In this section, we first establish a delay-dependent sufficient condition under which system (5) is asymptotically stable with a prescribed \(\mathcal{H_{\infty}}\) performance *γ*.

### Theorem 1

*For given scalars*
\(0< h_{1}< h_{2}\), \(\mu, \gamma>0\), \(K_{1}=\operatorname{diag}\{k^{-}_{1}, k^{-}_{2}, \ldots, k^{-}_{n}\}\), *and*
\(K_{2}=\operatorname{diag}\{k^{+}_{1}, k^{+}_{2}, \ldots, k^{+}_{n}\}\), *the error system* (5) *is globally asymptotically stable with*
\(\mathcal{H_{\infty}}\)
*performance*
*γ*
*if there exist real matrices*
\(P>0\), \(Q>0\), \(Z_{1}>0\), \(Z_{2}>0\), \(Z_{3}>0\), \(Z_{4}>0\), \(R>0\), \(T_{1}=\operatorname{diag}\{t_{11}, t_{12}, \ldots, t_{1n}\}>0\), \(T_{2}=\operatorname{diag}\{t_{21}, t_{22}, \ldots, t_{2n}\}>0\), *and matrices*
\(S_{11}\), \(S_{12}\), \(S_{21}\), \(S_{22}\), *M*, *G*
*with appropriate dimensions such that the following LMIs are satisfied*:

$$\begin{aligned}& \Pi^{*}_{[h(t)=h_{1}]}< 0,\quad\quad \Pi^{*}_{[h(t)=h_{2}]}< 0, \end{aligned}$$

(10)

$$\begin{aligned}& \begin{pmatrix} Z_{2} & 0 & S_{11} & S_{12}\\ \ast& 3Z_{2} & S_{21} & S_{22}\\ \ast& \ast& Z_{2} & 0\\ \ast& \ast& \ast& 3Z_{2} \end{pmatrix} >0, \end{aligned}$$

(11)

*where*

$$\begin{aligned}& P= \begin{pmatrix} P_{11} & P_{12} & P_{13}\\ \ast& P_{22} & P_{23}\\ \ast& \ast& P_{33} \end{pmatrix} ,\quad\quad R= \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} , \\& \Pi_{[h(t)]}^{*}= \begin{pmatrix} \Pi& \hat{H} \\ \ast& -I \end{pmatrix} ,\quad \Pi=[\Pi_{ij}]_{11\times11}, \hat{H}=[H \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0]^{T}, \\& \Pi_{11}=P_{12}+P^{T}_{12}-4Z_{1}+Z_{3}+ \mu R_{11}+R_{12}+R^{T}_{12}+R_{22} \\& \hphantom{\Pi_{11}=}{}-MA -(MA)^{T}-(1-\mu) \bigl(R_{12}+R^{T}_{12} \bigr)-(1-\mu )R_{22}-GC-(GC)^{T} \\& \hphantom{\Pi_{11}=}{} -2W^{T}K_{1}T_{1}K_{2}W, \\& \Pi_{12}=(1-\mu)R_{12}+(1-\mu)R_{22}-GD, \quad\quad\Pi _{13}=-P_{12}+P_{13}-2Z_{1} -R_{12}-R_{22}, \\& \Pi_{14}=-P_{13},\quad\quad \Pi _{15}=W^{T}(K_{1}+K_{2})T_{1}, \quad\quad \Pi_{16}=M,\quad\quad \Pi_{17}=h_{1}P^{T}_{22}+6Z_{1},\\& \Pi _{18}=\bigl(h(t)-h_{1}\bigr)P_{23}, \quad\quad \Pi_{19}=\bigl(h_{2}-h(t)\bigr)P_{23},\quad\quad \Pi_{1,10}=MB_{1}-GB_{2}, \\& \Pi _{1,11}=P_{11}+\bigl(h(t)-h_{1} \bigr)R_{11}+\bigl(h(t)-h_{1}\bigr)R_{12}+ \bigl(h(t)-h_{1}\bigr)R^{T}_{12}+\bigl(h(t)-h_{1}\bigr)R_{22} \\& \hphantom{\Pi_{1,11}=}{} -M-(MA)^{T}-(GC)^{T}, \\& \Pi_{22}=-(1-\mu )R_{22}-8Z_{2}+S_{11}+S^{T}_{11}+S_{12}+S^{T}_{12}-S_{21}-S^{T}_{21}-S_{22}-S^{T}_{22} \\& \hphantom{\Pi_{22}=}{} -2W^{T}K_{1}T_{2}K_{2}W, \\& \Pi_{23}=-2Z_{2}-S^{T}_{11}-S^{T}_{12}-S^{T}_{21}-S^{T}_{22}, \\& \Pi_{24}=Z_{2}-3Z_{2}-S_{11}+S_{12}+S_{21}-S_{22},\quad\quad \Pi _{26}=W^{T}(K_{1}+K_{2})T_{2}, \\& \Pi_{28}=6Z_{2}+2S^{T}_{21}+2S^{T}_{22},\quad\quad \Pi _{29}=6Z_{2}-2S_{12}+2S_{22}, \\& \Pi_{2,11}=-(GD)^{T},\quad\quad \Pi_{33}=-Z_{3}+Z_{4}-4Z_{1}-4Z_{2}+R_{22}, \\& \Pi_{34}=S_{11}-S_{12}+S_{21}-S_{22},\quad\quad \Pi _{37}=-h_{1}P^{T}_{22}+h_{1}P^{T}_{23}+6Z_{1}, \\& \Pi_{38}=-\bigl(h(t)-h_{1}\bigr)P_{23}+ \bigl(h(t)-h_{1}\bigr)P^{T}_{33}+6Z_{2}, \\& \Pi_{39}=-\bigl(h_{2}-h(t)\bigr)P_{23}+ \bigl(h_{2}-h(t)\bigr)P^{T}_{33}+2S_{12}+2S_{22}, \\& \Pi_{44}=-Z_{4}-4Z_{2},\quad\quad \Pi_{47}=-h_{1}P^{T}_{23}, \\& \Pi_{48}=-\bigl(h(t)-h_{1}\bigr)P^{T}_{33}-2S^{T}_{21}+2S^{T}_{22},\quad\quad \Pi _{49}=-\bigl(h_{2}-h(t)\bigr)P^{T}_{33}+6Z_{2}, \\& \Pi_{55}=Q-2T_{1}, \quad\quad\Pi_{66}=-(1- \mu)Q-2T_{2}, \quad\quad\Pi _{6,11}=M^{T}, \\& \Pi_{77}=-12Z_{1},\quad\quad \Pi_{7,11}=h_{1}P^{T}_{12},\quad\quad \Pi_{88}=-12Z_{2},\quad\quad \Pi _{89}=-4S_{22}, \\& \Pi _{8,11}=\bigl(h(t)-h_{1}\bigr)P^{T}_{13}- \bigl(h(t)-h_{1}\bigr)R^{T}_{12}- \bigl(h(t)-h_{1}\bigr)R_{22}, \\& \Pi_{99}=-12Z_{2},\quad\quad \Pi_{9,11}= \bigl(h_{2}-h(t)\bigr)P^{T}_{13}, \quad\quad\Pi _{10,10}=-\gamma^{2}, \\& \Pi_{10,11}=B^{T}_{1}M^{T}-B^{T}_{2}G^{T},\quad\quad \Pi _{11,11}=h^{2}_{1}Z_{1}+(h_{2}-h_{1})^{2}Z_{2}-M-M^{T}. \end{aligned}$$

