## 4 Main Contribution of the Control Design
### The Control Challenge
\) can be obtained from the evolution shown in (9), which yields  
\[\frac{d}{dt}\Delta_{t}=A\Delta+Bh\bigg{(}\Phi_{t}^{*},\frac{d}{dt}\Phi_{t}^{*} \bigg{)}-BF\bigg{(}\Phi_{t},\frac{d}{dt}\Phi_{t}\bigg{)}-BG(\Phi_{t})\tau_{t}- B\tilde{\tau}\bigg{(}\Phi_{t},\frac{d}{dt}\Phi_{t},t\bigg{)}. \tag{15}\]  
Using the selected controller \(\tau\) in (10) and considering that functions \(F\), \(G\), and \(h\) are known, the application of the proposed controller in (10) simplifies the dynamics of \(\Delta_{t}\) to  
\[\frac{d}{dt}\Delta_{t}=(A+K_{t})\Delta_{t}-B\tilde{\tau}\bigg{(}\Phi_{t}, \frac{d}{dt}\Phi_{t},t\bigg{)}. \tag{16}\]  
Consider the following time-dependent Lyapunov candidate function related to the acceleration tracking error as well as the state-dependent gains used to insert the position and velocity restrictions:  
\[V\big{(}\Delta,\tilde{K},t\big{)}=\ln\left(\frac{\Delta^{+}}{\Delta^{+}- \left[\left\|\Delta\right\|_{P}^{2}-\epsilon\right]_{+}^{2}}\right)+\gamma_{t }\cdot tr\Big{\{}\tilde{K}^{\top}\tilde{K}\Big{\}}. \tag{17}\]  
According to the Laypunov stability theory for non-autonomous systems [25], it is necessary to prove the following properties for the proposed Lyapunov candidate function \(V\big{(}\Delta,\tilde{K},t\big{)}\): (a) \(W_{1}\big{(}\Delta,\tilde{K}\big{)}\leq V\big{(}\Delta,\tilde{K},t\big{)}\leq W _{2}\big{(}\Delta,\tilde{K}\big{)}\), with \(W_{1}\) and \(W_{2}\) being continuous positive definite functions; and (b) the time derivative of \(V\big{(}\Delta,\tilde{K},t\big{)}\) must be negative definite. For this case, \(W_{1}=\lambda_{min}\{P\}\|\Delta\|+\min_{t}\{\gamma_{t}\}\cdot tr\{\bar{K}^{\top} \bar{K}\}\) and \(W_{2}=\lambda_{max}\{P\}\|\Delta\|+tr\{\bar{K}^{\top}\bar{K}\}\). The time derivative of \(V\big{(}\Delta,\bar{K},t\big{)}\) satisfies the following differential equation:  
\[ \begin{split}\frac{d}{dt}V\big{(}\Delta_{t},\bar{K}_{t},t\big{)}& =2\frac{\left[\|\Delta_{t}\|_{P}^{2}-\epsilon\right]_{+}}{\Delta^{+}- \left[\|\Delta_{t}\|_{P}^{2}-\epsilon\right]_{+}^{2}}\Delta^{\top}P\frac{d}{ dt}\Delta_{t}+\\ tr\bigg{\{}\bar{K}_{t}^{\top}\bigg{(}\bigg{(}\frac{d}{dt}\gamma _{t}\bigg{)}\bar{K}_{t}+2\gamma_{t}\frac{d}{dt}\bar{K}_{t}\bigg{)}\bigg{\}}. \end{split}  \tag{18}\]  
The term \(2\Delta_{t}^{\top}P\frac{d}{dt}\Delta_{t}\) admits the following equivalent form:  
\[ \begin{split} 2\Delta_{t}^{\top}P\frac{d}{dt}\Delta_{t}& =\Delta_{t}^{\top}\big{(}PA_{K}+A_{K}^{\top}P\big{)}\Delta_{t}+2 \Delta_{t}^{\top}P\bar{K}_{t}\Delta_{t}-\\ & 2\Delta_{t}^{\top}PB\bar{\xi}\bigg{(}\Phi_{t},\frac{d}{dt}\Phi_{t},t \bigg{)}.\end{split}  \tag{19}\]  
The application of the [PERSON]'s inequality (that is, for any matrices \(X\in\mathbb{R}^{n_{1}\times m_{1}}\), \(Y\in\mathbb{R}^{n_{1}\times m_{1}}\) there exists a positive definite matrix \(\Lambda\in\mathbb{R}^{n_{1}\times n_{1}}\) such that \(X^{\top}Y+Y^{\top}X\leq X^{\top}\Lambda^{-1}X+Y^{\top}\Lambda\gamma\)) to the term containing uncertainties and perturbations (\(2\Delta_{t}^{\top}PB\bar{\xi}(\Phi_{t},\frac{d}{dt}\Phi_{t},t)\)) in (19) leads to the following form for the right-hand side of \(2\Delta^{\top}P\frac{d}{dt}\Delta\):  
\[ \begin{split} 2\Delta_{t}^{\top}P\frac{d}{dt}\Delta_{t}& \leq\Delta_{t}^{\top}\big{(}PA_{K}+A_{K}^{\top}P\big{)}\Delta_{t}+2 \Delta_{t}^{\top}PB\bar{K}_{t}\Delta_{t}+\\ \Delta_{t}^{\top}PB\Lambda_{1}^{-1}B^{\top}P\Delta_{t}+\left\| \xi\bigg{(}\Phi_{t},\frac{d}{dt}\Phi_{t},t\bigg{)}\right\|_{\Lambda_{1}}.\end{split}  \tag{20}\]  
Using the results corresponding to the upper bound of \(\left\|\bar{\xi}\bigg{(}\Phi_{t},\frac{d}{dt}\Phi_{t},t\bigg{)}\right\|\) leads to  
\[ \begin{split} 2\Delta_{t}^{\top}P\frac{d}{dt}\Delta_{t}& \leq\Delta_{t}^{\top}\big{(}PA_{K}+A_{K}^{\top}P\big{)}\Delta_{t}+2 \Delta_{t}^{\top}PB\bar{K}_{t}\Delta_{t}+\\ \Delta_{t}^{\top}PB\Lambda_{1}^{-1}B^{\top}P\Delta_{t}& +\lambda_{max}\{\Lambda_{2}\}\xi^{0}.\end{split}