## 4 Approximate Solution of Event-Triggered HJI Equation
Then, the critic weight error dynamic is presented by  
\[\dot{\hat{W}}_{c}=-\alpha_{c}\bar{\Delta}\phi(t)\bar{\Delta}\phi^{\mathsf{T}} (t)\hat{W}_{c}+\alpha_{c}\bar{\Delta}\phi(t)me_{B}(t) \tag{48}\]  
where \(m=1/(1+\Delta\phi^{\mathsf{T}}\Delta\phi)\), \(\bar{\Delta}\phi=\Delta\phi/(1+\Delta\phi^{\mathsf{T}}\Delta\phi)\). **Remark 4**. : _This paper designs an online event-triggered ADP learning algorithm, the activation function must be continuously excited to ensure that the NN weights can be converged. Then, the value function and control laws are obtained. Therefore, the probing noise signal with different amplitudes and diverse frequencies is constructed as \(d(t)=0.5(\sin(0.08t)^{2}\cos(1.5t)+0.3\sin(2.3t)^{4}\cos(7t))\)._  
**Remark 5**. : _Under the triggering condition (23), the adaptive learning law of the critic network (47) updates its weight by the reinforcement signal from (46), then the control law (43), and perturbation law (44) are updated at the triggering instant. In order to intuitively clarify the main idea of the algorithm, Figure 1 shows the block diagram of the event-triggered ADP \(H_{\infty}\)-constraint control._  
**Assumption 4** ([26]). :
1. \(g(x)\) _and_ \(h(x)\) _are Lipschitz continuous and satisfy_ \(\|g(x)-g(\hat{x}_{k})\|\leq L_{g}\|e_{k}(t)\|\) _and_ \(\|h(x)-h(\hat{x}_{k})\|\leq L_{h}\|e_{k}(t)\|\)_._
2. \(\
abla\phi(x)\) _is Lipschitz continuous, that is_ \(\|\
abla\phi(x)-\
abla\phi(\hat{x}_{k})\|\leq L_{\phi}\|e_{k}(t)\|\)_,_ \(L_{\phi}\) _is a positive constant._  
**Assumption 5** ([34; 35; 36]). :
1. _The_ \(NN\) _activation function and its gradient are bounded by positive constants so that_ \(\|\phi(x)\|\leq\phi_{m,t}\|\
abla\phi(x)\|\leq\phi_{M}\)_._
2. _The critic NNs approximation error and its gradient are all bounded by a constant and satisfied_ \(\|e_{V}\|\leq e_{m,t}\ \|\
abla e_{V}\|\leq e_{M}\)_._  
The control law \(u(x)\) is admissible control. The approximate weight \(\hat{W}_{c}\) can be guaranteed to converge to ideal weight \(W_{c}\) under persistent excitation (PE) condition. Assume that the activation function \(\Delta\phi\) is continuously exciting in the interval \([t-T,t]\). Let the residual error due to the approximation error satisfies \(\|e_{B}\|\leq e_{M}\), and it can ensure that the critic network weight is UUB. The boundedness condition of critic NN weight is expressed at (50). **Theorem 3**. : _Let Assumption 1-5 be valid, and the control laws and the turning law of critic NN are implemented by (43), (44), and (47). The closed-loop system (1) is asymptotically stable and the weight approximation error (48) is UUB only when the triggering condition in (23) is used and the following inequalities condition holds_  
\[\big{(}\alpha_{c}-\frac{1}{2}\big{)}\underline{\lambda}(\bar{\Delta}\phi(t) \bar{\Delta}\phi^{\mathsf{T}}(t))-\alpha_{2}>0 \tag{49}\]  
_where \(\alpha_{2}\) and \(\alpha_{4}\) are constants from (60) and (63)._  
Proof. : The initial control for the dynamical system (1) is admissible. For the value function (4) of the system, the Lyapunov function is constructed as  
\[\mathcal{L}(t)=\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3} \tag{50}\]  
where \(\mathcal{L}_{1}=V^{*}(x)\), \(\mathcal{L}_{2}=V^{*}(\hat{x}_{k})\), \(\mathcal{L}_{3}=(1/2)\hat{W}_{c}^{\mathsf{T}}\hat{W}_{c}\). Figure 1: The structure diagram of Event-triggered ADP algorithm. The process of NN learning is mainly divided into two cases under event-triggered control: (1) During the flow dynamic on \(\mathfrak{t}\in[t_{k},t_{k+1})\). (2) At the triggering instant \(t=t_{k}\). **Case 1. ** When the triggering condition is not satisfied and the derivative of \(\mathcal{L}_{1}\) can be given as  
\[ \begin{split}\dot{\mathcal{L}}_{1}&=\
abla{V^{*}}^ {\mathsf{T}}(x)\big{(}f(x)+g(x)\hat{u}(\hat{x}_{k})+h(x)\hat{v}(\hat{x}_{k}) \big{)}\\ &=\
abla{V^{*}}^{\mathsf{T}}(x)g(x)(\hat{u}(\hat{x}_{k})-u^{*}(x) )+\
abla{V^{*}}^{\mathsf{T}}(x)h(x)(\hat{v}(\hat{x}_{k})-v^{*}(x))\\ &\quad+\
abla{V^{*}}^{\mathsf{T}}(x)\big{(}f(x)+g(x)u^{*}(x)+h(x )v^{*}(x)\big{)}\end{split}