# Non-Mypopic Beam Scheduling for Multiple Smart-Target Tracking in Phased Array Radar Networks
## 1 Introduction
Yet, [PERSON] conjectured that, when all projects are indexable, this index policy should approach optimality asymptotically under the average criterion as \(N\) and \(K\) grow to infinity in a fixed ratio [23]. [PERSON] further developed PCL-indexability conditions, named after satisfaction of partial conservation laws (PCLs) [22; 24; 25; 26]. The work of [22] guarantees the optimality of threshold policies as well as the indexability of the model and its Whittle index was given by an explicitly defined MP index. The RMABP formulation of multi-target tracking was first considered in [27], extending work in [28]. The work of [27; 28] addressed the greedy index policy and defined a greedy index for each target, a scalar function of its tracking error variance (TEV) state, and giving higher tracking priority at each time to targets with larger index values. The work of [29] addressed the multi-target tracking with the scalar Kalman filter and defined the MP index in TEV states. See also [30] for an application for multi-target tracking with jamming and non-detections. A heuristic approach was developed in [31], where scalar-state project approximations were considered by taking the trace of the channel estimation mean square error (MSE) as the system state. In [32], the problem of hunting hiding targets was formulated as an RMABP, where the state of a target is its posterior probability of being exposed. However, in the more practically relevant case of RMABP models with a multi-dimensional state Kalman filter, indexability remains an open problem. Additionally, the Whittle index policy has not yet been studied in practical multi-target tracking scenarios. Substantial research challenges persist in RMABP-based smart-target tracking. (1) In models with a multi-dimensional continuous tracking error covariance (TEC) state, the application of the Whittle index policy is at present elusive. (2) Tracking actions will increase the probability of targets employing evasive maneuvers. Meanwhile, in practice, the detection probability of the radar is not one, which complicates the assessment of long-termdecision-making performance. (3) The non-linear measurement function complicates the update of TEC states, which requires dynamic state information. When computing the MP-based Whittle index, the currently unknown dynamic states for future steps make it impossible to calculate long-term marginal benefits for TEC states. This paper considers a beam scheduling problem for tracking multiple smart targets to minimize the multi-target tracking error measured by the sum of the traces of TECs over an infinite time horizon. In the vein of the interacting multiple models (IMM) method, the TEC state and dynamic state of each target are updated based on dynamic model probability vectors. The Whittle index policy based on RMABP formulation is exploited to treat each target as a restless project. For multi-dimensional TECs, the MP index policy uses the trace of the TEC state as a threshold and computes the long-term marginal benefits of different actions through a threshold policy. The US method is employed to sample measurements, and the TEC states and dynamic states over future steps under different actions are updated through the non-linear filter. Consequently, an unscented sampling-based MP (US-MP) index policy is proposed for the multi-dimensional TEC state case. Yet, such a goal is hindered by the fact that no proof of indexability is at present available for this model. This paper partially circumvents this difficulty by presenting some numerical evidence supporting the conjecture that the RMAB model with scalar TEV states satisfies the aforementioned PCL-indexability conditions, and uses the proposed US-MP index policy resulting from them. Through extensive simulation results, the proposed index policy is shown to outperform myopic baseline policies in the discounted TEC cost and root mean square error (RMSE) metrics. The contributions of this paper are summarized as follows.