array} $$
The other parameters are the same as (10). According to Assumption 1, \( \varSigma_{i} (t) \) could be rewritten as
$$ \varSigma_{i} (t) = \hat{\varSigma }_{i} + \left[ {\begin{array}{*{20}c} {\varTheta_{1}^{i} } \\ 0 \\ 0 \\ \end{array} } \right]F(t)\left[ {\begin{array}{*{20}c} {\varTheta_{2}^{i} } & 0 & 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {(\varTheta_{2}^{i} )^{T} } \\ 0 \\ 0 \\ \end{array} } \right]F^{T} (t)\left[ {\begin{array}{*{20}c} {(\varTheta_{1}^{i} )^{T} } & 0 & 0 \\ \end{array} } \right], $$
(23)
where
$$ \begin{aligned} \varTheta_{1}^{i} = \left[ {\begin{array}{*{20}c} {H_{i}^{T} P_{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho H_{i}^{T} P_{i} } & { - \rho H_{i}^{T} P_{i} } \\ \end{array} } \right]^{T} , \hfill \\ \varTheta_{2}^{i} = \left[ {\begin{array}{*{20}c} { - M_{1}^{i} } & 0 & 0 & 0 & 0 & {M_{2}^{i} } & 0 & {M_{4}^{i} } & 0 & {M_{3}^{i} } \\ \end{array} } \right]. \hfill \\ \end{aligned} $$
By Lemma 2 and \( F_{i}^{T} (t)F_{i} (t) \le I \), \( \varSigma_{i} (t) < 0 \) holds if and only if there exists a positive scalar \( \varepsilon_{i} \) such that,
$$ \hat{\varSigma }_{i} + \varepsilon_{i} \left[ {\begin{array}{*{20}c} {\varTheta_{1}^{i} } \\ 0 \\ 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {(\varTheta_{1}^{i} )^{T} } & 0 & 0 \\ \end{array} } \right] + \varepsilon_{i}^{ - 1} \left[ {\begin{array}{*{20}c} {(\varTheta_{2}^{i} )^{T} } \\ 0 \\ 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varTheta_{2}^{i} } & 0 & 0 \\ \end{array} } \right] < 0. $$
(24)
Using Schur complement, (24) is equivalent to the following inequality
$$ \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\hat{\varSigma }_{i} + \varepsilon_{i} \left[ {\begin{array}{*{20}c} {\varTheta_{1}^{i} } \\ 0 \\ 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {(\varTheta_{1}^{i} )^{T} } & 0 & 0 \\ \end{array} } \right]} & {\left[ {\begin{array}{*{20}c} {(\varTheta_{2}^{i} )^{T} } \\ 0 \\ 0 \\ \end{array} } \right]} \\ * & { - \varepsilon_{i} I} \\ \end{array} } \right] < 0} \\ \end{array} . $$
(25)
Equation (25) could be rewritten as:
$$ \varOmega = \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\hat{\varXi }_{i} + \varepsilon_{i} \varTheta_{1}^{i} (\varTheta_{1}^{i} )^{T} } & {\tau_{1} G_{i} } & {\tau_{1} X_{i} } & {(\varTheta_{2}^{i} )^{T} } \\ * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } & 0 & 0 \\ * & * & { - \tau_{1} e^{{\alpha \tau_{1} }} \varLambda_{i} } & 0 \\ * & * & * & { - \varepsilon_{i} I} \\ \end{array} } \right] < 0} \\ \end{array} . $$
(26)
Set
$$ \begin{aligned}\varPi &= diag\left\{ {P_{i}^{ - 1} ,P_{i}^{ - 1} ,P_{i}^{ - 1} ,P_{i}^{ - 1} ,P_{i}^{ - 1} ,I,I,I,P_{i}^{ - 1} ,P_{i}^{ - 1} } \right\},\quad Y_{i} = P_{i}^{ - 1} ,\bar{X}_{i2} = P_{i}^{ - 1} X_{i2} P_{i}^{ - 1} ,\\ \bar{Q}_{i} &= P_{i}^{ - 1} Q_{i} P_{i}^{ - 1}, \quad \bar{R}_{i} = P_{i}^{ - 1} R_{i} P_{i}^{ - 1} ,\\ \bar{S}_{i} &= P_{i}^{ - 1} R_{i} P_{i}^{ - 1} , \quad \bar{U}_{i} = P_{i}^{ - 1} U_{i} P_{i}^{ - 1} ,\\ \bar{\varLambda }_{i} &= P_{i}^{ - 1} \varLambda_{i} P_{i}^{ - 1} , \quad \bar{G}_{i1} = P_{i}^{ - 1} G_{i1} P_{i}^{ - 1} ,\\ \bar{G}_{i2}& = P_{i}^{ - 1} G_{i2} P_{i}^{ - 1} , \quad \bar{X}_{i1} = P_{i}^{ - 1} X_{i1} P. \end{aligned}$$
