## 3 System Description
### Traffic State Dynamics  
In a mixed traffic flow environment, it is assumed that the real-time positions and speeds of every vehicle can be obtained. Traffic states of the roadway stretch are represented by the equations of motion at discrete time steps \(T\) indexed by \(k\), where (actual) time \(t=kT\). This discrete-time system calls for using constantly accelerated motion law for the interval \((k,k+1)\) (or \((t,t+T]\)), which is described by  
\[x_{i}(k+1)=x_{i}(k)+Tv_{i}(k)+\frac{1}{2}T^{2}a_{i}(k) \tag{10}\]  
\[v_{i}(k+1)=v_{i}(k)+Ta_{i}(k) \tag{11}\]  
Substituting the discrete-time form of (4) into (10) and (11), the following formulae are derived:  
\[x_{i}(k+1)=x_{i}(k)+Tv_{i}(k)+\frac{1}{2}T^{2}[\lambda_{x}(x_{i-1}(k)-x_{i}(k) -(d+hv_{i}(k)))+\lambda_{v}(v_{i-1}(k)-v_{i}(k))] \tag{12}\]  
\[v_{i}(k+1)=v_{i}(k)+T[\lambda_{x}(x_{i-1}(k)-x_{i}(k)-(d+hv_{i}(k)))+\lambda_ {v}(v_{i-1}(k)-v_{i}(k))] \tag{13}\]  
which, using matrix form, results in  
\[ \begin{bmatrix}x_{i}(k+1)\\ v_{i}(k+1)\end{bmatrix} = \begin{bmatrix}1-\frac{1}{2}T^{2}\lambda_{x}&T-\frac{ 1}{2}T^{2}\lambda_{x}h-\frac{1}{2}T^{2}\lambda_{v}\\ -T\lambda_{x}&1-T\lambda_{x}h-T\lambda_{v}\end{bmatrix}\begin{bmatrix}x_{i}(k )\\ v_{i}(k)\end{bmatrix} + \begin{bmatrix}\frac{1}{2}T^{2}\lambda_{x}&\frac{1}{2}T^{ 2}\lambda_{v}\\ T\lambda_{x}&T\lambda_{v}\end{bmatrix}\begin{bmatrix}x_{i-1}(k)\\ v_{i-1}(k)\end{bmatrix} + \begin{bmatrix}-\frac{1}{2}T^{2}\lambda_{x}\\ -T\lambda_{x}\end{bmatrix} d \tag{14}\]  
In the discrete system, \(X_{i}(k)\in\mathbb{R}^{2}\) the state vector of vehicle \(i\) at time step \(k\) is defined as:  
\[X_{i}(k)=[x_{i}(k),v_{i}(k)]^{T} \tag{15}\]where \(x_{i}\) and \(v_{i}\) are the position and speed of vehicle \(i\), respectively. Hence, the matrix (14) can be equivalently rewritten as:  
\[X_{i}(k+1)=\mathrm{a}_{n,1}X_{i-1}(k+1)+\mathrm{a}_{n,2}X_{i}(k+1)+b_{n,1}d \tag{16}\]  
Additionally, \(\mathrm{a}_{n,1}\) and \(\mathrm{a}_{n,2}\in\mathbb{R}^{2\times 2}(\mathbb{R})\) are the following vectors:  
\[\mathrm{a}_{n,1}= \begin{bmatrix}\frac{1}{2}T^{2}\lambda_{x}&\frac{1}{2}T^{2} \lambda_{v}\\ T\lambda_{x}&T\lambda_{v}\end{bmatrix} ,\ \mathrm{a}_{n,2}= \begin{bmatrix}1-\frac{1}{2}T^{2} \lambda_{x}&T-\frac{1}{2}T^{2}\lambda_{x}h-\frac{1}{2}T^{2}\lambda_{v}\\ -T\lambda_{x}&1-T\lambda_{x}h-T\lambda_{v}\end{bmatrix}  \tag{17}\]  
and \(b_{n,1}\in\mathbb{R}^{2}\) is represented as:  
\[b_{n,1}= \begin{bmatrix}-\frac{1}{2}T^{2}\lambda_{x}\\ -T\lambda_{x}\end{bmatrix}  \tag{18}\]  
From (5) and (15), we can obtain the following vector form:  
\[\dot{X}(k)=\left[\dot{x}_{i}(k),\dot{v}_{i}(k)\right]^{T}=\left[v_{i}(k),a_{i} (k)\right]^{T} \tag{19}\]  
In the optimal control scheme, the state-space system model can be formulated as follows:  
\[X(k+1)=AX(k)+BU(k) \tag{20}\]  
\[Y(k)=CX(k) \tag{21}\]  
where the state vector \(X(k)\) is as depicted in (15), and the control input vector \(U(k)\) is formed by the acceleration rate \(a\) and the minimum distance \(d\). Then, \(\left(X(k),U(k)\in\mathbb{R}^{2N}\right)\) can be expressed as:  
\[X(k)= \begin{bmatrix}X_{1}\\ X_{2}\\ \vdots\\ X_{N}\end{bmatrix} (k)= \begin{bmatrix}x_{1}\\ v_{1}\\ x_{2}\\ v_{2}\\ \vdots\\ x_{N}\end{bmatrix} (k),\ U(k)= \begin{bmatrix}a(k)\\ d\end{bmatrix}  \tag{22}\]  
where \(N\) represents the total number of vehicles, \(m\) signifies the number of vehicles implementing the JAD strategy, \(d\) denotes the minimum distance, and \(n\) is the number of human-driven vehicles (HDVs). The state matrix \(A\in\mathbb{R}^{2N\times 2N}\) is  
\[A=\left[ \begin{array}{cccccccc}a_{1,0}&&&&&&&\\ 0&a_{2,0}&&&&\\ &\ddots&\ddots&&&&\\ &&0&a_{m,0}&&&&\\ &&&a_{1,1}&a_{1,2}&&\\ &&&\ddots&\ddots&\\ &&&&a_{n,1}&a_{n,2}\end{array} \right]\,\ a_{m,0}= \begin{bmatrix}1&T\\ 0&1\end{bmatrix}  \tag{23}\]  
where \(a_{n,1}\) and \(a_{n,2}\) are given by (17), \(\mathrm{a}_{m,0}\in\mathbb{R}^{2\times 2}\) is a vector for the JAD strategy, and the input matrix \(B\in\mathbb{R}^{2N\times 2N}\) is \[\text{B}=