*Moreover*, *the gain matrix*
*K*
*of the state estimator of* (4) *can be designed as*
\(K=M^{-1}G\).

### Proof

Construct the following Lyapunov-Krasovskii functional:

$$ V(t)=V_{1}(t)+V_{2}(t)+V_{3}(t)+V_{4}(t)+V_{5}(t), $$

(12)

with

$$\begin{aligned}& V_{1}(t)= \begin{pmatrix} r(t) \\ \int_{t-h_{1}}^{t}r(s)\,ds \\ \int_{t-h_{2}}^{t-h_{1}}r(s)\,ds \end{pmatrix} ^{T} \begin{pmatrix} P_{11} & P_{12} & P_{13}\\ \ast& P_{22} & P_{23}\\ \ast& \ast& P_{33} \end{pmatrix} \begin{pmatrix} r(t) \\ \int_{t-h_{1}}^{t}r(s)\,ds \\ \int_{t-h_{2}}^{t-h_{1}}r(s)\,ds \end{pmatrix} , \\& V_{2}(t)= \int_{t-h(t)}^{t}g^{T}\bigl(Wr(s)\bigr)Qg \bigl(Wr(s)\bigr)\,ds, \\& V_{3}(t)=h_{1} \int_{-h_{1}}^{0} \int_{t+\theta}^{t}\dot {r}^{T}(s)Z_{1} \dot{r}(s)\,ds\,d\theta+h_{12} \int_{-h_{2}}^{-h_{1}} \int _{t+\theta}^{t}\dot{r}^{T}(s)Z_{2} \dot{r}(s)\,ds\,d\theta, \\& V_{4}(t)= \int_{t-h_{1}}^{t}r^{T}(s)Z_{3}r(s)\,ds+ \int _{t-h_{2}}^{t-h_{1}}r^{T}(s)Z_{4}r(s)\,ds, \\& V_{5}(t)= \int_{t-h(t)}^{t-h_{1}} \begin{pmatrix} r(t) \\ \int_{s}^{t}\dot{r}(u)\,du \end{pmatrix} ^{T} \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} \begin{pmatrix} r(t) \\ \int_{s}^{t}\dot{r}(u)\,du \end{pmatrix} \,ds, \end{aligned}$$

where \(h_{12}=h_{2}-h_{1}\).

The time derivative of \(V(t)\) along the trajectory of system (5) is given by

$$ \dot{V}(t)=\dot{V}_{1}(t)+\dot{V}_{2}(t)+ \dot{V}_{3}(t)+\dot {V}_{4}(t)+\dot{V}_{5}(t), $$

(13)