Using \( \varPhi = diag\{ \varPi ,P_{i}^{ - 1} ,P_{i}^{ - 1} ,I\} \) pre- and post- multiply the left term of (26), the following matrix inequalities are obtained:
$$ \varPhi \varOmega \varPhi = \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{i} } & {\tau_{1} \varPi G_{i} P_{i}^{ - 1} } & {\tau_{1} \varPi X_{i} P_{i}^{ - 1} } & {\varPi (\varTheta_{2}^{i} )^{T} } \\ * & { - \tau_{1} P_{i}^{ - 1} e^{{\alpha \tau_{1} }} \varLambda_{i} P_{i}^{ - 1} } & 0 & 0 \\ * & * & { - \tau_{1} P_{i}^{ - 1} e^{{\alpha \tau_{1} }} \varLambda_{i} P_{i}^{ - 1} } & 0 \\ * & * & * & { - \varepsilon_{i} I} \\ \end{array} } \right] < 0} \\ \end{array} $$
where
$$ \begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{i} = \left[ {\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{11}^{i} } & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{14}^{i} } & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{16}^{i} } & {Y_{i} \varGamma_{2} } & {D_{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{19}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{1,10}^{i} } \\ * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & { - e^{{ - \alpha h_{1} }} P_{i}^{ - 1} R_{i} P_{i}^{ - 1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{44}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{45}^{i} } & 0 & 0 & {Y_{i} \varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{69}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{6,10}^{i} } \\ * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{88}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{89}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{8,10}^{i} } \\ * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{99}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{10,10}^{i} } \\ \end{array} } \right]} \\ \end{array} , $$
$$ \begin{aligned} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{11}^{i} & = - A_{i} Y_{i} - Y_{i} A_{i}^{T} + \alpha Y_{i} + \bar{Q}_{i} + \bar{R}_{i} + \bar{S}_{i} + \bar{U}_{i} + e^{{ - \alpha \tau_{1} }} (\bar{G}_{i1}^{T} + \bar{G}_{i1} ) \\ & \quad - Y_{i} L_{1} Y_{i} - Y_{i} \varGamma_{1} Y_{i} + \varepsilon_{i} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{14}^{i} & = e^{{ - \alpha \tau_{1} }} (\bar{G}_{i2}^{T} - \bar{G}_{i1} )_{i} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{16}^{i} = B_{i} + Y_{i} L_{2} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{19}^{i} = - \rho_{i} Y_{i} A_{i}^{T} + \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{1,10}^{i} & = \rho_{i} Y_{i} A_{i}^{T} + C_{i} Y_{i} - \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{22}^{i} = - (1 - h)e^{{ - \alpha h_{1} }} \bar{U}_{i} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{44}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} \bar{S}_{i} + e^{{ - \alpha \tau_{1} }} ( - \bar{G}_{i2}^{T} - \bar{G}_{i2} + \bar{X}_{i1}^{T} + \bar{X}_{i1} ) - Y_{i} \varGamma_{1} Y_{i} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{45}^{i} & = e^{{ - \alpha \tau_{1} }} (\bar{X}_{i2}^{T} - \bar{X}_{i1} ),\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{55}^{i} = - e^{{ - \alpha \tau_{1} }} \bar{Q}_{i} - e^{{ - \alpha \tau_{1} }} (\bar{X}_{i2}^{T} + \bar{X}_{i2} ), \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{69}^{i} & = \rho_{i} B_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{6,10}^{i} = - \rho_{i} B_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{77}^{i} = W_{i} - I, \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{88}^{i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} W_{i} - I,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{89}^{i} = \rho_{i} D_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{8,10}^{i} = - \rho_{i} D_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{99}^{i} & = \tau_{1} \bar{\varLambda }_{i} -\,2\rho_{i} Y_{i} + \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} ,\;\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{9,10}^{i} = \rho_{i} Y_{i} - \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{10,10}^{i} & = -\,2\rho_{i} C_{i} Y_{i} + \varepsilon_{i} \rho_{i}^{2} H_{i} H_{i}^{T} , \\ \bar{G}_{i} & = [\bar{G}_{i1}^{T} \quad 0\quad 0\quad \bar{G}_{i2}^{T} \quad 0\quad 0\quad 0\quad 0\quad 0\quad 0]^{T} , \\ \bar{X}_{i} & = [0\quad 0\quad 0\quad \bar{X}_{i1}^{T} \quad \bar{X}_{i2}^{T} \quad 0\quad 0\quad 0\quad 0\quad 0]^{T} , \\ \bar{\varTheta }_{2}^{i} & = \left[ {\begin{array}{*{20}c} { - M_{1}^{i} Y_{i} } & 0 & 0 & 0 & 0 & {M_{2i}^{i} } & 0 & {M_{4i}^{i} } & 0 & {M_{3}^{i} Y_{i} } \\ \end{array} } \right]. \\ \end{aligned} $$
From \( L_{1} > 0, \) we can get
$$(P_{i}^{ - 1} - I)L_{1} (P_{i}^{ - 1} - I) \ge 0.$$
So, we have
$$ - P_{i}^{ - 1} L_{1} P_{i}^{ - 1} \le - P_{i}^{ - 1} L_{1} - \, L_{1} P_{i}^{ - 1} + I, $$
that is
$$ - Y_{i} L_{1} Y_{i} \le - Y_{i} L_{1} - L_{1} Y_{i} + I. $$
From (20), we have \( \varOmega < 0, \) so it follows
$$\varSigma_{i} (t) < 0.$$
From Lemma 3, we have
$$ \dot{V}_{i} (x(t)) \le - \alpha V_{i} (x(t)). $$
(27)
From (27), it follows that
$$ V_{{\sigma (t_{k} )}} (x(t)) \le e^{{ - \alpha (t - t_{k} )}} V_{{\sigma (t_{k} )}} (x(t_{k} )),\quad t \in [t_{k} ,t_{k + 1} ). $$
We have
$$ \begin{aligned} V_{{\sigma (t_{k} )}} (x(t)) & \le e^{{ - \alpha (t - t_{k} )}} V_{{\sigma (t_{k} )}} (x(t_{k} )) \\ & \le e^{{ - \alpha (t - t_{k} )}} \mu V_{{\sigma (t_{k - 1} )}} (x(t_{k} )) \\ & \le \cdots \\ & \le \mu^{{N_{\sigma } (t_{0} ,t)}} e^{{ - \alpha (t - t_{0} )}} V_{{\sigma (t_{0} )}} (x(t_{0} )) \\ & \le \mu^{{\frac{{t - t_{0} }}{{T_{a} }}}} e^{{ - \alpha (t - t_{0} )}} V_{{\sigma (t_{0} )}} (t_{0} ) \\ & \le e^{{ - (\alpha - \frac{ln\mu }{{T_{a} }})(t - t_{0} )}} V_{{\sigma (t_{0} )}} (t_{0} ). \\ \end{aligned} $$
(28)
From (8), it is easy to know that there exist scalars a, b such that
$$ \begin{aligned} a\left\| {x(t)} \right\|^{2} \le V_{\sigma (t)} (x(t)), \hfill \\ V_{{\sigma (t_{0} )}} (t_{0} ) \le b\left\| \phi \right\|_{c}^{2} , \hfill \\ \end{aligned} $$
