## 3 Design of Event-Triggered \(H_{\infty}\)-Constrained Optimal Control
: _Assuming that the optimal control law and the disturbance law are Lipschitz continuous on a compact set \(\Lambda\in\mathbb{R}^{n}\). There exist \(L_{u}>0\) and \(L_{v}>0\) satisfying_  
\[\|u^{*}(x)-u^{*}(\hat{x}_{k})\| \leq L_{u}\|e_{k}(t)\| \tag{21}\] \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]  
**Theorem 1**. : _Let \(V^{*}(x)\) be the optimal solution of (11). The optimal control laws \(u^{*}(\mathfrak{x}_{k})\) in (18) and \(v^{*}(x)\) in (19) are obtained to ensure that the dynamic system is uniformly ultimately bounded (UIUB) only when the following triggering condition is given as_  
\[\|e_{k}(t)\|\leq\sqrt{\frac{(1-\sigma^{2})}{L_{u}^{2}+\gamma^{2}L_{v}^{2}} \lambda(Q)\|x\|^{2}}\triangleq e_{T} \tag{23}\]  
_where \(\gamma>0\) is the level of disturbance attenuation from (3), \(\sigma\) is a positive design parameter. \(\underline{\lambda}(Q)\) is the minimum eigenvalue of the matrix \(Q\). The controller gets a new control law when the event error \(e_{k}\) exceeds the triggering threshold \(e_{T}\)._  
Proof. : The optimal value function \(V^{*}(x)\) is obtained from (11) and it is positive definite for any \(x\
eq 0\). Use \(V^{*}(x)\) as the Lyapunov function, and its derivative to time is defined as  
\[\dot{V}^{*}(x) =\
abla V^{*\mathsf{T}}(x)(f+gu^{*}(\mathfrak{x}_{k})+hv^{*}( \mathfrak{x}_{k})) \tag{24}\] \[=\
abla V^{*\mathsf{T}}(x)g(u^{*}(\mathfrak{x}_{k})-u^{*}(x))+ \
abla V^{*\mathsf{T}}(x)h(v^{*}(\mathfrak{x}_{k})-v^{*}(x))\] \[\quad+\
abla V^{*\mathsf{T}}(x)(f+gu^{*}(x)+hv^{*}(x))\]  
From (6), we can acquire  
\[\
abla V^{*\mathsf{T}}(x)(f+gu^{*}(x)+hv^{*}(x))=-x^{\mathsf{T}}Qx-M(u^{*}(x) )+\gamma^{2}{v^{*\mathsf{T}}}v^{*} \tag{25}\]  
Then, (8) and (9) can be transformed as  
\[\
abla V^{*\mathsf{T}}(x)g=-2u_{M}R\tanh^{-\mathsf{T}}\left(\frac{u^{*}(x)}{ u_{M}}\right) \tag{26}\]  
\[\
abla V^{*\mathsf{T}}(x)h=2\gamma^{2}{v^{*\mathsf{T}}}(x) \tag{27}\]  
According to (25)-(27), (24) can be rewritten as  
\[\dot{V}^{*}(x) =\underbrace{2u_{M}R\tanh^{-\mathsf{T}}\left(\frac{u^{*}(x)}{u_ {M}}\right)(u^{*}(x)-u^{*}(\mathfrak{x}_{k}))}_{s_{1}}-x^{\mathsf{T}}Qx \tag{28}\] \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]0  
By the weighted form of Young's inequality, we have  
\[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]1 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]2  
Similarly, we can obtain  
\[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]3 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]4  
Let Assumption 1-3 holding and note \(x^{\mathsf{T}}Qx\geq\underline{\lambda}(Q)\|x\|^{2}\), we have\[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]5 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]6 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]7 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]8 \[\|v^{*}(x)-v^{*}(\hat{x}_{k})\| \leq L_{v}\|e_{k}(t)\| \tag{22}\]9  
where \(\omega_{0}=(1/4)\eta_{0}^{2}g_{M}^{2}+(\eta_{0}^{2}h_{M}^{2})/(2\gamma^{2})\). From (10), we can obtain \(-M\big{(}u^{*}\big{)}\leq 0\). If the triggering condition is given by (23), we can obtain  
\[\|e_{k}(t)\|\leq\sqrt{\frac{(1-\sigma^{2})}{L_{u}^{2}+\gamma^{2}L_{v}^{2}} \lambda(Q)\|x\|^{2}}\triangleq e_{T} \tag{23}\]0  
Thus, it can conclude that \(\dot{V}^{*}(x)<0\) for \(\|x\|>\sqrt{\omega_{0}/(\sigma^{2}\underline{\lambda}(Q))}\). Therefore, the optimal laws \(u^{*}(\hat{x}_{k})\) and \(v^{*}(\hat{x}_{k})\) can ensure that the closed-loop system (1) is asymptotically stable by the Lyapunov theory. The proof is completed. From (23), we can acquire the triggering interval currently when the event is triggered. The difference between the two triggering instants is recorded as the triggering interval. The interval can be expressed as \(\tau_{k}=t_{k+1}-t_{k}\). However, the Zeno behavior occurs if there exists \((\tau_{k})_{min}=0\). **Remark 3**.