where

$$\begin{aligned} \dot{V}_{1}(t) =&2 \begin{pmatrix} r(t) \\ h_{1}u \\ ((h(t)-h_{1})v_{1} +(h_{2}-h(t))v_{2}) \end{pmatrix} ^{T} \begin{pmatrix} P_{11} & P_{12} & P_{13}\\ \ast& P_{22} & P_{23}\\ \ast& \ast& P_{33} \end{pmatrix} \\ &{}\times \begin{pmatrix} \dot{r}(t) \\ r(t)-r(t-h_{1}) \\ r(t-h_{1})-r(t-h_{2}) \end{pmatrix} \\ =&2r^{T}(t)P_{11} \dot {r}(t)+2r^{T}(t)P_{12}r(t)-2r^{T}(t)P_{12}r(t-h_{1}) \\ &{}+2r^{T}(t)P_{13}r(t-h_{1})-2r^{T}(t)P_{13}r(t-h_{2})+2h_{1}u^{T}P^{T}_{12} \dot {r}(t) \\ &{}+2h_{1}u^{T}P_{22}r(t)-2h_{1}u^{T}P_{22}r(t-h_{1})+2h_{1}u^{T}P_{23}r(t-h_{1}) \\ &{}-2h_{1}u^{T}P_{23}r(t-h_{2})+2 \bigl[h(t)-h_{1}\bigr]v^{T}_{1}P^{T}_{13} \dot {r}(t) \\ &{}+2\bigl[h_{2}-h(t)\bigr]v^{T}_{2}P^{T}_{13} \dot {r}(t)+2\bigl[h_{2}-h(t)\bigr]v^{T}_{2}P^{T}_{23}r(t) \\ &{}+2\bigl[h(t)-h_{1}\bigr]v^{T}_{1}P^{T}_{23}r(t)-2 \bigl[h(t)-h_{1}\bigr]v^{T}_{1}P^{T}_{23}r(t-h_{1}) \\ &{}-2\bigl[h_{2}-h(t)\bigr]v^{T}_{2}P^{T}_{23}r(t-h_{1})+2 \bigl[h(t)-h_{1}\bigr]v^{T}_{1}P_{33}r(t-h_{1}) \\ &{}-2\bigl[h(t)-h_{1}\bigr]v^{T}_{1}P_{33}r(t-h_{2})+2 \bigl[h_{2}-h(t)\bigr]v^{T}_{2}P_{33}r(t-h_{1}) \\ &{}-2\bigl[h_{2}-h(t)\bigr]v^{T}_{2}P_{33}r(t-h_{2}), \end{aligned}$$

(14)

where \(u=\frac{1}{h_{1}}\int_{t-h_{1}}^{t}r(s)\,ds\), \(v_{1}=\frac{1}{h(t)-h_{1}}\int_{t-h(t)}^{t-h_{1}}r(s)\,ds\), \(v_{2}=\frac {1}{h_{2}-h(t)}\int_{t-h_{2}}^{t-h(t)}r(s)\,ds\),

$$\begin{aligned}& \dot{V}_{2}(t) \leq g^{T}\bigl(Wr(t)\bigr)Qg\bigl(Wr(t) \bigr) \\& \hphantom{\dot{V}_{2}(t) \leq}{}-(1-\mu)g^{T}\bigl(Wr\bigl(t-h(t)\bigr)\bigr)Qg\bigl(Wr\bigl(t-h(t) \bigr)\bigr), \end{aligned}$$

(15)

$$\begin{aligned}& \dot{V}_{3}(t) = h^{2}_{1} \dot{r}^{T}(t)Z_{1}\dot {r}(t)+(h_{2}-h_{1})^{2} \dot{r}^{T}(t)Z_{2}\dot{r}(t) \\& \hphantom{\dot{V}_{3}(t) =}{} -h_{1} \int_{t-h_{1}}^{t}\dot{r}^{T}(s)Z_{1} \dot {r}(s)\,ds-(h_{2}-h_{1}) \int_{t-h_{2}}^{t-h_{1}}\dot{r}^{T}(s)Z_{2} \dot{r}(s)\,ds, \end{aligned}$$

(16)

based on Lemma 2, one can have

$$\begin{aligned}& -h_{1} \int_{t-h_{1}}^{t}\dot{r}^{T}(s)Z_{1} \dot{r}(s)\,ds \\& \quad\leq -\bigl[r(t)-r(t-h_{1})\bigr]^{T}Z_{1} \bigl[r(t)-r(t-h_{1})\bigr] \\& \quad\quad{}-\bigl[r(t)+r(t-h_{1})-2u\bigr]^{T}3Z_{1} \bigl[r(t)+r(t-h_{1})-2u\bigr], \end{aligned}$$

(17)

by employing Lemma 1 and Lemma 2, we can derive

$$\begin{aligned}& -(h_{2}-h_{1}) \int_{t-h_{2}}^{t-h_{1}}\dot{r}^{T}(s)Z_{2} \dot {r}(s)\,ds \\& \quad= -(h_{2}-h_{1}) \int_{t-h(t)}^{t-h_{1}}\dot{r}^{T}(s)Z_{2} \dot {r}(s)\,ds-(h_{2}-h_{1}) \int_{t-h_{2}}^{t-h(t)}\dot{r}^{T}(s)Z_{2} \dot{r}(s)\,ds \\& \quad\leq -\alpha _{1}\bigl[r(t-h_{1})-r\bigl(t-h(t)\bigr) \bigr]^{T}Z_{2}\bigl[r(t-h_{1})-r\bigl(t-h(t) \bigr)\bigr] \\& \quad\quad{}-3\alpha _{1}\bigl[r(t-h_{1})+r\bigl(t-h(t) \bigr)-2v_{1}\bigr]^{T}Z_{2}\bigl[r(t-h_{1})+r \bigl(t-h(t)\bigr)-2v_{1}\bigr] \\& \quad\quad{}-\alpha _{2}\bigl[r\bigl(t-h(t)\bigr)-r(t-h_{2}) \bigr]^{T}Z_{2}\bigl[r\bigl(t-h(t)\bigr)-r(t-h_{2}) \bigr] \\& \quad\quad{}-3\alpha _{2}\bigl[r\bigl(t-h(t)\bigr)+r(t-h_{2})-2v_{2} \bigr]^{T}Z_{2}\bigl[r\bigl(t-h(t)\bigr)+r(t-h_{2})-2v_{2} \bigr] \\& \quad\leq -\beta^{T}(t) \begin{pmatrix} Z_{2} & 0 & S_{11} & S_{12}\\ \ast& 3Z_{2} & S_{21} & S_{22}\\ \ast& \ast& Z_{2} & 0\\ \ast& \ast& \ast& 3Z_{2} \end{pmatrix} \beta(t), \end{aligned}$$

