## 3 Main Results
### Algorithm Feasibility and Stability Analysis  
In the designed prescribed performance adaptive cruise control, the solution \((z,v)\) of Equation (1) is limited by the open set \(\mathcal{D}\). For a fixed \(k>0\), the set \(\{(z,v)|(k,z,v)\in\mathcal{D}\}\subseteq\mathbb{R}^{2}\) is nonconvex, as shown in Figure 2. Therefore, the following proof method is provided, and the specific conclusion is shown in Theorem 1. **Theorem 1**. : _For the vehicle dynamics system described in Equation (1), if the desired running speed of the vehicle \(v_{\text{ref}}(k)\) and the leader vehicle's positions \(z^{L}\) and \(\dot{z}^{L}\) are bounded, substituting the controller (14) into the vehicle dynamics system (1) can obtain a solved closed-loop system. Each solution of the closed-loop system can be generalized to a maximum solution \((z,v):[0,w)\rightarrow\mathbb{R}^{2}\), \(w\in(0,\infty]\), and the solution has the following properties:_  
* _The speed_ \(v(k)\) _and control input_ \(F(k)\) _are bounded;_
* _There exists_ \(\varepsilon>0\)_, for all_ \(k>0\)_, if the following conditions are satisfied._  
\[\epsilon_{v}(k)\leq\phi_{v}(k)^{-1}-\varepsilon,\epsilon_{d}(k)\leq\phi_{d}(k )^{-1}-\varepsilon,\]  
\[\varepsilon\leq\left\{\max\Bigl{\{}0,\phi_{v}(k)^{-1}+\epsilon_{v}(k)\Bigr{\}} +\max\Bigl{\{}0,\phi_{d}(k)^{-1}+\epsilon_{d}(k)\Bigr{\}}\right\}\]  
Proof. : Step 1: Explanation of the existence of the maximum solution. Definition:  
\[\psi_{v}(t):=\phi_{v}(k)^{-1},\psi_{d}(k):=\phi_{d}(t)^{-1}\] \[f:\mathbb{R}_{\geq 0}\times\mathbb{R}\rightarrow\mathbb{R},(k,v) \mapsto-\frac{1}{m}\left(R_{i}(k)+R_{\text{w}}(k,v)+R_{f}(v)-\delta(k)\right)\] \[F:\mathcal{D}\rightarrow\mathbb{R}^{2},(k,z,v)\mapsto\] \[\left\{ \begin{array}{l}\left(\begin{array}{c}v\\ -\frac{1}{m}\frac{v-v_{\text{ref}}(k)}{1-\phi_{v}(k)^{2}|v-v_{\text{ref}}(k)|^{ 2}}+f(k,v)\\ \end{array}\right),\text{if}(k,z,v)\in\mathcal{D}_{v}\\ \left(\begin{array}{c}-\frac{1}{m}\frac{v-v_{\text{ref}}(k)}{1-\phi_{v}(k) ^{2}|v-v_{\text{ref}}(k)|^{2}}-\frac{1}{m}\frac{z(k)-z^{L}(k)+d_{\text{g}}+ \text{ln}(k)+\psi_{d}(t)}{1-\psi_{d}(k)^{2}|z(k)-z^{L}(k)+d_{\text{g}}+\text{ ln}(k)+\psi_{d}(k)|^{2}}+f(k,v)\\ \end{array}\right),\text{if}(k,z,v)\in\mathcal{D}_{d}\\ \left(\begin{array}{c}\frac{1}{m}\min\left\{\begin{array}{c}-\frac{v-v_{ \text{ref}}(k)}{1-\phi_{v}(k)^{2}|v-v_{\text{ref}}(k)|^{2}},\\ -\frac{1}{m}\frac{v-v_{\text{ref}}(k)}{1-\phi_{v}(k)^{2}|v-v_{\text{ref}}(k)|^{ 2}}-\frac{1}{m}\frac{z(k)-z^{L}(k)+d_{\text{g}}+\text{ln}(k)+\psi_{d}(k)}{1- \psi_{d}(k)^{2}|z(t)-z^{L}(t)+d_{\text{g}}+\text{ln}(k)+\psi_{d}(k)|^{2}}\\ \end{array}\right\}+f(k,v)\\ \text{if}(k,z,v)\in\mathcal{D}_{\overline{g}}^{d}\end{array}\right\}+f(k,v) \end{array} \right), \tag{15}\]  
* If \(\epsilon_{v}(k)>0\) and \(\epsilon_{d}(k)>0\), there exists \(\psi_{v}(k)-\epsilon_{v}(k)\geq\epsilon\) and \(\psi_{d}(k)-\epsilon_{d}(k)\geq\epsilon\) such that \(\kappa_{v}(k)\leq\frac{1}{\epsilon\phi_{v}(k)}\) and \(\kappa_{d}^{1}(k)\leq\frac{1}{\epsilon\phi_{d}(k)}\); then, \(-\kappa_{d}^{1}(k)\epsilon_{d}(k)-\kappa_{d}^{2}(k)\epsilon_{v}(k)\geq-\frac{1 }{\epsilon\phi_{v}(k)^{2}}-\frac{1}{\epsilon\phi_{v}(k)^{2}}\), \(-\kappa_{v}(k)\epsilon_{v}(k)\geq-\frac{1}{\epsilon\phi_{v}(k)^{2}}\). Furthermore, \[0\geq F(k)=\min\Bigl{\{}-\kappa_{v}(k)\epsilon_{v}(k),-\kappa_{d}^{1}(k) \epsilon_{d}(k)-\kappa_{d}^{2}(k)\epsilon_{v}(k)\Bigr{\}}\geq-\frac{1}{ \epsilon\phi_{d}(k)^{2}}-\frac{1}{\epsilon\phi_{v}(k)^{2}}\geq C_{k_{0}}\] Therefore, we have \(F(k)\leq C_{k_{0}}\). * If \(\epsilon_{v}(k)<0\) and \(\epsilon_{d}(k)<0\), there exists \(\epsilon\leq\max\{0,\psi_{v}(k)+\epsilon_{v}(k)\}+\max\{0,\psi_{d}(k)+\epsilon _{d}(k)\}\). When \((k,z,v)\in\mathcal{D}_{v}^{d}\), we have \(\max\{\psi_{v}(k)+\epsilon_{v}(k),\psi_{d}(k)+\epsilon_{d}(k)\}\geq\frac{ \epsilon}{2}\). Thus, we have \(\psi_{v}(k)+\epsilon_{v}(k)\geq\frac{\epsilon}{2}\) or \(\psi_{d}(k)+\epsilon_{d}(k)\geq\frac{\epsilon}{2}\).