## 2 The Assumptions and Definitions  
### Definitions  
Several definitions and assumptions are introduced before presenting our main results. \(\angle(\mathbf{v}_{1},\mathbf{v}_{2})\) is the angle formed by two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\). Mathematically, \(\angle(\mathbf{v}_{1},\mathbf{v}_{2})=\arccos(\frac{\mathbf{v}_{1}\cdot\mathbf{ v}_{2}}{\|\mathbf{v}_{1}\|\|\mathbf{v}_{2}\|})\). Here, \(\angle(\mathbf{v}_{1},\mathbf{v}_{2})\) exists between \(0\) and \(\pi\). \(T\) is the sampling interval of our chasing controller in discrete-time systems. \(\mathbf{r}_{k}^{r}\) is the evader's location at time index \(k\). \(\mathbf{r}_{k}^{p}\) is the pursuer's location at time index \(k\). The pursuer is _in the motion camouflage state_ at time index \(k+1\) in the case where \(\angle(\mathbf{r}_{k+1}^{p}-\mathbf{r}_{k+1}^{e},\mathbf{r}_{k}^{p}-\mathbf{ r}_{k}^{e})=0\). \(v_{k}^{p}\) is the pursuer's speed at time index \(k\). \(v_{k}^{e}\) is the evader's speed at time index \(k\). The subscript \(k\) implies the time index \(k\). The pursuer's motion model is  
\[\mathbf{r}_{k+1}^{p}=\mathbf{r}_{k}^{p}+Tv_{k}^{p}\mathbf{u}_{k}. \tag{1}\]  
Here, \(\mathbf{u}_{k}\) is a unit vector and indicates the pursuer's heading at time index \(k\). \(\mathbf{u}_{k}\) is determined at every time index \(k\) so that the pursuer is in the motion camouflage state at time index \(k\). In addition, the evader's motion model is  
\[\mathbf{r}_{k+1}^{e}=\mathbf{r}_{k}^{e}+T\mathbf{v}_{k}^{e}. \tag{2}\]  
Here, \(\mathbf{v}_{k}^{e}\) indicates the evader's velocity vector at time index \(k\). In order to capture the evader, it is necessary that the evader's speed is slower than the pursuer's speed ([1,2] introduced a variation of PNG controls to capture evaders that are of higher speeds than the pursuer. But, considering a high-speed evader which moves away from a slowly moving pursuer, it is impossible to capture the evader). We control the pursuer's speed so that it is always bigger than that of the evader. This implies that \(v_{k}^{e}<v_{k}^{p}\) at every time index \(k\). Let \(a_{m}\) denote the pursuer's maximum acceleration. In addition, let \(\epsilon>0\) denote a small constant. The required time interval to decrease the pursuer's speed from \(v_{k}^{p}\) to \(v_{k}^{e}+\epsilon\) is  
\[T_{r}=(v_{k}^{p}-v_{k}^{e}-\epsilon)/a_{m}. \tag{3}\]  
The traversal distance of the pursuer as it decreases its speed from \(v_{k}^{p}\) to \(v_{k}^{e}+\epsilon\) is  
\[D_{k}=v_{k}^{p}*T_{r}-0.5*a_{m}*T_{r}*T_{r}. \tag{4}\]  
We say that the pursuer is in the _terminal phase_ in the case where  
\[\|\mathbf{r}_{k}^{e}-\mathbf{r}_{k}^{p}\|<D_{k}. \tag{5}\]  
is met. \(L_{k}\) is the infinite line (LOS) intersecting both \(\mathbf{r}_{k}^{p}\) and \(\mathbf{r}_{k}^{e}\). We draw one infinite line (LOS) \(\bar{L}_{k+1}\) intersecting \(\mathbf{r}_{k+1}^{e}\), such that \(\bar{L}_{k+1}\) is parallel to \(L_{k}\). Note that there exists only one infinite line intersecting \(\mathbf{r}_{k+1}^{e}\), such that the line is parallel to \(L_{k}\). \(\mathbf{c}_{k}\) is the point on \(\bar{L}_{k+1}\), which is the closest to \(\mathbf{r}_{k}^{p}\). Let \(d_{k}^{c}=\|\mathbf{r}_{k}^{p}-\mathbf{c}_{k}\|\), and let \(\chi_{k}^{e}=\angle(\mathbf{r}_{k}^{e}-\mathbf{r}_{k}^{p},\mathbf{r}_{k+1}^{e} -\mathbf{r}_{k}^{e})\). Let \(d_{k}=\|\mathbf{r}_{k}^{p}-\mathbf{r}_{k}^{e}\|\). See Figure 1 for an illustration of these concepts. As depicted in Figure 1, \(d_{k}^{c}=v_{k}^{e}T\sin(\chi_{k}^{c})\leq v_{k}^{e}T\). Since \(v_{k}^{e}<v_{k}^{p}\), we have  
\[d_{k}^{c}<v_{k}^{p}T. \tag{6}\]  
Utilizing (6), we define \(\delta_{k}>0\) as follows. \[\delta_{k}=\sqrt{(v_{k}^{p}T)^{2}-(d_{k}^{c})^{2}}. \tag{7}\]  
We also define the _heading point_\(\mathbf{h}_{k}\) as follows:  
\[\mathbf{h}_{k} = \mathbf{c}_{k}+\frac{\mathbf{r}_{k}^{e}-\mathbf{r}_{k}^{p}}{ \|\mathbf{r}_{k}^{e}-\mathbf{r}_{k}^{p}\|}\delta_{k}. \tag{8}\]  
Here, \(\delta_{k}\) satisfies that \(\|\mathbf{h}_{k}-\mathbf{r}_{k}^{p}\|\) is \(v_{k}^{p}T\).