(18)

where

$$\begin{aligned}& \alpha_{1}=(h_{2}-h_{1})/ \bigl(h(t)-h_{1}\bigr),\\& \alpha _{2}=(h_{2}-h_{1})/ \bigl(h_{2}-h(t)\bigr), \\& \beta^{T}(t)=\bigl[r^{T}(t-h_{1})-r^{T} \bigl(t-h(t)\bigr), r^{T}(t-h_{1})+r^{T} \bigl(t-h(t)\bigr)-2v^{T}_{1}, \\& \hphantom{\beta^{T}(t)=} r^{T}\bigl(t-h(t)\bigr)-r^{T}(t-h_{2}), r^{T}\bigl(t-h(t)\bigr)+r^{T}(t-h_{2})-2v^{T}_{2} \bigr]. \end{aligned}$$

Calculating \(\dot{V}_{4}(t)\), \(\dot{V}_{5}(t)\) yields

$$\begin{aligned}& \dot{V}_{4}(t) = r^{T}(t)Z_{3}r(t)-r^{T}(t-h_{1})Z_{3}r(t-h_{1}) \\& \hphantom{\dot{V}_{4}(t) =}{}+r^{T}(t-h_{1})Z_{4}r(t-h_{1})-r^{T}(t-h_{2})Z_{4}r(t-h_{2}), \end{aligned}$$

(19)

$$\begin{aligned}& \dot{V}_{5}(t) \leq \begin{pmatrix} r(t)\\ \int_{t-h_{1}}^{t}\dot{r}(u)\,du \end{pmatrix} ^{T} \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} \begin{pmatrix} r(t)\\ \int_{t-h_{1}}^{t}\dot{r}(u)\,du \end{pmatrix} \\& \hphantom{\dot{V}_{5}(t) \leq}{} -(1-\mu) \begin{pmatrix} r(t)\\ \int_{t-h(t)}^{t}\dot{r}(u)\,du \end{pmatrix} ^{T} \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} \begin{pmatrix} r(t)\\ \int_{t-h(t)}^{t}\dot{r}(u)\,du \end{pmatrix} \\& \hphantom{\dot{V}_{5}(t) \leq}{}+2 \int_{t-h(t)}^{t-h_{1}} \begin{pmatrix} r(t) \\ \int_{s}^{t}\dot{r}(u)\,du \end{pmatrix} ^{T} \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} \begin{pmatrix} \dot{r}(t) \\ \dot{r}(t) \end{pmatrix} \,ds \\& \hphantom{\dot{V}_{5}(t) }{}= r^{T}(t)R_{11}r(t)+r^{T}(t)R_{12}r(t)-r^{T}(t)R_{12}r(t-h_{1})+r^{T}(t)R^{T}_{12}r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-r^{T}(t-h_{1})R^{T}_{12}r(t)+r^{T}(t)R_{22}r(t)-r^{T}(t)R_{22}r(t-h_{1}) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-r^{T}(t-h_{1})R_{22}r(t)+r^{T}(t-h_{1})R_{22}r(t-h_{1}) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-(1-\mu)r^{T}(t)R_{11}r(t)-(1-\mu)r^{T}(t)R_{12}r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+(1-\mu)r^{T}(t)R_{12}r\bigl(t-h(t)\bigr)-(1- \mu)r^{T}(t)R^{T}_{12}r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+(1-\mu)r^{T}\bigl(t-h(t)\bigr)R^{T}_{12}r(t)-(1- \mu)r^{T}(t)R_{22}r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+(1-\mu)r^{T}(t)R_{22}r\bigl(t-h(t)\bigr)+(1-\mu )r^{T}\bigl(t-h(t)\bigr)R_{22}r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-(1-\mu)r^{T}\bigl(t-h(t)\bigr)R_{22}r\bigl(t-h(t)\bigr) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+2 \bigl[h(t)-h_{1}\bigr]r^{T}(t)R_{11}\dot {r}(t)+2\bigl[h(t)-h_{1}\bigr]r^{T}(t)R_{12} \dot{r}(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+2 \int _{t-h(t)}^{t-h_{1}}\bigl[r(t)-r(s)\bigr]^{T}R^{T}_{12} \dot{r}(t)\,ds+2 \int_{t-h(t)}^{t-h_{1}}\bigl[r(t)-r(s)\bigr]^{T}R_{22} \dot{r}(t)\,ds \\& \hphantom{\dot{V}_{5}(t) }{}= r^{T}(t)\bigl[\mu R_{11}+R_{12}+R^{T}_{12}+R_{22}-(1- \mu ) \bigl(R_{12}+R^{T}_{12}\bigr)-(1-\mu)R_{22}\bigr]r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{} +(1-\mu)r^{T}(t) (R_{12}+R_{22})r\bigl(t-h(t)\bigr) +(1-\mu )r^{T}\bigl(t-h(t)\bigr) \bigl(R^{T}_{12}+R_{22} \bigr)r(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-r^{T}(t) (R_{12}+R_{22})r(t-h_{1})-r^{T}(t-h_{1}) \bigl(R^{T}_{12}+R_{22} \bigr)r(t)+r^{T}(t-h_{1})R_{22}r(t-h_{1}) \\& \hphantom{\dot{V}_{5}(t) \leq}{}+2\bigl[h(t)-h_{1}\bigr]r^{T}(t)\bigl[R_{11}+R_{12}+R^{T}_{12}+R_{22} \bigr]\dot {r}(t) \\& \hphantom{\dot{V}_{5}(t) \leq}{}-2\bigl[h(t)-h_{1}\bigr]v^{T}_{1} \bigl[R^{T}_{12}+R_{22}\bigr]\dot{r}(t). \end{aligned}$$

(20)

