## 3 Nonlinear State-Feedback VRFT for Approximate ORM Tracking Control Using Neural Networks
_A6_: There exists a set of nonlinear parameterized state-feedback continuously differentiable controllers \(\left\{\mathbf{C}(\mathbf{x}_{k}^{E},\mathbf{\theta})\right\}\), a \(\overset{\wedge}{\mathbf{\theta}}\) for which \(\overset{\wedge}{\mathbf{u}}_{k}=\mathbf{C}(\mathbf{x}_{k}^{E},\mathbf{\theta})\), and an \(\varepsilon>0\) for which  
\[J_{\mathrm{V}R}^{N}(\mathbf{\theta})=\sum_{k=0}^{N-1}\left\|\widetilde{ \mathbf{u}}_{k}-\mathbf{C}(\widetilde{\mathbf{x}}_{k}^{E},\mathbf{\theta}) \right\|^{2}<\varepsilon^{2}, \tag{8}\]  
\[\|\frac{\partial\mathbf{C}(\mathbf{x}_{k}^{E},\mathbf{\theta})}{\partial \mathbf{x}_{k}^{E}}\|<B_{\mathrm{c}x}, \tag{9}\]  
where \(\widetilde{\mathbf{x}}_{k}^{E}=\left[\left(\widetilde{\mathbf{x}}_{k}\right)^ {T}\left(\widetilde{\mathbf{x}}_{k}^{m}\right)^{T}\left(\widetilde{\mathbf{r} }_{k}\right)^{T}\right]^{T}\), \(\widetilde{\mathbf{x}}_{k}^{E}=\left[\left(\overset{\wedge}{\mathbf{x}}_{k} \right)^{T}\left(\widetilde{\mathbf{x}}_{k}^{m}\right)^{T}\left(\widetilde{ \mathbf{r}}_{k}\right)^{T}\right]^{T}\) and \(\widetilde{\mathbf{x}}_{k}^{m}=\left(\mathbf{s}_{x}^{m}\right)^{-1}( \widetilde{\mathbf{y}}_{k})\), \(\widetilde{\mathbf{r}}_{k-1}=\left(\mathbf{s}_{r}^{m}\right)^{-1}( \widetilde{\mathbf{y}}_{k})\). Technically, \(\left\{\overset{\wedge}{\mathbf{u}}_{k},\overset{\wedge}{\mathbf{x}}_{k}, \overset{\wedge}{\mathbf{y}}_{k}\right\}\) are generated with \(\overset{\wedge}{\mathbf{u}}_{k}=\mathbf{C}(\widetilde{\mathbf{x}}_{k}^{E}, \overset{\wedge}{\mathbf{\theta}})\) in closed-loop, by processing the virtual signals \(\widetilde{\mathbf{x}}_{k}^{-m}\), \(\widetilde{\mathbf{r}}_{k-1}\) obtained from \(\widetilde{\mathbf{y}}_{k}\). _Theorem 1_: Under assumptions _A3-A6_, there exists a finite \(B>0\) such that  
\[J_{\mathrm{MR}}^{N}(\mathbf{\theta})=\sum_{k=1}^{N}\left\|\widetilde{\mathbf{y }}_{k}-\widetilde{\mathbf{y}}_{k}\right\|^{2}=\sum_{k=1}^{N}\left\|\mathbf{s} (\widehat{\mathbf{x}}_{k^{\prime}},\widehat{\mathbf{u}}_{k-1})-\mathbf{s}^{m} (\widetilde{\mathbf{x}}_{k}^{m},\widetilde{\mathbf{r}}_{k-1})\right\|_{2}^{2}< B\varepsilon^{2}. \tag{10}\]  
Proof:: See Appendix A. _Corollary 1_. The controller \(\mathbf{C}(\mathbf{x}_{k}^{E},\overset{\wedge}{\mathbf{\theta}})\) obtained by minimizing the c.f. (6) is stabilizing and admissible for \(J_{\mathrm{MR}}^{\infty}\) in Equation (5) with \(\gamma<1\). Proof:: By Equation (8), properly identified \(\mathbf{C}(\mathbf{x}_{k}^{E},\overset{\wedge}{\mathbf{\theta}})\) renders the finite-time \(J_{\mathrm{MR}}^{N}(\overset{\wedge}{\mathbf{\theta}})\) (10) arbitrarily small. Secondly, a good exploration of \(U\times X\times Y\) ensured by \(D=\left\{\widetilde{\mathbf{u}}_{k},\widetilde{\mathbf{x}}_{k},\widetilde{ \mathbf{y}}_{k}\right\}\) reflects in good exploration of domains \(R_{m}\), \(X_{m}\) by \(\widetilde{\mathbf{r}}_{k},\widetilde{\mathbf{x}}_{k}^{m}\) respectively. In (10), \(\overset{\wedge}{\mathbf{u}}_{k},\overset{\wedge}{\mathbf{x}}_{k},\overset{ \wedge}{\mathbf{x}}_{k},\overset{\wedge}{\mathbf{y}}_{k},\overset{\wedge}{ \mathbf{x}}_{k},\overset{\wedge}{\mathbf{x}}_{k}^{E}\) are all generated from the same \(\widetilde{\mathbf{r}}_{k}\). If (10) holds for many combinations \(\widetilde{\mathbf{r}}_{k},\widetilde{\mathbf{x}}_{k}^{m}\) rendered form exploratory data, then by the arguments of continuous differentiability and bounded derivatives of the maps (7) and by assumption _A4_, they will hold for any possible combination of \(\mathbf{r}_{k}\) and \(\mathbf{x}_{k}^{m}=\mathbf{f}^{m}(\mathbf{x}_{k-1}^{m},\mathbf{r}_{k-1})\) generated from any \(\mathbf{r}_{k}\). To show this, note that both \(\widetilde{\mathbf{y}}_{k}=\mathbf{y}_{k}^{m}=\mathbf{s}^{m}(\widetilde{\mathbf{ x}}_{k}^{m},\widetilde{\mathbf{r}}_{k-1})\) and \(\overset{\wedge}{\mathbf{y}}_{k}=\mathbf{y}_{k}=\mathbf{s}(\overset{\wedge}{ \mathbf{x}}_{k},\mathbf{u}_{k-1})=\mathbf{s}(\overset{\wedge}{\mathbf{x}}  
sense of bounded output when \(\mathbf{r}_{k}\) is bounded. Then, it is an admissible one for the infinite horizon c.f. \(J^{\infty}_{MR}\) with \(\gamma<1\).