(29)
where
$$ \begin{aligned} a & = \mathop {\hbox{min} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{min} } (P_{i} )} \right\}, \\ b & = \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (P_{i} )} \right\} + \tau_{1} \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (Q_{i} )} \right\} + h_{1} \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (R_{i} )} \right\}\\ &\quad + \tau_{1} \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (S_{i} )} \right\} + h_{1} \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (U_{i} )} \right\} \\ & \quad + (\tau_{1}^{2} /2)\mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (\varLambda_{i} )} \right\} + \frac{4}{3}\tau_{1} \mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left( {(\gamma_{i}^{ + } + \gamma_{i}^{ - } )^{2} - \gamma_{i}^{ + } \gamma_{i}^{ - } } \right)\mathop {\hbox{max} }\limits_{{i \in {\mathbb{N}}}} \left\{ {\lambda_{\hbox{max} } (W_{i} )} \right\}. \\ \end{aligned} $$
From (27)–(29), we obtain
$$ \left\| {x(t)} \right\| \le \sqrt {\frac{b}{a}} e^{{ - \frac{1}{2}(\alpha - \frac{ln\mu }{{T_{a} }})(t - t_{0} )}} \left\| \phi \right\|_{c} . $$
(30)
This completes the proof.□
3.2 Exponentiall Stabilization of Uncertain Neutral Neural Networks
For switched neutral neural networks (1), we consider the following state feedback controller
$$ u(t) = K_{\sigma (t)} x(t). $$
(31)
Under the controller (31), the corresponding closed-loop system is given by
$$ \begin{aligned} & \dot{x}(t) - C_{\sigma (t)} (t)\dot{x}(t - h(t)) = - (A_{\sigma (t)} (t) - E_{\sigma (t)} (t)K_{\sigma (t)} )x(t) + B_{1} (t)f(x(t)) \\ & \quad \quad \quad \quad + B_{2} (t)f(x(t - \tau (t))), \\ & x(t) = \varphi (t),\quad \quad \forall t \in [ - \bar{\tau },0], \\ \end{aligned} $$
(32)
Theorem 2
Under Assumption 1, for given constants\( \alpha > 0,\mu \ge 1 \)\( \rho_{i} > 0,\varepsilon_{i} > 0,i \in {\mathbb{N}} \), the switched neutral neural networks (1) is exponentially stabilizable under feedback controller (31) for any switching signal with the average dwell time satisfying\( T_{a} > \left( {ln\mu } \right)/\alpha \), if there exist symmetric and positive definite matrices\( Y_{i} ,\;\bar{Q}_{i} ,\bar{R}_{i} ,\bar{S}_{i} ,\bar{U}_{i} ,\bar{\varLambda }_{i} ,W_{i} , \)and any matrices\( Z_{i} ,\bar{G}_{i1} , \)\( \bar{G}_{i2} ,\bar{X}_{i1} \)and\( \bar{X}_{i2} \)such that the following LMIs hold for all\( i,j \in {\mathbb{N}},i \ne j \):
$$ \tilde{\varSigma }_{i} = \left[ {\begin{array}{*{20}c} {\tilde{\varXi }_{i} } & {\tau_{1} \bar{G}_{i} } & {\tau_{1} \bar{X}_{i} } & {(\tilde{\varTheta }_{2}^{i} )^{T} } \\ * & { - \tau_{1} e^{{\alpha \tau_{1} }} \bar{\varLambda }_{i} } & 0 & 0 \\ * & * & { - \tau_{1} e^{{\alpha \tau_{1} }} \bar{\varLambda }_{i} } & 0 \\ * & * & * & { - \varepsilon I} \\ \end{array} } \right] < 0, $$
(33)
$$ Y_{i} \le \mu Y_{j} ,\bar{Q}_{i} \le \mu \bar{Q}_{j} ,\bar{R}_{i} \le \mu \bar{R}_{j} ,\bar{S}_{i} \le \mu \bar{S}_{j} ,\bar{U}_{i} \le \mu \bar{U}_{j} ,\bar{\varLambda }_{i} \le \mu \bar{\varLambda }_{j} ,W_{i} \le \mu W_{j} , $$
(34)
where