According to (3), for any positive definite diagonal matrices \(T_{1}\), \(T_{2}\), the following inequalities hold:

$$\begin{aligned}& -2g^{T}\bigl(Wr(t)\bigr)T_{1}g\bigl(Wr(t) \bigr)+2r^{T}(t)W^{T}(K_{1}+K_{2})T_{1}g \bigl(Wr(t)\bigr) \\& \quad{} -2r^{T}(t)W^{T}K_{1}T_{1}K_{2}Wr(t) \geq0, \end{aligned}$$

(21)

$$\begin{aligned}& -2g^{T}\bigl(Wr\bigl(t-h(t)\bigr)\bigr)T_{2}g\bigl(Wr \bigl(t-h(t)\bigr)\bigr) +2r^{T}\bigl(t-h(t)\bigr)W^{T}(K_{1}+K_{2})T_{2}g \bigl(Wr\bigl(t-h(t)\bigr)\bigr) \\& \quad{}-2r^{T}\bigl(t-h(t)\bigr)W^{T}K_{1}T_{2}K_{2}Wr \bigl(t-h(t)\bigr)\geq0. \end{aligned}$$

(22)

Furthermore, for any matrix *M* with appropriate dimension, the following equation holds:

$$\begin{aligned}& \bigl(2r^{T}(t)+2\dot{r}^{T}(t)\bigr)M\bigl[-\dot {r}(t)-(A+KC)r(t)-KDr\bigl(t-h(t)\bigr) \\& \quad{}+g\bigl(Wr\bigl(t-h(t)\bigr)\bigr)+(B_{1}-KB_{2})w(t) \bigr]=0. \end{aligned}$$

(23)

Under the zero-initial condition, it is obvious that \(V(r(t))|_{t=0}=0\). For convenience, let

$$ J_{\infty}= \int_{0}^{\infty}\bigl[\bar{z}^{T}(t) \bar{z}(t)-\gamma^{2}w^{T}(t)w(t)\bigr]\,dt. $$

(24)

Then for any nonzero \(w(t)\in\mathcal{L}_{2}[0,\infty)\), we obtain

$$\begin{aligned} J_{\infty} \leq& \int_{0}^{\infty}\bigl[\bar{z}^{T}(t) \bar{z}(t)-\gamma ^{2}w^{T}(t)w(t)\bigr]\,dt+V\bigl(r(t)\bigr) \big|_{t\rightarrow\infty}-V\bigl(r(t)\bigr)\big| _{t=0} \\ =& \int_{0}^{\infty}\bigl[\bar{z}^{T}(t) \bar{z}(t)-\gamma ^{2}w^{T}(t)w(t)+\dot{V}(t)\bigr]\,dt. \end{aligned}$$

(25)

From (13)-(25), one has

$$ \bar{z}^{T}(t)\bar{z}(t)-\gamma^{2}w^{T}(t)w(t)+ \dot{V}(t)\leq\xi ^{T}(t)\Phi\xi(t), $$

(26)

where

$$\begin{aligned} \xi^{T}(t) =&\bigl[r^{T}(t), r^{T}\bigl(t-h(t) \bigr), r^{T}(t-h_{1}), r^{T}(t-h_{2}), \\ &g^{T}\bigl(Wr(t)\bigr),g^{T}\bigl(Wr\bigl(t-h(t)\bigr)\bigr), u^{T}, v^{T}_{1}, v^{T}_{2}, w^{T}(t), \dot {r}^{T}(t)\bigr], \end{aligned}$$

and \(\Phi=[\Phi_{ij}]_{11\times11}\) with \(\Phi_{11}=\Pi_{11}+H^{T}H\), \(\Phi_{ij}=\Pi_{ij} \) (\(i\leq j\), \(1\leq i \leq11\), \(2 \leq j \leq11\)). By applying the Schur complement, \(\Phi<0\) is equivalent to \(\Pi^{*}<0\). Then, if (10) holds, we can ensure the error system (5) with the guaranteed \(\mathcal{H_{\infty}}\) performance defined by Definition 1.

In the sequel, we will show that the equilibrium point of (5) with \(w(t)=0\) is globally asymptotically stable if (10) holds. When \(w(t)=0\), the error system (5) becomes

$$ \dot{r}(t) = -(A+KC)r(t)-KDr\bigl(t-h(t)\bigr)+g\bigl(Wr\bigl(t-h(t)\bigr) \bigr). $$

(27)

We still consider the Lyapunov-Krasovskii functional candidate (12) and calculate its time-derivative along the trajectory of (27). We can easily obtain

$$ \dot{V}(t)\leq\bar{\xi}^{T}(t)\bar{\Pi}^{\ast}\bar{\xi}(t), $$

(28)

where

$$\begin{aligned} \bar{\xi}^{T}(t) =&\bigl[r^{T}(t), r^{T} \bigl(t-h(t)\bigr), r^{T}(t-h_{1}), r^{T}(t-h_{2}), \\ &g^{T}\bigl(Wr(t)\bigr),g^{T}\bigl(Wr\bigl(t-h(t)\bigr)\bigr), u^{T}, v^{T}_{1}, v^{T}_{2}, \dot {r}^{T}(t)\bigr], \end{aligned}$$

