## 4 Control Design
### ADP Trajectory Control
Due to  
\[I_{s2d}\ =\ g^{+}(x_{d})\big{(}\dot{x}_{d}-f(x_{d})\big{)} \tag{37}\]3  
we obtain  
\[I_{s2d}\ =\ g^{+}(x_{d})\big{(}\dot{x}_{d}-f(x_{d})\big{)} \tag{37}\]4  
Therefore, the error update rate of the weight estimation is  
\[I_{s2d}\ =\ g^{+}(x_{d})\big{(}\dot{x}_{d}-f(x_{d})\big{)} \tag{37}\]5  
The ideal optimal feedback control law and the corresponding iterative control law are  
\[ \begin{array}{rcl}I_{s2v}\ =\ -\frac{1}{2}R^{-1}g^{\mathrm{T}}(x)((\
abla \sigma_{\tau}(e))^{\mathrm{T}}w_{\tau}+\
abla\varepsilon_{\tau})\\ I_{s2v}\ =\ -\frac{1}{2}R^{-1}g^{\mathrm{T}}(x)(\
abla\sigma_{\tau}(e))^{\mathrm{T}} \hat{w}_{\tau}\end{array}  \tag{58}\]  
**Assumption 4**. : _The desired control law \(I_{s2d}\) and \(\eta\) both have unknown upper bounds, which are_  
\[\|I_{s2d}\|\leq w_{ud}\qquad\|\eta\|\leq\eta_{M} \tag{59}\]  
**Theorem 1**. : _Under the conditions of Assumptions 3 and 4, if the solution of the neural network-based_ HJB _equation exists, considering the state space model (30) in SSS and the evaluation network weight update rate (57), if the optimal control law of the system trajectory tracking is chosen as_  
\[I_{s2}\ =\ I_{s2d}+\hat{I}_{s2v}\ =\ g^{+}(x_{d})\big{(}\dot{x}_{d}-f(x_{d}) \big{)}-\frac{1}{2}R^{-1}g^{\mathrm{T}}(x)(\
abla\sigma_{\tau}(e))^{\mathrm{T }}\hat{w}_{\tau} \tag{60}\]  
_this ensures that both the weight approximation error and the system trajectory tracking error are eventually consistent and bounded._  
Proof. : Considering the Lyapunov theory, we define a positive definite energy function  
\[V_{1}\ =\ \frac{1}{2}e^{\mathrm{T}}e+J^{*}(e)+\frac{1}{2\alpha_{\tau}} \widetilde{w}_{\tau}^{\mathrm{T}}\widetilde{w}_{\tau} \tag{61}\]  
Derivative of time, we obtain  
\[ \begin{array}{rcl}\dot{V}_{1}&=&\frac{1}{\alpha_{\tau}} \widetilde{w}_{\tau}^{\mathrm{T}}\dot{\widetilde{w}}_{\tau}+(\
abla J^{*}(e))^ {\mathrm{T}}\dot{e}+e^{\mathrm{T}}\dot{e}\\ &=&\frac{1}{\alpha_{\tau}}\widetilde{w}_{\tau}^{\mathrm{T}}\dot{ \widetilde{w}}_{\tau}+(\
abla J^{*}(e))^{\mathrm{T}}\dot{e}+e^{\mathrm{T}}[f_{ e}+[g(x)-g(x_{d})]I_{s2}+g(x_{d})I_{s2v}]\\ &=&\frac{1}{\alpha_{\tau}}\widetilde{w}_{\tau}^{\mathrm{T}}\dot{ \widetilde{w}}_{\tau}-e^{\mathrm{T}}Qe-I_{s2v}^{\mathrm{T}}RI_{s2v}+e^{ \mathrm{T}}[f_{e}+[g(x)-g(x_{d})]I_{s2}+g(x_{d})I_{s2v}]\end{array}  \tag{62}\]  
Because \(f(x)\) is a Lipschitz function, there must exist \(L_{f}>0\). When \(L_{f}>0\), it holds \(\|f_{e}\|\leq L_{f}\|e\|\). According to assumption 1, we know that \(g(x)\) and \(g(x_{d})\) are bounded, so we can set  
\[\|g(x)\|\leq w_{g},\ \|g(x_{d})\|\leq w_{gd} \tag{63}\]and then  
\[\|g(x)-g(x_{d})\|\leq\Delta w_{g} \tag{64}\]  
Using the trigonometric inequality, we have  
\[ \begin{split}\dot{V}_{1}&\leq L_{f}\|e\|^{2}+\Delta w_{ g}\|l_{2}\|\|e\|+w_{gd}\|l_{2 zv}\|e\|e-e^{T}Qee-I_{2 zv}^{T}Rl_{2 zv}+\frac{1}{\pi_{ \tau}}\tilde{v}_{\tau}^{T}\tilde{w}_{\tau}\\ &\leq L_{f}\|e\|^{2}+\Delta w_{g}\|l_{2 zd}+l_{2 zv}\|\|e\|+w_{gd}\|l _{2 zv}\|e\|-e^{T}Qe-I_{2 zv}^{T}Rl_{2 zv}+\tilde{v}_{\tau}^{T}(e_{pH}-\tilde{w}_ {\tau}^{T}\eta)\eta\\ &\leq L_{f}\|e\|^{2}+\Delta w_{g}\|l_{2 zd}\|\|e\|+\frac{1}{2} \Delta w_{g}^{2}\|l_{2 zv}\|^{2}+\frac{1}{2}\|e\|^{2}+\tilde{w}_{\tau}^{T}e_{pH} \eta\\ &+\frac{1}{2}w_{gd}^{2}\|e\|^{2}-\left\|\tilde{w}_{\tau}^{T}\eta \right\|^{2}+\frac{1}{2}\|L_{2v}\|^{2}-\lambda_{\min}(R)\|L_{2v}\|^{2}-\lambda _{\min}(\Omega)\|e\|^{2}\\ &\leq-\left[\left(\lambda_{\min}(Q)-L_{f}-\frac{1}{2}w_{gd}^{2}- \frac{1}{2}\right)\|e\|-\Delta w_{g}\|l_{2 zd}\right]\|e\|\\ &-\left(\lambda_{\min}(R)-\frac{1}{2}\Delta w_{g}^{2}-\frac{1}{2} \right)\|L_{2v}\|^{2}+\frac{1}{2}\tilde{e}_{PH}^{2}-\frac{1}{2}\left\|\tilde{ v}_{\tau}^{T}\eta\right\|^{2}\\ &\leq-\left[\left(\lambda_{\min}(Q)-L_{f}-\frac{1}{2}w_{gd}^{2}- \frac{1}{2}\right)\|e\|-\Delta w_{g}w_{ud}\right]\|e\|\\ &-\left(\lambda_{\min}(R)-\frac{1}{2}\Delta w_{g}^{2}-\frac{1}{2} \right)\|L_{2v}\|^{2}+\frac{1}{2}\tilde{e}_{PH}^{2}-\frac{1}{2}\left\|\tilde{ w}_{\tau}^{T}\eta\right\|^{2}\end{split}  \tag{65}\]  
Clearly, in the collections  
\[