# Non-Mypopic Beam Scheduling for Multiple Smart-Target Tracking in Phased Array Radar Networks
## 2 Model Description and Problem Formulation
### A US-Based MP Index for Multi-Dimensional State Targets  
In the case of target dynamic states moving over in an \(L\)-dimensional state space \(\mathbb{R}^{L}\), with \(L\geq 2\), the TEC matrix indicates the tracking accuracy of the target state's components, e.g., location and velocity. Since the TEC state is an \(L\times L\) matrix, a convenient scalar measure of tracking performance is given by its trace. This section leverages the past success of the PCL-indexability and extends a new heuristic approach to define a US-MP index for the multi-dimensional case.  
In particular, the cost metric \(F(\mathbf{P},\hat{\boldsymbol{x}},\pi)\) and the work metric \(G(\mathbf{P},\hat{\boldsymbol{x}},\pi)\) are defined by replacing the one-dimensional \(P\) with the TEC \(\mathbf{P}\) and the dynamic state \(\hat{\boldsymbol{x}}\) in (36) and (37). The target's optimal tracking subproblem (18) becomes  
\[\min_{\pi\in\Pi}\,F(\mathbf{P},\hat{\boldsymbol{x}},\pi)+\lambda G(\mathbf{P},\hat{\boldsymbol{x}},\pi). \tag{43}\]  
The meaning of threshold policies is defined next in the present setting. For such a purpose, the \(Z\)-threshold policy tracks the target in states \(\mathbf{P}\) and \(\hat{\boldsymbol{x}}\) if and only if \(\text{tr}(\mathbf{P})/L>Z\). Then, the corresponding cost metric \(F(\mathbf{P},\hat{\boldsymbol{x}},Z)\) and work metric \(G(\mathbf{P},\hat{\boldsymbol{x}},Z)\) are the following functional equations:  
\[F(\mathbf{P},\hat{\boldsymbol{x}},Z)=\begin{cases}C(\mathbf{P},1)+\beta p_{ \text{d}}F(\psi^{1}(\mathbf{P},\hat{\boldsymbol{x}},\boldsymbol{u}^{1}),\hat {\boldsymbol{x}}^{*},Z)\\ \qquad\quad+\beta(1-p_{\text{d}})F(\phi^{0}(\mathbf{P},\boldsymbol{u}^{1}), \hat{\boldsymbol{x}}^{*},Z),&\text{tr}(\mathbf{P})/L>Z\\ C(\mathbf{P},0)+\beta F(\phi^{0}(\mathbf{P},\boldsymbol{u}^{0}),\hat{ \boldsymbol{x}}^{*},Z),&\text{tr}(\mathbf{P})/L\leq Z.\end{cases} \tag{44}\]  
\[G(\mathbf{P},\hat{\boldsymbol{x}},Z)=\begin{cases}1+\beta p_{\text{d}}G(\psi^ {1}(\mathbf{P},\hat{\boldsymbol{x}},\boldsymbol{u}^{1}),\hat{\boldsymbol{x}} ^{*},Z)\\ \qquad+\beta(1-p_{\text{d}})G(\phi^{0}(\mathbf{P},\boldsymbol{u}^{1}),\hat{ \boldsymbol{x}}^{*},Z),&\text{tr}(\mathbf{P})/L>Z\\ \beta G(\phi^{0}(\mathbf{P},\boldsymbol{u}^{0}),\hat{\boldsymbol{x}}^{*},Z),& \text{tr}(\mathbf{P})/L\leq Z.\end{cases} \tag{45}\]  
where \(\hat{\boldsymbol{x}}^{*}\) denotes the subsequent dynamic state. Again, such solutions can be approximated through a value-iteration approach using a truncated time horizon \(\tau\).  
Now, similar to the scalar case above, the marginal cost metric \(f(\mathbf{P},\hat{\mathbf{x}},Z)\) and the marginal work metric \(g(\mathbf{P},\hat{\mathbf{x}},Z)\) are defined through the \(\langle a,Z\rangle\) method in (44) and (45). Then, the US-MP metric is defined by  
\[\begin{split}\text{mp}(\mathbf{P},\hat{\mathbf{x}},Z)&= \frac{f(\mathbf{P},\hat{\mathbf{x}},Z)}{g(\mathbf{P},\hat{\mathbf{x}},Z)}\\ &=\frac{F(\mathbf{P},\hat{\mathbf{x}},\langle 0,Z\rangle)-F(\mathbf{P}, \hat{\mathbf{x}},\langle 1,Z\rangle)}{G(\mathbf{P},\hat{\mathbf{x}},\langle 1,Z\rangle)-G( \mathbf{P},\hat{\mathbf{x}},\langle 0,Z\rangle)},\end{split} \tag{46}\]  
provided that \(g(\mathbf{P},\hat{\mathbf{x}},Z)>0\).  
The corresponding US-MP index is defined by  
\[\text{mp}^{*}(\mathbf{P},\hat{\mathbf{x}})\triangleq\text{mp}(\mathbf{P},\hat{ \mathbf{x}},\text{tr}(\mathbf{P})/L), \tag{47}\]  
provided that \(g(\mathbf{P},\hat{\mathbf{x}},\text{tr}(\mathbf{P})/L)>0\) for all \(\mathbf{P}\).  
The cost metric is defined by the trace of TEC states in (11), and \(\text{tr}(\mathbf{P})/L\) is considered in the Z-threshold policy in (44) and (45). In particular, the determinant metric of TEC states can also be considered, provided these metric settings are consistent.  
However, there is currently no theory supporting any relation of the index \(\text{mp}^{*}(\mathbf{P},\hat{\mathbf{x}})\) with the Whittle index in the multi-dimensional case, assuming that both are well defined.