and \(\bar{\Pi}^{*}=[\bar{\Pi}^{*}_{ij}]_{10\times10}\) with

$$\begin{aligned}& \bar{\Pi}^{\ast}_{11}=P_{12}+P^{T}_{12}-4Z_{1}+Z_{3}+ \mu R_{11}+R_{12}+R^{T}_{12}+R_{22} \\& \hphantom{\bar{\Pi}^{\ast}_{11}=}{}-MA-(MA)^{T}-(1-\mu) \bigl(R_{12}+R^{T}_{12} \bigr)-(1-\mu )R_{22}-GC-(GC)^{T} \\& \hphantom{\bar{\Pi}^{\ast}_{11}=}{} -2W^{T}K_{1}T_{1}K_{2}W, \\& \bar{\Pi}^{\ast}_{12}=(1-\mu)R_{12}+(1- \mu)R_{22}-GD,\quad\quad \bar{\Pi}^{\ast }_{13}=-P_{12}+P_{13}-2Z_{1} -R_{12}-R_{22}, \\& \bar{\Pi}^{\ast}_{14}=-P_{13},\quad\quad \bar{\Pi}^{\ast }_{15}=W^{T}(K_{1}+K_{2})T_{1},\quad\quad \bar{\Pi}^{\ast}_{16}=M, \\& \bar{\Pi}^{\ast}_{17}=h_{1}P^{T}_{22}+6Z_{1},\quad\quad \bar{\Pi}^{\ast }_{18}=\bigl(h(t)-h_{1} \bigr)P_{23}, \quad\quad\bar{\Pi}^{\ast }_{19}= \bigl(h_{2}-h(t)\bigr)P_{23}, \\& \bar{\Pi}^{\ast }_{1,10}=P_{11}+\bigl(h(t)-h_{1} \bigr)R_{11}+\bigl(h(t)-h_{1}\bigr)R_{12}+ \bigl(h(t)-h_{1}\bigr)R^{T}_{12}+\bigl(h(t)-h_{1}\bigr)R_{22} \\& \hphantom{\bar{\Pi}^{\ast}_{1,10}=}{}-M-(MA)^{T}-(GC)^{T}, \\& \bar{\Pi}^{\ast}_{22}=-(1-\mu )R_{22}-8Z_{2}+S_{11}+S^{T}_{11}+S_{12}+S^{T}_{12}-S_{21}-S^{T}_{21}-S_{22}-S^{T}_{22} \\& \hphantom{ \bar{\Pi}^{\ast}_{22}=}{}-2W^{T}K_{1}T_{2}K_{2}W, \\& \bar{\Pi}^{\ast }_{23}=-2Z_{2}-S^{T}_{11}-S^{T}_{12}-S^{T}_{21}-S^{T}_{22}, \\& \bar{\Pi}^{\ast}_{24}=Z_{2}-3Z_{2}-S_{11}+S_{12}+S_{21}-S_{22},\quad\quad \bar {\Pi}^{\ast}_{26}=W^{T}(K_{1}+K_{2})T_{2}, \\& \bar{\Pi}^{\ast}_{28}=6Z_{2}+2S^{T}_{21}+2S^{T}_{22},\quad\quad \bar{\Pi}^{\ast }_{29}=6Z_{2}-2S_{12}+2S_{22},\quad\quad \bar{\Pi}^{\ast }_{2,10}=-(GD)^{T}, \\& \bar{\Pi}^{\ast}_{33}=-Z_{3}+Z_{4}-4Z_{1}-4Z_{2}+R_{22},\quad\quad \bar{\Pi }^{\ast}_{34}=S_{11}-S_{12}+S_{21}-S_{22}, \\& \bar{\Pi}^{\ast}_{37}=-h_{1}P^{T}_{22}+h_{1}P^{T}_{23}+6Z_{1},\quad\quad \bar {\Pi}^{\ast}_{38}=-\bigl(h(t)-h_{1} \bigr)P_{23}+\bigl(h(t)-h_{1}\bigr)P^{T}_{33} +6Z_{2}, \\& \bar{\Pi}^{\ast }_{39}=- \bigl(h_{2}-h(t)\bigr)P_{23}+\bigl(h_{2}-h(t) \bigr)P^{T}_{33}+2S_{12}+2S_{22}, \\& \bar{\Pi}^{\ast}_{44}=-Z_{4}-4Z_{2},\quad\quad \bar{\Pi}^{\ast }_{47}=-h_{1}P^{T}_{23},\quad\quad \bar{\Pi}^{\ast }_{48}=-\bigl(h(t)-h_{1} \bigr)P^{T}_{33}-2S^{T}_{21}+2S^{T}_{22}, \\& \bar{\Pi}^{\ast}_{49}=-\bigl(h_{2}-h(t) \bigr)P^{T}_{33}+6Z_{2}, \quad\quad\bar{\Pi}^{\ast }_{55}=Q-2T_{1},\quad\quad \bar{\Pi}^{\ast}_{66}=-(1-\mu)Q-2T_{2}, \\& \bar{\Pi}^{\ast}_{6,10}=M^{T},\quad\quad \bar{ \Pi}^{\ast}_{77}=-12Z_{1},\quad\quad \bar { \Pi}^{\ast}_{7,10}=h_{1}P^{T}_{12},\quad\quad \bar{\Pi}^{\ast }_{88}=-12Z_{2}, \\& \bar{\Pi}^{\ast}_{89}=-4S_{22},\quad\quad \bar{ \Pi}^{\ast }_{8,10}=\bigl(h(t)-h_{1} \bigr)P^{T}_{13}-\bigl(h(t)-h_{1} \bigr)R^{T}_{12}-\bigl(h(t)-h_{1} \bigr)R_{22}, \\& \bar{\Pi}^{\ast}_{99}=-12Z_{2}, \quad\quad\bar{ \Pi}^{\ast }_{9,10}=\bigl(h_{2}-h(t) \bigr)P^{T}_{13}, \\& \bar{\Pi}^{\ast }_{10,10}=h^{2}_{1}Z_{1}+(h_{2}-h_{1})^{2}Z_{2}-M-M^{T}. \end{aligned}$$

