## 3 Methodology
### Distributed Cooperative Control Model for AVs
Accordingly, a smaller value of \(\mathcal{J}_{3}\) indicates that the speed of the vehicle is closer to the desired speed, which benefits smoother traffic flows and higher throughputs. Third, we expect to reduce the fuel consumption of each AV by optimizing the local control cost. It is commonly believed that a lower cost of control inputs results in a higher fuel economy [30]. Hence, the index of the control cost is defined as follows  
\[\mathcal{J}_{i4}=\sum_{k=0}^{N_{p}-1}||u_{i}(k|t)||^{2}. \tag{8}\]  
Combining Equations (3), (4), (6) and (8), we define an objective function with a weighted sum of the four indexes  
\[\mathcal{J}_{i}=\omega_{1}\mathcal{J}_{i1}+\omega_{2}\mathcal{J}_{i2}+\omega_ {3}\mathcal{J}_{i3}+\omega_{4}\mathcal{J}_{i4} \tag{9}\]  
where \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\), and \(\omega_{4}\) are weight factors to penalize the virtual position errors, relative speed errors, variances of the desired speed, and control cost, respectively. These weight coefficients can be adjusted according to actual traffic demand in practice. (2) _Constraints_: In what follows, we consider the constraints of the mobility characteristics of vehicles and the safety condition. The vehicles have to follow the car-following models according to their type, i.e., AVs or MVs. The trajectory of ego vehicle \(i\), \(i\in\Omega_{A}(t)\), is subject to satisfying the following constraints:  
\[\left\{ \begin{aligned} \bar{p}_{ih}(k+1|t)&=\bar{p}_{ih}(k|t)- \delta\bar{\sigma}_{ih}(k|t)-\frac{\delta^{2}}{2}u_{i}(k|t)\\ \bar{\sigma}_{ih}(k+1|t)&=\bar{\sigma}_{ih}(k|t)+ \delta u_{i}(k|t),i\in\Omega_{A}(t),h\in\mathcal{L}\end{aligned} \right. \tag{10}\]  
\[\left\{ \begin{aligned} \bar{p}_{lf}(k+1|t)&=\bar{p}_{lf}(k|t)- \delta\bar{\sigma}_{lf}(k|t)\\ \bar{\sigma}_{lf}(k+1|t)&=\mathcal{A}_{a}\{\min\{v_{ jl}/(k|t)+a_{max},\gamma_{max},v_{pf}(k|t)+\frac{g-v_{lf}(k|t)\tau}{(v_{pf}(k|t)+v_{ jl}(k|t))/2b_{max}+\tau}\},\\ &(1-\epsilon)\min\{v_{jl}/(k|t)+a_{max},\gamma_{max},v_{pf}(k|t) +\frac{g-v_{lf}(k|t)\tau}{(v_{pf}(k|t)+v_{jl}(k|t))/2b_{max}+\tau}\}\\ &\quad+\epsilon(v_{if}(k|t)-a_{max})\},\;j\in\mathcal{N}_{i}(t),f\in\mathcal{C}_{h}\end{aligned} \right. \tag{11}\]  
In addition, the trajectory of neighboring vehicle \(j\) is subject to either of the constraints (10) and (11) depending on its vehicle type, i.e., AV or MV, respectively. In Equation (11), \(\mathcal{R}_{a}\) is defined as a random selection function, \(v_{pf}\) denotes the speed of the immediately preceding vehicle, and the definitions of other parameters are given in Algorithm 1. Considering the physical limitations of vehicles, road restrictions, and comfort of passengers, the speed and acceleration of AVs should be confined to a range. Denote \(\mathcal{A}_{min}\), \(\mathcal{A}_{max}\), \(\mathcal{V}_{min}\), and \(\mathcal{V}_{max}\) as the minimum acceleration, maximum acceleration, minimum speed, and maximum speed. The following constraints should be satisfied for each AV:  
\[\mathcal{A}_{min}\leq u_{i}(k|t)\leq\mathcal{A}_{max}. \tag{12}\]  
\[\mathcal{V}_{min}\leq v_{ih}(k+1|t)\leq\mathcal{V}_{max}. \tag{13}\]  
To avoid rear-end collisions of vehicles on the same lane, we add a constraint to guarantee that vehicle \(i\) stays a safe distance from its immediately preceding vehicle \(q\). Denote \(d_{m}\) as the minimum gap, and \(l_{q}\) is the length of vehicle \(q\). The constraint of the safety condition is defined as follows:  
\[\bar{p}_{ih}(k+1|t)-\bar{p}_{qh}(k+1|t)\geq d_{m}+l_{q}. \tag{14}\](3) _Problem_: We aim to minimize the objective function (9) of all AVs in the CZ in a prediction horizon while guaranteeing that each AV satisfies the constraints (12)-(14).