## 4 Approximate Solution of Event-Triggered HJI Equation
\begin{split} o_{2}&\leq\left\|\frac{1}{2}\gamma^{ -2}h^{\mathsf{T}}(x)\big{(}\
abla\phi^{\mathsf{T}}(x)W_{c}+\
abla\varepsilon_ {V}(x)\big{)}-\frac{1}{2}\gamma^{-2}h^{\mathsf{T}}(\hat{x}_{k})\
abla\phi^{ \mathsf{T}}(\hat{x}_{k})\tilde{W}_{c}\right\|^{2}\\ &\leq\frac{1}{2}\|\
abla\phi(x)h(x)-\
abla\phi(\hat{x}_{k})h(\hat {x}_{k})\|^{2}\|W_{c}\|^{2}+\|h^{\mathsf{T}}(x)\
abla\varepsilon_{V}(x)\|^{2}+ \|\
abla\phi(\hat{x}_{k})h(\hat{x}_{k})\tilde{W}_{c}\|^{2}\\ &\leq(L_{\phi}^{2}h_{M}^{2}+\phi_{M}^{2}L_{h}^{2})\|e_{k}\|^{2}\| W_{c}\|^{2}+h_{M}^{2}\varepsilon_{M}^{2}+h_{M}^{2}\phi_{M}^{2}\big{\|}\tilde{W}_{c} \big{\|}^{2}\end{split}  \tag{59}\]  
According to (53), (58) and (59), we have  
\[ \begin{split}\hat{\mathcal{L}}_{1}&\leq-x^{\mathsf{ T}}Qx-M\big{(}u^{*}(x)\big{)}+\alpha_{1}\|e_{k}\|^{2}\|W_{c}\|^{2}+\alpha_{2}\| \tilde{W}_{c}\|^{2}+\alpha_{3}\\ &\leq-(1-\sigma^{2})\underline{\lambda}(Q)\|x\|^{2}-\sigma^{2} \underline{\lambda}(Q)\|x\|^{2}+\alpha_{1}\|e_{k}\|^{2}\|W_{c}\|^{2}+\alpha_{2 }\|\tilde{W}_{c}\|^{2}+\alpha_{3}\end{split}  \tag{60}\]  
where \(\alpha_{1}=\|R^{-1}\|^{2}\left(L_{\phi}^{2}g_{M}^{2}+\phi_{M}^{2}L_{g}^{2} \right)+\gamma^{2}(L_{\phi}^{2}h_{M}^{2}+\phi_{M}^{2}L_{h}^{2})\), \(\alpha_{2}=\|R^{-1}\|^{2}g_{M}^{2}\phi_{M}^{2}+\gamma^{2}h_{M}^{2}\phi_{M}^{2}\), \(\alpha_{3}=\frac{1}{4}g_{M}^{2}\eta_{0}+2\gamma^{2}v_{M}^{2}+\|R^{-1}\|^{2}g_ {M}^{2}\varepsilon_{M}^{2}+\gamma^{2}h_{M}^{2}\varepsilon_{M}^{2}\). If the event is not triggered, we have \(\hat{\mathcal{L}}_{2}=0\). Note that \(1/(1+\Delta\phi^{\mathsf{T}}\Delta\phi)\leq 1\), thus, \(m\), \(\bar{\Delta}\phi(t)\) from (48) are bounded. Then, according to \(W_{c}=\tilde{W}_{c}+\tilde{W}_{c}\), the part of \(\hat{\mathcal{L}}_{3}\) can be given as  
\[ \begin{split}\hat{\mathcal{L}}_{3}&=\tilde{W}_{c}^{ \mathsf{T}}\hat{W}_{c}=-\alpha_{c}\tilde{W}_{c}^{\mathsf{T}}\bar{\Delta}\phi(t )\bar{\Delta}\phi^{\mathsf{T}}(t)\tilde{W}_{c}+\alpha_{c}\tilde{W}_{c}^{ \mathsf{T}}\bar{\Delta}\phi(t)m\varepsilon_{B}(t)\\ &\leq-\alpha_{c}\underline{\lambda}(\bar{\Delta}\phi(t)\bar{\Delta }\phi^{\mathsf{T}}(t))\|\tilde{W}_{c}\|^{2}+\frac{1}{2}\alpha_{c}^{2} \varepsilon_{M}^{2}+\frac{1}{2}\underline{\lambda}(\bar{\Delta}\phi(t)\bar{ \Delta}\phi^{\mathsf{T}}(t))\|\tilde{W}_{c}\|^{2}\\ &\leq-(\alpha_{c}-\frac{1}{2})\underline{\lambda}(\bar{\Delta}\phi( t)\bar{\Delta}\phi^{\mathsf{T}}(t))\|\tilde{W}_{c}\|^{2}+\frac{1}{2}\alpha_{c}^{2} \varepsilon_{M}^{2}\end{split}  \tag{61}\]  
Combining (60) and (61), we have  
\[ \begin{split}\hat{\mathcal{L}}\leq&-(1-\sigma^{2}) \underline{\lambda}(Q)\|x\|^{2}-\sigma^{2}\underline{\lambda}(Q)\|x\|^{2}+ \alpha_{1}\|e_{k}\|^{2}\|W_{c}\|^{2}+\alpha_{2}\|\tilde{W}_{c}\|^{2}\\ &-\big{(}\alpha_{c}-\frac{1}{2}\big{)}\underline{\lambda}(\bar{ \Delta}\phi(t)\bar{\Delta}\phi^{\mathsf{T}}(t))\|\tilde{W}_{c}\|^{2}+\frac{1}{ 2}\alpha_{c}^{2}\varepsilon_{M}^{2}+\alpha_{3}\end{split}  \tag{62}\]  
Let the triggering condition (23) holding the derivative of \(\mathcal{L}\) become  
\[\hat{\mathcal{L}}\leq-\sigma^{2}\underline{\lambda}(Q)\|x\|^{2}-\left((\alpha_{ c}-\frac{1}{2})\underline{\lambda}(\bar{\Delta}\phi(t)\bar{\Delta}\phi^{\mathsf{T}}(t))- \varpi_{2}\right)\|\tilde{W}_{c}\|^{2}+\varpi_{4} \tag{63}\]  
where \(\varpi_{4}=(1/2)\alpha_{c}^{2}\varepsilon_{M}^{2}+\varpi_{3}\). Let the triggering condition (23) and the inequality condition (49) hold. We can conclude that \(\hat{\mathcal{L}}<0\) if the \(\|x(t)\|\) and \(\|\tilde{W}_{c}\|\) are satisfied with  
\[\|x\|>\sqrt{\frac{\varpi_{4}}{\sigma^{2}\underline{\lambda}(Q)}} \tag{64}\]\[\left\|\tilde{W}_{c}\right\|>\sqrt{\frac{2\varpi_{4}}{(2\alpha_{c}-1)\underline{ \lambda}(\tilde{\Delta}\phi(t)\tilde{\Delta}\phi^{\mathsf{T}}(t))-2\varpi_{2}}} \tag{65}\]  
Therefore, by the Lyapunov theory, the state \(x\) and the NN weight error \(\tilde{W}_{c}\) are uniformly ultimately bounded. **Case 2. ** The event is triggered at \(t=t_{k+1}\).