It is obvious that if \(\Pi^{*}_{[h(t)=h_{1}]}<0\), \(\Pi ^{*}_{[h(t)=h_{2}]}<0\), then \(\bar{\Pi}^{*}_{[h(t)=h_{1}]}<0\), \(\bar{\Pi }^{*}_{[h(t)=h_{2}]}<0\). So system (27) is globally asymptotically stable. Moreover, if (10) holds, the state estimator (4) for the static neural networks (1) has the guaranteed \(\mathcal{H_{\infty}}\) performance and guarantees the globally asymptotically stable of the error system (5). This completes the proof. □

### Remark 2

The time-varying delay in [28–33] was always assumed to satisfy \(0\leq h(t)\leq h\), which is a special case of the condition (2) in this paper. Therefore, compared with [28–33], the time-varying delay discussed in this paper is less restrictive. In [30, 31], for the sake of converting a nonlinear matrix inequality into a linear matrix inequality, some inequalities such as \(-PT^{-1}P\leq-2P+T\), which lack freedom and may lead to some conservativeness for the derived results, were utilized in the discussion of the guaranteed \(\mathcal{H}_{\infty}\) performance state estimation problem. In our paper, the zero equality (23) is used to avoid this problem, which can give much flexibility in solving LMIs. In [32], Jensen’s integral inequality, which ignored some terms and may introduce conservativeness to some extent, was employed to estimate the upper bound of the time derivative of the Lyapunov-Krasovskii functional. In this paper, Wirtinger’s integral inequality, which takes information not only on the state and the delayed state of a system, but also on the integral of the state over the period of the delay into account, is exploited to give an estimation of the time derivative of the Lyapunov-Krasovskii functional.

### Remark 3

Based on a Lyapunov-Krasovskii functional with triple integrals involving augmented terms, the guaranteed \(\mathcal{H}_{\infty}\) performance state estimation problem of static neural networks with interval time-varying delay was investigated in [34], and a sufficient criterion guaranteeing the globally asymptotical stability of the error system (5) for a given \(\mathcal{H}_{\infty}\) performance index was obtained [34]. Since the augmented Lyapunov-Krasovskii functional contained more information, the criterion derived in [34] had less conservativeness than most of the previous results [28–33]. However, the computational burden increased at the same time because of the augmented Lyapunov-Krasovskii functional. Compared with the results in [34], the advantages of the method used in this paper mainly rely on two aspects. First, the Lyapunov-Krasovskii functional is simpler than that in [34], since the triple integrals and other augmented terms in [34] are not needed, which will reduce the computational complexity. Second, in the proof of Theorem 1, Wirtinger’s integral inequality, which includes Jensen’s integral inequality, and a reciprocally convex approach are employed to estimate the upper bound of the derivative of the Lyapunov-Krasovskii functional, which will yield less conservative results.

When \(0\leq h(t)\leq h\), that is, the lower bound of the time-varying delay is 0, we introduce the Lyapunov-Krasovskii functional as follows:

$$ V(t)=\sum_{i=1}^{5}V_{i}(t), $$

(29)

with

$$\begin{aligned}& V_{1}(t) = \begin{pmatrix} r(t) \\ \int_{t-h}^{t}r(s)\,ds \end{pmatrix} ^{T} \begin{pmatrix} P_{11} & P_{12}\\ \ast& P_{22} \end{pmatrix} \begin{pmatrix} r(t) \\ \int_{t-h}^{t}r(s)\,ds \end{pmatrix} , \\& V_{2}(t) = \int_{t-h(t)}^{t}g^{T}\bigl(Wr(s)\bigr)Qg \bigl(Wr(s)\bigr)\,ds, \\& V_{3}(t) = h \int_{-h}^{0} \int_{t+\theta}^{t}\dot{r}^{T}(s)Z_{2} \dot {r}(s)\,ds\,d\theta, \\& V_{4}(t) = \int_{t-h}^{t}r^{T}(s)Z_{4}r(s)\,ds, \\& V_{5}(t) = \int_{t-h(t)}^{t} \begin{pmatrix} r(t) \\ \int_{s}^{t}\dot{r}(u)\,du \end{pmatrix} ^{T} \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} \begin{pmatrix} r(t) \\ \int_{s}^{t}\dot{r}(u)\,du \end{pmatrix} \,ds. \end{aligned}$$

By a similar method to that employed in Theorem 1, we can obtain the following corollary.

### Corollary 1

*For given scalars*
*h*, *μ*, *and*
\(\gamma>0\), *the error system* (5) *is globally asymptotically stable with the*
\(\mathcal{H_{\infty}}\)
*performance*
*γ*
*if there exist real matrices*
\(P>0\), \(Q>0\), \(Z_{2}>0\), \(Z_{4}>0\), \(R>0\), \(T_{1}=\operatorname{diag}\{t_{11}, t_{12}, \ldots, t_{1n}\}>0\), \(T_{2}=\operatorname{diag}\{t_{21}, t_{22}, \ldots, t_{2n}\}>0\), *and matrices*
\(S_{11}\), \(S_{12}\), \(S_{21}\), \(S_{22}\), *M*, *G*
*with appropriate dimensions such that the following LMIs are satisfied*:

$$\begin{aligned}& \Xi^{*}_{[h(t)=0]}< 0, \quad\quad\Xi^{*}_{[h(t)=h]}< 0, \end{aligned}$$

(30)

$$\begin{aligned}& \begin{pmatrix} Z_{2} & 0 & S_{11} & S_{12}\\ \ast& 3Z_{2} & S_{21} & S_{22}\\ \ast& \ast& Z_{2} & 0\\ \ast& \ast& \ast& 3Z_{2} \end{pmatrix} >0, \end{aligned}$$

(31)

