## 4 Approximate Solution of Event-Triggered HJI Equation
 \tag{51}\]  
According to (25)-(27) and (51) can be transformed as  
\[ \begin{split}\dot{\mathcal{L}}_{1}&=2u_{M}R\tanh ^{-\mathsf{T}}\bigg{(}\frac{u^{*}(x)}{u_{M}}\bigg{)}(\hat{u}(\hat{x}_{k})-u^{*} (x))+2\gamma{2^{\prime}}{v^{*}}^{\mathsf{T}}(x)(\hat{v}(\hat{x}_{k})-v^{*}(x) )\\ &\quad-x^{\mathsf{T}}Qx-M\big{(}u^{*}(x)\big{)}+\gamma^{2}v^{*}(x )^{\mathsf{T}}v^{*}(x)\end{split}  \tag{52}\]  
Similarly, using [PERSON]'s inequality, we have  
\[ \begin{split}\dot{\mathcal{L}}_{1}&\leq\Big{\|}u_{M }R\tanh^{-\mathsf{T}}\bigg{(}\frac{u^{*}(x)}{u_{M}}\bigg{)}\Big{\|}^{2}+\|\hat{ u}(\hat{x}_{k})-u^{*}(x)\|^{2}+\gamma^{2}\|v^{*}(x)\|^{2}\\ &\quad+\gamma^{2}\|\hat{v}(\hat{x}_{k})-v^{*}(x)\|^{2}-x^{ \mathsf{T}}Qx-M\big{(}u^{*}(x)\big{)}+\gamma^{2}{v^{*}}^{\mathsf{T}}(x)v^{*}( x)\\ &\leq\frac{1}{4}\Big{\|}\
abla{V^{*}}^{\mathsf{T}}(x)\Big{\|}^{2 }+2\gamma^{2}\|v(x)^{*}\|^{2}-x^{\mathsf{T}}Qx-M\big{(}u^{*}(x)\big{)}\\ &\quad+\underbrace{\|\hat{u}(\hat{x}_{k})-u^{*}(x)\|^{2}}_{o_{1} }+\gamma^{2}\underbrace{\|\hat{v}(\hat{x}_{k})-v^{*}(x)\|^{2}}_{o_{2}}\end{split}  \tag{53}\]  
Combining (8) with (38), then we obtain  
\[ \begin{split} u^{*}(x)&=-u_{M}\tanh\Big{(}\frac{1}{2 u_{M}}R^{-1}g^{\mathsf{T}}(x)\
abla\
abla^{*}(x)\Big{)}\\ &=-u_{M}\tanh\Big{(}\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}(x)\
abla \phi^{\mathsf{T}}(x)\big{\{}W_{c}+\
abla\varepsilon_{V}(x)\big{\}}\Big{)} \end{split}  \tag{54}\]  
Note that \(|\tanh(x)|\leq 1\) and \(|x-y|\leq|x|+|y|\), Then, we have  
\[ \begin{split} o_{1}&=\Big{\|}u_{M}\tanh\Big{(}\frac {1}{2u_{M}}R^{-1}g^{\mathsf{T}}(x)\big{(}\
abla\phi^{\mathsf{T}}(x)W_{c}+ \
abla\varepsilon_{V}(x)\big{)}\Big{)}\\ &\quad-u_{M}\tanh\Big{(}\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}(\hat{ x}_{k})\
abla\phi^{\mathsf{T}}(\hat{x}_{k})(W_{c}-\tilde{W}_{c})\Big{)}\Big{\|}^{2} \\ &\leq u_{M}^{2}\Big{\|}\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}(x) \big{(}\
abla\phi^{\mathsf{T}}(x)W_{c}+\
abla\varepsilon_{V}(x)\big{)}+\frac{1 }{2u_{M}}R^{-1}g^{\mathsf{T}}(\hat{x}_{k})\
abla\phi^{\mathsf{T}}(\hat{x}_{k} )(\tilde{W}_{c}-W_{c})\Big{\|}^{2}\\ &\leq\frac{1}{4}\|R^{-1}\|^{2}\Big{\|}\big{(}\
abla\phi(x)g(x)- \
abla\phi(\hat{x}_{k})g(\hat{x}_{k})\big{)}W_{c}+g^{\mathsf{T}}(x)\
abla \varepsilon_{V}(x)+\
abla\phi(\hat{x}_{k})g(\hat{x}_{k})\tilde{W}_{c}\Big{\|}^ {2}\\ &\leq\frac{1}{2}\|R^{-1}\|^{2}\big{(}\underbrace{\|\
abla\phi(x)g(x )-\
abla\phi(\hat{x}_{k})g(\hat{x}_{k})\|^{2}}_{\mathcal{F}_{1}}\|W_{c}\|^{2}+ \underbrace{\Big{\|}g^{\mathsf{T}}(x)\
abla\varepsilon_{V}(x)+\
abla\phi(\hat{ x}_{k})g(\hat{x}_{k})\tilde{W}_{c}\Big{\|}^{2}}_{\mathcal{F}_{2}}\Big{)} \end{split}  \tag{55}\]  
According to Assumption 4 and Assumption 5, we obtain  
\[ \begin{split}\mathcal{F}_{1}&=\|\
abla\phi(x)g(x)- \
abla\phi(\hat{x}_{k})g(x)+\
abla\phi(\hat{x}_{k})g(x)-\
abla\phi(\hat{x}_{k} )g(\hat{x}_{k})\|^{2}\\ &=\|(\
abla\phi(x)-\
abla\phi(\hat{x}_{k}))g(x)+\
abla\phi(\hat{x} _{k})(g(x)-g(\hat{x}_{k}))\|^{2}\\ &\leq 2\|(\
abla\phi(x)-\
abla\phi(\hat{x}_{k}))g(x)\|^{2}+2\|\
abla \phi(\hat{x}_{k})(g(x)-g(\hat{x}_{k}))\|^{2}\\ &\leq 2\Big{(}L_{q}^{2}g_{SM}^{2}+\phi_{M}^{2}L_{g}^{2}\Big{)}\| \varepsilon_{k}\|^{2}\end{split}  \tag{56}\]Similarly, it can be obtained  
\[ \begin{split}\mathcal{F}_{2}&=\left\|g^{\mathsf{T}}(x) \
abla\varepsilon_{V}(x)+\
abla\phi(\hat{x}_{k})g(\hat{x}_{k})\tilde{W}_{c} \right\|^{2}\\ &\leq 2\left\|g^{\mathsf{T}}(x)\
abla\varepsilon_{V}(x)\right\|^{2}+2 \left\|\
abla\phi(\hat{x}_{k})g(\hat{x}_{k})\tilde{W}_{c}\right\|^{2}\\ &\leq 2\left(\hat{g}_{M}^{2}\varepsilon_{M}^{2}+\hat{g}_{M}^{2} \phi_{M}^{2}\big{\|}\tilde{W}_{c}\big{\|}^{2}\right)\end{split}  \tag{57}\]  
According to (55)-(57), \(o_{1}\) rewrite as  
\[ \begin{split} o_{1}&\leq\|R^{-1}\|^{2}\left(L_{ \phi}^{2}g_{M}^{2}+\phi_{M}^{2}L_{g}^{2}\right)\|e_{k}\|^{2}\|W_{c}\|^{2}+\|R^{ -1}\|^{2}(g_{M}^{2}\varepsilon_{M}^{2}+\hat{g}_{M}^{2}\phi_{M}^{2}\big{\|} \tilde{W}_{c}\big{\|}^{2})\end{split}  \tag{58}\]  
Lisewise, combining (9) with (37), \(o_{2}\) can be given as  
\[