## 3 Problem Formulation
### System Model  
**Environment model:** The mission region \(\mathbb{O}\) is regarded as a bounded two-dimensional rectangular plane and uniformly discretized into \(L_{x}\times L_{y}\) square cells, which can be denoted as:  
\[\mathbb{O}=\left\{G_{x,y}\mid x\in\{1,2,\ldots,L_{x}\},y\in\left\{1,2,\ldots, L_{y}\right\}\right\}, \tag{1}\]  
where \(L_{x}\) and \(L_{y}\) represent the number of cells in the rows and columns of mission region, respectively. In our model, we use \(G_{x,y}\) to represent each cell and the coordinate of cell \(G_{x,y}\) is indicated as \(g_{x,y}=(x,y)\), where \(x\in\{1,2,\ldots,L_{x}\},y\in\{1,2,\ldots,L_{y}\}\). **Target model:** Let \(V=\left\{V_{1},V_{2},V_{3}\ldots,V_{N_{V}}\right\}\) be the set of \(N_{V}\) moving targets. And we use \(\
u_{k,t}=(x,y)\) to denote the coordinate of the target \(V_{k}(1\leq k\leq N_{V})\) at timestep \(t(1\leq t\leq T)\). The mission period is divided into \(T\) discrete timesteps in our model. UAVs and targets make movement decisions only at the beginning of each timestep. At each timestep, each target can only take up a single cell. and the target presence state of each cell is denoted as \(\theta_{x,y}\). There are two possible values of \(\theta_{x,y}\), namely \(\theta_{x,y}=1\) means that a  
Figure 1: The scenario of multi-UAV cooperative searching and tracking for moving targets. target is located in the cell \(G_{x,y}\) and \(\theta_{x,y}=0\) means that there is no target. And it is possible to have more than one target in a single cell. **UAV model:** The multi-UAV system \(U=\{U_{1},U_{2},U_{3}\ldots,U_{N_{U}}\}\) consists of \(N_{U}\) homogeneous fixed-wing UAVs. The location of UAV can be determined by its projection onto the mission region \(\mathbb{O}\) and the coordinate of UAV \(U_{i}(1\leq i\leq N_{U})\) at timestep \(t(1\leq t\leq T)\) is represented as \(\mathbf{\mu}_{i,t}=(x,y)\). The UAV can choose to move into one of its eight neighbor cells or just stay at its current cell at each timestep. Each UAV is equipped with a sensor and the UAV \(U_{i}\) can only detect cells within its sensing area \(\mathbb{S}_{i,t}\) at timestep \(t\) with sensing radius \(R_{s}\) (A cell is regarded to be completely covered if its center is covered), where  
\[\mathbb{S}_{i,t}=\big{\{}G_{x,y}\in\mathbb{O}\ |\ \|\mathbf{\mu}_{i,t}-\mathbf{g}_{x,y}\| \leq R_{s}\big{\}}. \tag{2}\]  
The detection result of UAV \(U_{i}\) at timestep \(t\) for the cell \(G_{x,y}\) is represented as \(Z^{x,y}_{i,t}\). It has two possible values, namely \(Z^{x,y}_{i,t}=1\) means that a target is detected in the cell and \(Z^{x,y}_{i,t}=0\) means that no target is detected. The sensor of UAV can only sense the presence of target but cannot distinguish the identity. Due to the imperfect detection ability of sensors, the detection model's conditional probability of positive and negative properties is constructed as follows:  
\[P\Big{(}Z^{x,y}_{i,t}|\theta_{x,y}\Big{)}=\left\{ \begin{array}{l}P\Big{(}Z^{ x,y}_{i,t}=1|\theta_{x,y}=1\Big{)}=P_{D}\\ PZ^{x,y}_{i,t}=0|\theta_{x,y}=1\Big{)}=1-P_{D}\\ P\Big{(}Z^{x,y}_{i,t}=1|\theta_{x,y}=0\Big{)}=P_{F}\\ P\Big{(}Z^{x,y}_{i,t}=0|\theta_{x,y}=0\Big{)}=1-P_{F}\end{array} \right., \tag{3}\]  
where \(P_{D}\) and \(P_{F}\) denote the true detection probability and false alarm probability respectively. Each UAV is limited to interacting with its neighboring UAVs within the communication range. In our model, we only take into account the restricted communication range, disregarding any communication delays, bandwidth restrictions or interruptions. The group of neighbors for UAV \(U_{i}\) at timestep \(t\) is denoted as  
\[\mathbb{N}_{i,t}=\big{\{}U_{j}\in U\ |\ \|\mathbf{\mu}_{i,t}-\mathbf{\mu}_{j,t}\|\leq R _{c}\big{\}}, \tag{4}\]  
where \(R_{c}\) represents the communication range. And we use \(d_{i,t}=|\mathbb{N}_{i,t}|\) to denote the number of neighbors of UAV \(U_{i}\) at timestep \(t\) (a UAV is assumed to be a neighbor of itself).