*where*

$$\begin{aligned}& P= \begin{pmatrix} P_{11} & P_{12} \\ \ast& P_{22} \end{pmatrix} ,\quad\quad R= \begin{pmatrix} R_{11} & R_{12}\\ \ast& R_{22} \end{pmatrix} , \\& \Xi_{[h(t)]}^{*}= \begin{pmatrix} \Xi& \bar{H} \\ \ast& -I \end{pmatrix} , \quad\Xi=[\Xi_{ij}]_{9\times9}, \bar{H}=[H \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0]^{T}, \\& \Xi_{11}=P_{12}+P^{T}_{12}-4Z_{2}+Z_{4}\\& \hphantom{\Xi_{11}=}{}-MA-(MA)^{T}-GC-(GC)^{T}+ \mu R_{11}-(1-\mu) \bigl(R_{12}+R^{T}_{12}\bigr)-(1-\mu )R_{22} \\& \hphantom{\Xi_{11}=}{} -2W^{T}K_{1}T_{1}K_{2}W, \\& \Xi_{12}=-2Z_{2}-S_{11}-S_{12}-S_{21}-S_{22}-GD+(1- \mu)R_{12}+(1-\mu )R_{22}, \\& \Xi_{13}=-P_{12}+S_{11}-S_{12}+S_{21}-S_{22},\quad\quad \Xi _{14}=W^{T}(K_{1}+K_{2})T_{1}, \\& \Xi_{15}=M,\quad\quad \Xi_{16}=h(t)P^{T}_{22}+6Z_{2},\quad\quad \Xi _{17}=\bigl(h-h(t)\bigr)P^{T}_{22}+2S_{12}+2S_{22}, \\& \Xi_{18}=MB_{1}-GB_{2}, \\& \Xi _{19}=P_{11}-M-(MA)^{T}-(GC)^{T}+h(t)R_{11} +h(t)R_{12}+h(t)R^{T}_{12}+h(t)R_{22}, \\& \Xi _{22}=-8Z_{2}+S_{11}+S^{T}_{11}+S_{12}+S^{T}_{12}-S_{21}-S^{T}_{21}-S_{22}-S^{T}_{22} -(1-\mu)R_{22} \\& \hphantom{\Xi _{22}=}{}-2W^{T}K_{1}T_{2}K_{2}W, \\& \Xi_{23}=-2Z_{2}-S_{11}+S_{12}+S_{21}-S_{22},\quad\quad \Xi _{25}=W^{T}(K_{1}+K_{2})T_{2}, \\& \Xi_{26}=6Z_{2}+2S^{T}_{21}+2S^{T}_{22},\quad\quad \Xi _{27}=6Z_{2}-2S_{12}+2S_{22}, \\& \Xi_{29}=-(GD)^{T},\quad\quad \Xi_{33}=-4Z_{2}-Z_{4},\quad\quad \Xi _{36}=-h(t)P^{T}_{22}-2S^{T}_{21}+2S^{T}_{22}, \\& \Xi_{37}=-\bigl(h-h(t)\bigr)P^{T}_{22}+6Z_{2},\quad\quad \Xi_{44}=Q-2T_{1},\quad\quad \Xi _{55}=-(1- \mu)Q-2T_{2}, \\& \Xi_{59}=M^{T},\quad\quad \Xi_{66}=-12Z_{2},\quad\quad \Xi_{67}=-4S_{22}, \quad\quad\Xi _{69}=h(t)P^{T}_{12}-h(t)R^{T}_{12} -h(t)R_{22}, \\& \Xi_{77}=-12Z_{2},\quad\quad \Xi_{79}=\bigl(h-h(t)\bigr)P^{T}_{12},\quad\quad \Xi _{88}=-\gamma^{2}, \\& \Xi_{89}=(MB_{1})^{T}-(GB_{2})^{T},\quad\quad \Xi _{99}=h^{2}Z_{2}-M-M^{T}. \end{aligned}$$

*Moreover*, *the gain matrix*
*K*
*of the state estimator of* (4) *can be designed as*
\(K=M^{-1}G\).

### Remark 4

As an effective approach to establish the delay-dependent stability criteria for delayed neural networks, the complete delay-decomposing approach was proposed in [21], which significantly reduced the conservativeness of the derived stability criteria. A novel Lyapunov-Krasovskii functional decomposing the delay in all integral terms was constructed. Since delay information can be taken fully into account by dividing the delay interval into several subintervals, less conservative results may be obtained. The computational burden for the complete delay-decomposing approach will increase with the increasing number of subintervals. In order to get less conservative results as well as less computational burden, the number of the subintervals should be chosen properly. Jensen’s inequality was used to estimate the derivative of the Lyapunov-Krasovskii functional in [21]. The conservativeness of the derived result in this paper can be further reduced by our method with the complete delay-decomposing approach [21].

### Remark 5

The integral inequality method and the free-weighting matrix method are two main techniques to deal with the bounds of the integrals that appear in the derivative of Lyapunov-Krasovskii functional for stability analysis of delayed neural networks. A free-matrix-based integral inequality was developed and was applied to a stability analysis of systems with time-varying delay [23]. A free-matrix-based integral inequality implied Wirtinger’s inequality as a special case. The free matrices can provide freedom in reducing the conservativeness of the inequality. This new inequality was used to derive improved delay-dependent stability criteria although the computational burden increased because of the introduction of free-weighting matrices. The free-matrix-based integral inequality in [23] made use of information as regards only a single integral of the system state. Different from the free-matrix-based integral inequality, a new integral inequality was developed basing on information as regards a double integral of the system state in [24]. It also included the Wirtinger-based integral inequality. By employing a free-matrix-based integral inequality [23] or the novel integral inequality in [24], less conservative results than those obtained in our paper may be further derived.