# Non-Mypopic Beam Scheduling for Multiple Smart-Target Tracking in Phased Array Radar Networks
## 2 Model Description and Problem Formulation
### PCL-Indexability Approach for Scalar-State Targets  
This subsection next outlines the PCL-indexability approach for scalar-state restless bandit projects as it applies to the present model (so \(L=1\)). Accordingly, the dynamic and measurement models are one-dimensional and linear. The TEV state \(P_{n,t}\) transitions are redefined as  
\[P_{n,t}=\begin{cases}\mu^{1}_{n}(P_{n,t-1},\mathbf{u}^{1}_{n}),&a_{n,t-1}=1,\text{ detected}\\ \mu^{0}_{n}(P_{n,t-1},\mathbf{u}^{1}_{n}),&a_{n,t-1}=1,\text{ not detected}\\ \mu^{0}_{n}(P_{n,t-1},\mathbf{u}^{0}_{n}),&a_{n,t-1  
\[\hat{P}_{n,t}^{m}=\left(1-K_{n,t}^{m}H_{n}\right)\hat{P}_{n,t|-1}^{m}, \tag{33}\]  
\[K_{n,t}^{m}=\hat{P}_{n,t|t-1}^{m}H_{n}^{\prime}\left(H_{n}\hat{P}_{n,t|t-1}^{m}H _{n}^{\prime}+R_{n}^{m}\right)^{-1}, \tag{34}\]  
\[\hat{P}_{n,t|t-1}^{m}=F_{n}^{m}p_{n,t-1}(F_{n}^{m})^{\prime}+Q_{n}^{m}. \tag{35}\]  
And \(F_{n}^{m}\), \(Q_{n}^{m}\), and \(R_{n}^{m}\) denote the TEV state transition ratio, process noise variance, and measurement noise variance of dynamic model \(m\), respectively. In the sequel, this subsection pays attention to the MP index of a single target with multiple dynamic models and the detection probability \(p_{\text{d}}\) as described above. This subsection drops the subscript \(n\) from notations. Starting from an initial TEV state \(P_{0}=P\in\mathbb{S}_{++}\triangleq(0,\infty)\), for a policy \(\pi\), define the cost metric and work metric  
\[F(P,\pi)\triangleq\mathrm{E}_{P}^{\pi}\left[\sum_{t=0}^{\infty}\beta^{t}C(P_{ t},a_{t})\right], \tag{36}\]  
\[G(P,\pi)\triangleq\mathrm{E}_{P}^{\pi}\left[\sum_{t=0}^{\infty}\beta^{t}a_{t} \right]. \tag{37}\]  
The subproblem (18) can be reformulated in scalar TEV states as  
\[\min_{\pi\in\Pi}F(P,\pi)+\lambda G(P,\pi). \tag{38}\]  
The indexability of subproblem (38) is studied under deterministic stationary policies and threshold policies. Deterministic stationary policies are represented as (Borel measurable) subsets of TEV states, where the corresponding target is tracked. An active set denoted as \(B\subseteq\mathbb{S}_{++}\) is employed to characterize the \(B\)-active policy. In particular, for any given threshold level \(Z\in\mathbb{S}\triangleq\mathbb{S}_{++}\cup\{-\infty,\infty\}\), this is designated as the \(Z\)-threshold policy. This means that, for a target in TEV state \(P\), if \(P>Z\), the target is tracked; otherwise, it is not tracked. Therefore, a \(Z\)-threshold policy has active set \(B(Z)\triangleq\{P\in\mathbb{S}_{++}:P>Z\}\). It is clear that, if \(0<Z<\infty\), then \(B(Z)=(Z,\infty)\); if \(Z\leq 0\), then \(B(Z)=\mathbb{S}_{++}\); and if \(Z=\infty\), then \(B(Z)=\emptyset\). Denote the corresponding cost and work metrics in (36) and (37) by \(F(P,Z)\) and \(G(P,Z)\), respectively. Given a threshold \(Z\), the cost and work metrics are characterized by the functional equations. \[F(P,Z)= \begin{cases}C(P,1)+\beta p_{\text{d}}F\big{(}\mu^{1}(P,\mathbf{u}^{1}),Z \big{)}+\beta(1-p_{\text{d}})F\big{(}\mu^{0}(P,\mathbf{u}^{1}),Z\big{)},&P>Z\\ C(P,0)+\beta F\big{(}\mu^{0}(P,\mathbf{u}^{0}),Z\big{)},&P\leq Z.\end{cases}  \tag{39}\]  
and  
\[G(P,Z)= \begin{cases}1+\beta p_{\text{d}}G\big{(}\mu^{1}(P,\mathbf{u}^{1}),Z\big{)} +\beta(1-p_{\text{d}})G\big{(}\mu^{0}(P,\mathbf{u}^{1}),Z\big{)},&P>Z\\ \beta G\big{(}\mu^{0}(P,\mathbf{u}^{0}),Z\big{)},&P\leq Z.\end{cases}  \tag{40}\]  
In practice, such metrics can be approximately computed by a value-iteration scheme with a finite truncated time horizon \(\tau\), as discussed in (Section 11 in [22]). Next, \(\langle a,Z\rangle\) represents the policy that executes action \(a\) at time \(t=0\) and subsequently follows the \(Z\)-threshold policy from \(t=1\) onwards in (39) and (40). Accordingly, the corresponding marginal metrics for the previously discussed measures are defined specifically: marginal cost metric \(f(P,Z)\triangleq F(P,\langle 0,Z\rangle)-F(P,\langle 1,Z\rangle)\) and the marginal work metric \(g(P,Z)\triangleq G(P,\langle 1,Z\rangle)-G(P,\langle 0,Z\rangle)\).