## 3 Modeling Methodology
### Solvability Analysis of the Proposed CLC Control Strategy  
MPC-based control strategies should be online feasible, and sequential solvability is one way to measure online feasibility. Specifically, it means the optimization problem (39) and (40) can find a feasible solution at each step if their initial states are feasible. The constraints in (39) and (40) consider two perspectives: safety and comfort. The following analysis proves that if the longitudinal safety distance constraints of controlled vehicles and their preceding vehicles are satisfied at the \(k\) step, we can always find a solution that satisfies the constraints of comfort to make the longitudinal safety distance constraints hold at the \(k+1\) step. Due to the different preceding vehicle switching functions and safety constraints designed in different lane change stages, we will first investigate the vehicles C and M, and vehicles B and A adopting the same preceding vehicle switching function \(LPFA(*)\) (i.e., At the LC-O stage) from Lemma 1 to Lemma 2. Then, the vehicles B and M, and vehicles M and A (i.e., at the LC-T stage) adopting the same preceding vehicle switching function \(LPFB(*)\) will be investigated from Lemma 3 to Lemma 4. Note that the mathematical method for analyzing the sequential solvability of each controlled vehicle and its preceding vehicle are similar. However, due to the difference in the constraints' mathematical format (such as preceding vehicle switching function \(LPFA(*)\) and \(LPFB(*)\)), the investigation processes will be present one by one as each controlled vehicle and its preceding vehicle in two stages. **Lemma 1. **_Given \(S_{CM}(k)\geq S_{safe,CM}(k)LPFA(k)\) is satisfied at step, a feasible solution \(a_{M}(k)\ =\ a_{min}\) can make \(S_{CM}(k+1)\geq S_{safe,CM}(k+1)LPFA(k+1)\) hold, if the speed difference between vehicles C and M satisfies \(v_{C}(k)-v_{M}(k)\geq a_{min}dt/2\) for any step \((k=1,\ 2,\ \ldots,\ N)\)._  
**Proof. ** Since \(LPFA(*)\) is a segmented function, it needs to be discussed separately. When \(0\leq y_{M}(k)\leq d/2\) and because all critical longitudinal safety distances is a fixed value \(d_{0}\), the following relationship exists. \[S_{CM}(k)\geq S_{safe,CM}(k)LPFA(k)=d_{0}L_{1} \tag{41}\]  
The above Equation (41) can be rewritten in the following form. \[S_{CM}(k)-S_{safe,CM}(k)LPFA(k)=S_{CM}(k)-d_{0}L_{1}\geq 0 \tag{42}\]  
By substituting (4) into (42), the following inequality (43) can be obtained. \[ \begin{array}{ll}S_{CM}(k+1)-&S_{safe,CM}(k+1)LPFA(k+1)=S_{CM}(k+1)-d_{0}L_{ 1}\\ &=S_{CM}(k)+[v_{C}(k)-v_{M}(k)]dt-a_{M}(k)dt^{2}/2-d_{0}L_{1}\\ &\geq[v_{C}(k)-v_{M}(k)]dt-a_{M}(k)dt^{2}/2\end{array}  \tag{43}\]  
Let \(a_{M}(k)=a_{min}\) and if the speed difference between vehicles C and M satisfies \(v_{C}(k)-v_{M}(k)\geq a_{min}dt/2\) at the same time, the following inequality (44) will be held after multi-step amplification. \[ \begin{array}{ll}S_{CM}(k+1)-&S_{safe,CM}(k+1)LPFA(k+1)\\ &\geq[v_{C}(k)-v_{M}(k)]dt-a_{M}(k)dt^{2}/2\\ &=a_{min}dt^{2}/2-a_{min}dt^{2}/2=0\end{array}  \tag{44}\]  
Thus, when \(0\leq y_{M}(k)\leq d/2\), the longitudinal distance constraint \(S_{CM}(k+1)\geq S_{safe,CM}(k+1)LPFA(k+1)\) will be satisfied at \(k+1\) step. When \(y_{M}(k)>d/2\), the following relationship exists. \[S_{CM}(k)\geq S_{safe,CM}(k)LPFA(k)=d_{0}LPFA(k) \tag{45}\]  
Equation (45) can also be rewritten as following; Equation (46). \[S_{CM}(k)-S_{safe,CM}(k)LPFA(k)=S_{CM}(k)-d_{0}LPFA(k)\geq 0 \tag{46}\]  
When \(y_{M}(k)>d/2\), the function \(LPFA(*)\) is monotonically decreasing, so we have \(LPFA(k+1)<LPFA(k)\), then by substituting (4) and (46), the following inequality (47) can be obtained. \[ \begin{array}{ll}S_{CM}(k+1)-&S_{safe,CM}(k+1)LPFA(k+1)=S_{CM}(k+1)-d_{0} LPFA(k+1)\\ &>S_{CM}(k+1)-d_{0}LPFA(k)\\ &=S_{CM}(k)+[v_{C}(k)-v_{M}(k)]dt-a_{M}(k)dt^{2}/2-d_{0}LPFA(k)\\ &\geq[v_{C}(k)-v_{M}(k)]dt-a_{M}(k)dt^{2}/2\end{array}