$$ \begin{array}{*{20}c} {\tilde{\varXi }_{i} = \left[ {\begin{array}{*{20}c} {\tilde{\varXi }_{11}^{i} } & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{14}^{i} } & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{16}^{i} } & {Y_{i} \varGamma_{2} } & {D_{i} } & {\tilde{\varXi }_{19}^{i} } & {\tilde{\varXi }_{1,10}^{i} } \\ * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{22}^{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & { - e^{{ - \alpha h_{1} }} \bar{R}_{i} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & {\bar{\varXi }_{44}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{45}^{i} } & 0 & 0 & {Y_{i} \varGamma_{2} } & 0 & 0 \\ * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{55}^{i} } & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & { - I} & 0 & 0 & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{69}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{6,10}^{i} } \\ * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{77}^{i} } & 0 & 0 & 0 \\ * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{88}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{89}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{8,10}^{i} } \\ * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{99}^{i} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{9,10}^{i} } \\ * & * & * & * & * & * & * & * & * & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{10,10}^{i} } \\ \end{array} } \right]} \\ \end{array} , $$
(35)
$$ \begin{aligned} \tilde{\varXi }_{11}^{i} & = - A_{i} Y_{i} - Y_{i} A_{i}^{T} + \alpha Y_{i} + \bar{Q}_{i} + \bar{R}_{i} + \bar{S}_{i} + \bar{U}_{i} + e^{{ - \alpha \tau_{1} }} (\bar{G}_{i1}^{T} + \bar{G}_{i1} ) - L_{1} Y_{i} - Y_{i} L_{1} \\ & \quad - \varGamma_{1} Y_{i} - Y_{i} \varGamma_{1} + 2I + \varepsilon_{i} H_{i} H_{i}^{T} + E_{i} Z_{i} + Z_{i}^{T} E_{i}^{T} , \\ \tilde{\varXi }_{19}^{i} & = - \rho_{i} Y_{i} A_{i}^{T} + \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} + \rho_{i} Z_{i}^{T} E_{i}^{T} , \\ \tilde{\varXi }_{1,10}^{i} & = \rho_{i} Y_{i} A_{i}^{T} + C_{i} Y_{i} - \varepsilon_{i} \rho_{i} H_{i} H_{i}^{T} - \rho_{i} Z_{i}^{T} E_{i}^{T} , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varXi }_{44}^{-i} & = - (1 - \tau )e^{{ - \alpha \tau_{1} }} \bar{S}_{i} + e^{{ - \alpha \tau_{1} }} ( - \bar{G}_{i2}^{T} - \bar{G}_{i2} + \bar{X}_{i1}^{T} + \bar{X}_{i1} ) - \varGamma_{1} Y_{i} - Y_{i} \varGamma_{1} + I, \\ \tilde{\varTheta }_{2}^{i} & = \left[ {\begin{array}{*{20}c} { - M_{1}^{i} Y_{i} + M_{5}^{i} Z_{i} } & 0 & 0 & 0 & 0 & {M_{2}^{i} } & 0 & {M_{4}^{i} } & 0 & {M_{3}^{i} Y_{i} } \\ \end{array} } \right]. \\ \end{aligned} $$
and the other elements in (33) and (35) are given by (20). Moreover, the controller gains are constructed by
$$ K_{i} = Z_{i} Y_{i}^{ - 1} ,\quad i \in {\mathbb{N}}. $$
Proof
Consider the system (32). Using Theorem 1, replace \( {\text{A}}_{i} \left( {\text{t}} \right) \) with \( A_{i} \left( t \right) - E_{i} \left( t \right)K_{i} \) and notice \( K_{i} = ZY_{i}^{ - 1} \), (33) can be get. This completes the proof.□