## 3 Methodology
### Distributed Cooperative Control Model for AVs
To this end, we construct a distributed multi-objective optimization model as follows:  
\[\min_{u_{i}(0|t),\ldots,u_{i}(N_{p}-1|t)}\mathcal{J}_{i}(\bar{p}_{\mathit{alt}}, \bar{v}_{\mathit{alt}},\bar{p}_{\mathit{jf}},\bar{v}_{\mathit{jf}},u_{i}) \tag{15}\]  
subject to Equations (10)-(14), where the optimal control inputs of AVs are computed by minimizing the cost metric \(\mathcal{J}_{i}\) at each time step. In the proposed distributed optimization control model, the performance index (9) and the inequality constraints in (12)-(14) are convex functions with respect to the decision variable \(u_{i}(k|t)\). The equality constraint in (10) is an affine function of \(u_{i}(k|t)\). Hence, the problem in (15) is a convex optimization problem. In addition, note that the feasible domain of the decision variable \(u_{i}(k|t)\) is non-empty and the performance index function (9) is a coercive function during each time step. According to the Weierstrass lemma [31], feasible solutions to the optimization problem (15) can be always found in the feasible domain. When analyzing the driving behavior of an MV approaching an intersection, it is important to consider two different situations. In the first situation, when the MV is not the leading vehicle in its lane, the Krauss model can be used to predict its trajectory. However, in the second situation, when the MV is the leading vehicle in its lane, its behavior is affected by various real-world factors such as visibility and the positions of conflicting vehicles in other lanes. To simplify this problem while characterizing the driving behavior of leading MVs, we propose using the Krauss model to simulate a virtual car-following behavior for the leading MV at the intersection. This involves assuming that a neighboring vehicle \(j\) of the leading MV \(m\) is closer to the intersection and can be treated as the virtual preceding vehicle of the leading MV. The virtual gap between vehicles \(m\) and \(j\) can be expressed as  
\[g_{\mathit{mj}}=\bar{p}_{\mathit{mh}}-\bar{p}_{\mathit{jf}}-l_{j}-d_{\mathit{mj }}+c_{\mathit{mj}}, \tag{16}\]  
where the variables and parameters are defined in accordance with Equations (3) and (5). The gap \(g\) in the Krauss model for the leading MV \(m\) can be obtained as  
\[g=\min_{j}g_{\mathit{mj}}. \tag{17}\]  
In situations where there is no virtual preceding vehicle, the leading MV will accelerate to its maximum velocity, using maximum acceleration, and continue to cross the intersection. The idea of the proposed coordination scheme in this paper is inspired by the concept of consensus control of multi-agent systems [11, 18, 21, 32]. Each AV is regulated to cross the intersection while staying a desired virtual safe distance from its neighboring vehicles in conflicting lanes while improving traffic mobility and fuel economy. In other words, each AV tends to reach a consensus with its neighboring vehicles by solving the local optimization problem (15). Therefore, through information sharing, the ego AV and its neighboring vehicles (if they are AVs) cooperatively optimize their trajectories, which pushes the distributed solutions towards the global optimality. The current studies on vehicle consensus coordination at signal-free intersections, such as [11, 18], are subject to a fully connected AV environment. All the vehicles must be controllable to achieve the objectives. The scheme presented in this paper can coordinate mixed-automated vehicles to cross the signal-free intersection. When potential conflicts occur between AVs and MVs, AVs use the predicted future trajectories of MVs to optimize their movements to avoid collisions. Note that if two or more MVs with conflict relationships approach the intersection simultaneously, they may slow down or even stop for safe crossing, resulting in a decrease in traffic efficiency. To alleviate this situation, we expect to apply our scheme under high penetration rates of AVs for potential future implementation.