## 2 Problem Formulation
### Target Motion Model  
In this work, we consider a target motion model that conforms to the Markov process [24], moving randomly through a series of grids, with each grid representing a fixed time step and each time step determining the target's position in the grid. The search area is divided into grids and this search problem is simplified to a path planning problem in the discrete domain range. As shown in Figure 1a, a \(3\times 3\) grid is arranged according to numerical order. As shown in Figure 1b, assuming that the target is located in grid 5 at time \(t\), the target can only move to the neighboring grids at the time \(t+1\) or remain stationary. The probability of the target moving from grid 5 to the other grid is shown in Figure 1c.  
Before the search begins, the target location is modeled from the available information using a probability distribution function. Supposing that the search space is \(S\), let the unknown variable indicate the position of the target at the time \(t\). The initial probability distribution of the target at time \(t=0\) is given by the following equation:  
\[\rho(x_{0})=P(x(0)=x),x_{0}\in S \tag{1}\]  
where \(P(x(0)=x)\) denotes the probability that the initial position of the target is located at \(x\) and satisfies \(\sum\limits_{x\in S}\rho(x_{0})=1\).  
The position of the target motion is represented according to a certain transfer probability, and the conditional probability distribution of its future state depends only on the current state. Then, the single-step transfer probability at time \(t\) is:  
\[P_{i,j}(t)=P[x_{t+1}=j|x_{t}=i] \tag{2}\]  
where \(P_{i,j}(t)\) is the single-step transfer probability conditional on \(x_{t}=i\) to \(x_{t+1}=j\) at time \(t\) to time \(t+1\). It is characterized by two distinct attributes:  
1. \(P_{i,j}(t)\geq 0,i,j\in S\) denotes the nonnegativity of the elements in the single-step transfer probability matrix;
2. \(\sum\limits_{i\in S}P_{i,j}(t)=1,i\in S\) denotes that the sum of the elements in the single-step transfer probability matrix is 1.  
Based on the single-step transfer probability of the moving target, the k-step transfer probability of the target can be further obtained as:  
\[P_{i,j}^{k}(t)=P[x_{t+k}=j|x_{t}=i] \tag{3}\]  
where \(P_{i,j}^{k}(t)\) denotes the probability that the target is transferred from \(x=i\) to \(x=j\) at time \(t\) to time \(t+k\). When \(k=1\), it is equivalent to the single-step transfer probability.  
The k-step transfer probability of a Markov process satisfies the Kolmogorov-Chapman equation:  
\[P_{i,j}^{(k+r)}(t)=\sum\limits_{x\in S}P_{i,x}^{(k)}(t)P_{x,j}^{(r)}(t+k),i,j\in S \tag{4}\]  
Figure 1: (**a**) Grid segmentation; (**b**) target position transfer direction; (**c**) target position transfer probability.  
Based on the Markov motion model, the available information regarding the initial position distribution of the target can be employed to consistently estimate and revise the target's motion position. This allows for the acquisition of a precise posterior distribution for the target. Subsequently, a search algorithm can be employed to determine the optimal search node for the subsequent step.