## 3 Control Law
### Capturability Analysis  
Besides achieving motion camouflage, the pursuer must capture the evader in finite time. We prove that the distance between the evader and the pursuer monotonically decreases as time passes.  
**Theorem 2**.: _Under our chasing controller, the distance between the evader and the pursuer monotonically decreases until the evader is captured._  
Proof.: Before proving the capturability of our chasing controller, the relationship between \(d_{k}\) and \(d_{k+1}\) is introduced.  
First, we handle the case where \(\chi_{k}^{e}>\pi/2\). This case, the evader at time index \(k\) maneuvers to decrease the distance between the evader and the pursuer. See Figure 1 for an illustration of \(d_{k}\) and \(d_{k+1}\). Since \(L_{k}\) is parallel to \(\bar{L}_{k+1}\), the geometry in Figure 1 leads to  
\[d_{k+1}=d_{k}-\alpha_{k}-\delta_{k}. \tag{15}\]  
This results in  
\[d_{k+1}<d_{k}. \tag{16}\]Next, we handle the case where \(\chi^{e}_{k}\leq\pi/2\). This case, the evader at time index \(k\) maneuvers to increase the distance between the evader and the pursuer. See Figure 2 for an illustration of \(d_{k}\) and \(d_{k+1}\). Since \(L_{k}\) is parallel to \(\bar{L}_{k+1}\), we get  
\[d_{k+1}=d_{k}+\alpha_{k}-\delta_{k}. \tag{17}\]  
Since \(v^{p}_{k}>v^{e}_{k}\), \(\alpha_{k}-\delta_{k}\) in (17) is negative. Hence, Ref. (16) is obtained.  
Utilizing (16), \(d_{k}\) monotonically decreases as \(k\) increases. Therefore, there exists a time index \(k^{\prime}\) such that \(d_{k^{\prime}}>0\) and that \(d_{k^{\prime}+1}<0\). We next prove that the distance between \(\mathbf{r}^{p}_{k^{\prime}}\) and \(\mathbf{r}^{e}_{k^{\prime}+1}\) is less than \(v^{p}_{k}T\).  
First, handle the case where \(\chi^{e}_{k^{\prime}}>\pi/2\). Utilizing both \(d_{k^{\prime}+1}<0\) and (15), the following equation is derived.  
\[d_{k^{\prime}}-\alpha_{k^{\prime}}<\delta_{k^{\prime}}. \tag{18}\]  
\(v^{p}_{k}>v^{e}_{k}\) leads to  
\[\alpha_{k^{\prime}}-\delta_{k^{\prime}}<0. \tag{19}\]  
Since \(d_{k^{\prime}}\) is positive, Ref. (19) further results in  
\[d_{k^{\prime}}-\alpha_{k^{\prime}}>-\delta_{k^{\prime}}. \tag{20}\]  
Utilizing (18) and (20),  
\[(d_{k^{\prime}}-\alpha_{k^{\prime}})^{2}<(\delta_{k^{\prime}})^{2} \tag{21}\]  
is derived.  
Utilizing (9), the distance between \(\mathbf{r}^{p}_{k^{\prime}}\) and \(\mathbf{r}^{e}_{k^{\prime}+1}\) is calculated as  
\[\|\mathbf{r}^{p}_{k^{\prime}}-\mathbf{r}^{e}_{k^{\prime}+1}\|=\|(d_{k^{\prime} }-\alpha_{k^{\prime}},-d^{c}_{k^{\prime}})\|. \tag{22}\]  
Utilizing (7), (22) and (21),  
\[\|\mathbf{r}^{p}_{k^{\prime}}-\mathbf{r}^{e}_{k^{\prime}+1}\|<v^{p}_{k}T \tag{23}\]  
is calculated.  
Next, we handle the case where \(\chi^{e}_{k^{\prime}}\leq\pi/2\). Utilizing both \(d_{k^{\prime}+1}<0\) and (17), we obtain  
\[d_{k^{\prime}}+\alpha_{k^{\prime}}<\delta_{k^{\prime}}. \tag{24}\]  
Since \(d_{k^{\prime}}+\alpha_{k^{\prime}}\) is positive,  
\[(d_{k^{\prime}}+\alpha_{k^{\prime}})^{2}<(\delta_{k^{\prime}})^{2} \tag{25}\]  
is calculated.  
Utilizing (10), the distance between \(\mathbf{r}^{p}_{k^{\prime}}\) and \(\mathbf{r}^{e}_{k^{\prime}+1}\) is calculated as  
\[\|\mathbf{r}^{p}_{k^{\prime}}-\mathbf{r}^{e}_{k^{\prime}+1}\|=\|(d_{k^{\prime }}+\alpha_{k^{\prime}},-d^{c}_{k^{\prime}})\|. \tag{26}\]  
Utilizing (7), (26), and (25),  
\[\|\mathbf{r}^{p}_{k^{\prime}}-\mathbf{r}^{e}_{k^{\prime}+1}\|<v^{p}_{k}T \tag{27}\]is derived.  
We proved that the distance between \(\mathbf{r}_{k^{\prime}}^{p}\) and \(\mathbf{r}_{k^{\prime}+1}^{e}\) is less than \(v_{k}^{p}T\). At time index \(k^{\prime}\), the pursuer heads towards the evader directly. Thereafter, the pursuer captures the evader at time index \(k^{\prime}+1\).