## 4 Approximate Solution of Event-Triggered HJI Equation  
An ADP-based event-triggered algorithm is shown before. In this section, an event-based single neural network with the approximation function is constructed, namely critic NN, to approximately solve the HJI equation. Based on the general approximation of neural networks. The value function \(V(x)\) and its gradient \(\
abla V(x)\) can be obtained by the neural network as  
\[V(x)=W_{c}^{\mathrm{T}}\phi(x)+\varepsilon_{V}(x) \tag{37}\]  
\[\
abla V(x)=\
abla\phi(x)^{\mathrm{T}}W_{c}+\
abla\varepsilon_{V}(x) \tag{38}\]  
where \(W_{c}\in\mathbb{R}^{L_{1}}\) is the ideal weight the critic NN, \(L_{1}\) is the number of neurons, \(\phi(x)\in\mathbb{R}^{L_{1}}\) is the suitable activation function of NN. \(\varepsilon_{V}(x)\) indicates the reconstruction error. The approximate value function can be obtained from (37), and substituted it into (12) to obtain the following Bellman equation  
\[\int_{t-T}^{t}\Big{(}x^{\mathrm{T}}Qx+M(u)-\gamma^{2}v^{\mathrm{T}}v\Big{)}d \tau+W_{c}^{\mathrm{T}}\Delta\phi(x)\triangleq e_{B}(t) \tag{39}\]  
where \(\Delta\phi(x)=\phi(x(t))-\phi(x(t-T))\). Due to the existence of the NN approximation error, the Bellman equation error is  
\[\varepsilon_{B}(t)=\varepsilon_{V}(t)-\varepsilon_{V}(t-T)=-\int_{t-T}^{t} \Big{(}\
abla\varepsilon_{V}^{\mathrm{T}}(f(x)+g(x)u+h(x)v)\Big{)}d\tau \tag{40}\]  
For (39), the optimal value is obtained only when \(\varepsilon_{B}\) goes to zero indefinitely. However, the ideal weights \(W_{c}\) are unknown, they can be estimated by the current weights \(\hat{W}_{c}\). The actual value function of the approximate network can be expressed as  
\[\hat{V}(x)=\hat{W}_{c}^{\mathrm{T}}\phi(x) \tag{41}\]\[\
abla\hat{V}(x)=\
abla\phi(x)^{\mathsf{T}}\hat{W}_{c} \tag{42}\]  
To solve the optimal control problem, the value function is obtained by solving the Bellman equation in the policy evaluation, and then the new control laws are obtained by the present value function in the policy improvement. Combined with (20), (21), and (42), the event-triggered control law and disturbance law become  
\[\hat{u}(\hat{x}_{k})=-u_{M}\tanh\left(\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}( \hat{x}_{k})\
abla\phi^{\mathsf{T}}(\hat{x}_{k})\hat{W}_{c}\right) \tag{43}\]  
\[\hat{v}(\hat{x}_{k})=\frac{1}{2}\gamma^{-2}h^{\mathsf{T}}(\hat{x}_{k})\
abla \phi^{\mathsf{T}}(\hat{x}_{k})\hat{W}_{c} \tag{44}\]  
Combining (12) with (41), the approximate Bellman equation is described as  
\[\hat{q}(t)+\hat{W}_{c}^{\mathsf{T}}\Delta\phi(x)\triangleq e_{c} \tag{45}\]  
where the reinforcement signal during the integral interval can be defined as  
\[\hat{q}(t)=\int_{t-T}^{t}\big{(}x^{\mathsf{T}}Qx+M(\hat{u}(\hat{x}_{k}))- \gamma^{2}v^{\mathsf{T}}(\hat{x}_{k})v(\hat{x}_{k})\big{)}d\tau \tag{46}\]  
In the iterative process of NN, its purpose is to minimize the value function. The gradient-descent method is used in this paper. Then, the critic network weights are adjusted online by minimizing the objective function \(E=(1/2)e_{c}^{\mathsf{T}}e_{c}\). Thus, the turning law of the critic NN weight can be expressed as  
\[ \begin{split}\dot{\hat{W}}_{c}&=-\alpha_{c}\frac{ \partial E}{\partial\hat{W}_{c}}=-\alpha_{c}\frac{\partial E}{\partial e_{c}} \frac{\partial e_{c}}{\partial\hat{W}_{c}}=-\alpha_{c}\frac{\Delta\phi}{(1+ \Delta\phi^{\mathsf{T}}\Delta\phi)^{2}}\\ &\quad\times\Big{\{}\int_{t-T}^{t}\big{(}x^{\mathsf{T}}Qx+M( \hat{u}(\hat{x}_{k}))-\gamma^{2}v^{\mathsf{T}}(\hat{x}_{k})v(\hat{x}_{k}) \big{)}d\tau+\Delta\phi^{\mathsf{T}}(x)\hat{W}_{c}\Big{\}}\end{split}  \tag{47}\]  
where \(\alpha_{c}>0\) is an adaptive learning rate and \((1+\Delta\phi^{\mathsf{T}}\Delta\phi)^{2}\) is used for normalization. Define the critic weight approximation error as \(\hat{W}_{c}=W_{c}-\hat{W}_{c}\), and the square of the denominator in (47) is used to guarantee \(\hat{W}_{c}\) is bounded.