## 3 Design of Adaptive-Impulsive Controller  
**Assumption 1**.: _For any \(x_{i}(t)=(x_{i1}(t),\,x_{i2}(t),\,\ldots,\,x_{in}(t))^{T}\) and \(y_{i}(t)=(y_{i1}(t),\,y_{i2}(t),\,\ldots,\,y_{in}(t))^{T}\), there exists a positive constant \(L_{i}>0\), such that \(\|F_{i}(t,y_{i}(t),\alpha_{i})-F_{i}(t,x_{i}(t),\alpha_{i})\|\leq L_{i}\|y_{i} (t)-x_{i}(t)\|\)._  
_Our goal is to design an adaptive pulse controller user interface(UII) that allows the actual network to be asymptotically synchronized with the scheduled network to reduce costs [21; 42]. The response network is designed as_  
\[\left\{\begin{array}{rl}\dot{y}_{i}(t)=f_{i1}(y_{i}(t))+f_{i2}(y_{i}(t)) \overline{\alpha}+\sum\limits_{j=N_{i}}{b_{ij}\Gamma y_{j}(t)+u_{i}\,t\
eq t_{ k}},\\ \Delta y_{i}(t^{+})=B_{ik}e_{i}(t),\ \ t=t_{k},k=1,2,\cdots,\\ y_{i}(t_{0}^{+})=y_{i0}\end{array}\right. \tag{9}\]  
_where \(\overline{\alpha}\) is the estimation of the unknown \(\alpha\), \(B_{ik}\) is the feedback matrix of node \(i\) received at impulsive moment \(t_{k}\), \(u_{i}\) is the adaptive controller received by the node \(i\), and:_  
\(y_{i}(t_{k}^{+})=\underset{t\to t_{k}^{+}}{\lim}y_{i}(t)\)_, \(y_{i}(t_{k}^{-})=\underset{t\to t_{k}^{-}}{\lim}y_{i}(t)\), \(\Delta y_{i}(t_{k}^{+})=y_{i}(t_{k}^{+})-y_{i}(t_{k}^{-})\). Here, \(t_{k}\) satisfies \(0\leq t_{1}<t_{2}<\ldots<t_{k}<\ldots\), and \(\underset{k\to\infty}{\lim}t_{k}=+\infty\)._  
**Theorem 1**.: _Supposing that Assumption 1 holds, then:_  
Figure 5: Measurement of resilient recovery utilizing outer synchronization error.  
where \(L=max\{L_{i}\}\).  
Noticing that \(\sum\limits_{i=1}^{N}{{{e_{i}}^{T}}\frac{{{e_{i}}}}{{{\|{E\|}}\|}^{2}}}=1\), we have  
\[\begin{array}{l}-\sum\limits_{i=1}^{N}{{{e_{i}}^{T}}{{g_{i}}{e_{i}}}}-\mu \sum\limits_{i=1}^{N}{{{\|{\overline{\alpha}}-\alpha\|}^{2}}{{e_{i}}^{T}}\frac{ {{e_{i}}}}{{{\|{E\|}}^{2}}}}\\ =\sum\limits_{i=1}^{N}{{-g\|{e_{i}}\|}^{2}}-\mu\sum\limits_{i=1}^{N}{{{\|{ \overline{\alpha}}-\alpha\|}^{2}}{{e_{i}}^{T}}\frac{{{e_{i}}}}{{{\|{E\|}}^{2}}}} \\ =\sum\limits_{i=1}^{N}{{-g\|{e_{i}}\|}^{2}}-\mu\sum\limits_{i=1}^{N}{{{\|{ \overline{\alpha}}-\alpha\|}^{2}}}\end{array} \tag{14}\]  
Equation (11) can be simplified as  
\[\begin{array}{l}\dot{V}\leq L\sum\limits_{i=1}^{N}{{{e_{i}}^{T}}{e_{i}}}+ \sum\limits_{i=1}^{N}{{\sum\limits_{j=1}^{N}{{e_{i}}^{T}}{b_{ij}}\Gamma{e_{i}} }}(t)+\sum\limits_{i=1}^{N}{{\sum\limits_{j=1}^{N}{{e_{i}}^{T}}{a_{ij}}\Gamma{ e_{i}}}}(t)+\sum\limits_{i=1}^{N}{{-g\|{e_{i}}\|}^{2}}-\mu\sum\limits_{i=1}^{N}{{{\|{ \overline{\alpha}}-\alpha\|}^{2}}}\\ \\ \leq(L-g)E^{T}(t)E(t)+E^{T}(t)(B\otimes\Gamma)E(t)+E^{T}(t)(A\otimes\Gamma)E(t )-\mu\sum\limits_{i=1}^{N}{{{\|{\overline{\alpha}}-\alpha\|}^{2}}}\end{array} \tag{15}\]  
where \(\otimes\) is the Kronecker product.  
\[\begin{array}{l}\dot{V}(t)\leq(L-g)E^{T}(t)E(t)+[\frac{(B\otimes\Gamma)+(B \otimes\Gamma)^{T}}{2}+\frac{(A\otimes\Gamma)+(A\otimes\Gamma)^{T}}{2}]\times E ^{T}(t)E(t)-\mu\sum\limits_{i=1}^{N}{{{\|{\overline{\alpha}}-\alpha\|}^{2}}} \\ \leq(L-g+\kappa+\iota)\times E^{T}(t)E(t)-\mu\sum\limits_{i=1}^{N}{{{\|{ \overline{\alpha}}-\alpha\|}^{2}}}\\ \leq 2\chi V(t)\end{array}\]  
where \(\kappa=\lambda_{\We have \(V(t)\to 0,\left(k\rightarrow\infty\right).\) That is, \(e_{i}(t)\to 0\) and \(\overline{\alpha}\rightarrow\alpha\). This means the outer synchronization is realized between the drive network in Equation (4) and the response network in Equation (5). Then, the proof work is completed. \(\Box\)