## 3 Design of Event-Triggered \(H_{\infty}\)-Constrained Optimal Control  
An event-triggered control is introduced into ADP to solve the zero-sum game problem and then develops an event-triggered HJI equation. Then, a triggering condition is designed to limit the number of sampling states which can avoid the Zeno behavior. In the ETC, the event-triggered instant sequences are defined as \(\{t_{k}\}_{k=0}^{\infty}\), \(t_{k}<t_{k+1}\) and \(k\in\mathbb{N}^{+}\). A novel sampling state can be obtained at the triggering instant when the triggering condition is violated. The event error determines the appearance of the sampling state, and it can be expressed as  
\[e_{k}(t)=\hat{x}_{k}-x(t),t\in[t_{k},t_{k+1}) \tag{15}\]  
where \(\hat{x}_{k}=x(t_{k})\) is the sampling state at the triggering instant \(t_{k}\) and \(x(t)\) represents the current state, respectively. **Remark 2**. : _The event is triggered depending on two basic conditions, which are the event-triggered error \(e_{k}\) and the triggering threshold \(e_{T}\). An event occurs and the controller updates when the error violates the triggering threshold, this is, the error is \(e_{k}=0\) at \(t=t_{k}\). The holding of the sampling state in the non-triggered interval \(t\in[t_{k},t_{k+1})\) is realized by the zero-order holder (ZOH). The ZOH is used to store the control law of the previous moment and convert the sampled signal into a continuous signal. Then, the law is kept until the next triggering instant._  
It is assumed that there is no delay between the sensor and the controller. According to (15), the closed-loop system (1) is rewritten as  
\[\dot{x}=f(x)+g(x)u(\hat{x}_{k})+h(x)v(\hat{x}_{k}),t\in[t_{k},t_{k+1}) \tag{16}\]  
Then, the event-based value function is shown as  
\[V\big{(}x(t-T)\big{)}=\int_{t-T}^{t}\Big{(}x^{\mathsf{T}}Qx+M(u(\hat{x}_{k}))- \gamma^{2}v^{\mathsf{T}}(\hat{x}_{k})v(\hat{x}_{k})\Big{)}d\tau+V\big{(}x(t) \big{)} \tag{17}\]  
Therefore, the optimal event-triggered control law and the disturbance law in \(t\in[t_{k},t_{k+1})\) can be transformed into  
\[u^{*}(\hat{x}_{k})=-u_{M}\tanh\Big{(}\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}(\hat {x}_{k})\
abla V^{*}(\hat{x}_{k})\Big{)} \tag{18}\]  
\[v^{*}(\hat{x}_{k})=\frac{1}{2}\gamma^{-2}h^{\mathsf{T}}(\hat{x}_{k})\
abla V^{ *}(\hat{x}_{k}) \tag{19}\]  
Similar to (11), the event-triggered HJI equation becomes  
\[ \begin{split}&\
abla{V^{*}}^{\mathsf{T}}(x)f(x)+x^{\mathsf{T}}Qx+M(u^{*}( \hat{x}_{k}))-u_{M}{V^{*}}^{\mathsf{T}}(x)g\tanh\Big{(}\frac{1}{2u_{M}}R^{-1} g^{\mathsf{T}}(\hat{x}_{k})\
abla V^{*}(\hat{x}_{k})\Big{)}\\ &+\frac{1}{2}\gamma^{-2}\
abla{V^{*}}^{\mathsf{T}}(x)h(x)h^{ \mathsf{T}}(\hat{x}_{k})\
abla V^{*}(\hat{x}_{k})-\frac{1}{4}\gamma^{-2} \
abla{V^{*}}^{\mathsf{T}}(\hat{x}_{k})h(\hat{x}_{k})h^{\mathsf{T}}(\hat{x}_{ k})\
abla V^{*}(\hat{x}_{k})=0\end{split}  \tag{20}\]  
To sum up, in order to reduce communication load and computational cost, we introduce the event-triggered ADP in Algorithm 2 as follows  
```
1: Choose the initial admissible control laws \(u_{0}\), \(v_{0}\). 2:Policy Evaluation. During the interval time, the value function \(V(x)\) can be obtain by solving the event-based Bellman equation (17). 3:Policy Improvement. (18) and (19) can obtain the optimal control policy \(u^{*}(\hat{x}_{k})\) and the optimal disturbance policy \(v^{*}(\hat{x}_{k})\) separately. ```  
**Algorithm 2** Event-triggered ADP algorithm. **Assumption 1** ([15]). : _For the optimal value function \(V^{*}(x)\) is continuously differentiable, \(V^{*}(x)\) and \(\
abla V^{*}(x)\) are bounded on \(\Lambda\), such that the solution \(V^{*}(x)\) and its partial derivative \(\
abla V^{*}(x)\) can be denoted as \(\max\{\|V^{*}(x)\|,\|\
abla V^{*}(x)\|\}\leq\eta_{0}\) with a positive constant \(\eta_{0}\)._  
**Assumption 2** ([26]). : _For \(\forall x\in\mathbb{R}^{n}\), there exist \(g_{M}>0\) and \(h_{M}>0\) and can be satisfied with \(\|g(x)\|\leq g_{M}\) and \(\|h(x)\|\leq h_{M}\)._  
**Assumption 3**.