## 4 Main Results
### Flight-Time Control Part  
In this part, the guidance law for the followers in the LOS direction is designed to realize simultaneous arrival. Equation (17) shows that there is an uncertain disturbance caused by the target maneuver. Designing a disturbance observer to estimate the maneuver of the target is the first step. In light of [30] and the ESO theory in Lemma 1, a fixed-time distributed disturbance observer (DDOB) is presented as follows: Let \(Z_{1}\) denote the estimation of the consensus error of the \(t_{go}\) between multiple vehicles, and let \(Z_{2}\) denote the estimation of the uncertain disturbance. Then, the observer is designed as follows:  
\[\begin{array}{l}\dot{Z}_{1i}=h_{1}\phi_{1}\left(\frac{\sum\limits_{j=1}^{N}a _{ij}(t_{gai}-t_{goj})-Z_{1}}{\epsilon}\right)+\sum\limits_{j=1}^{N}a_{ij}( \frac{x_{1i}^{2}x_{3i}!}{x_{2i}!}^{2}-\frac{x_{1j}^{2}x_{3i}!}{x_{2i}! }^{2})+ \\ \sum\limits_{j  
where \(h_{1}\) and \(h_{2}\) are the design parameters, satisfying \(h_{1},h_{2}>0\) and \(h_{1}\geq 2\sqrt{h_{2}}\), \(\varepsilon\) is the amplification factor satisfying \(\varepsilon\in(0,1)\), and the concrete form of \(\phi_{1}\) and \(\phi_{2}\) is:  
\[ \begin{array}{l}\phi_{1}(x)=sig^{\alpha}(x)+sig^{\beta}(x)\\ \phi_{2}(x)=sig^{2\alpha-1}(x)+sig^{2\beta-1}(x)\end{array}  \tag{25}\]  
Moreover, \(\alpha\), \(\beta\) satisfies \(\alpha\in(0.5,1)\), \(\beta\in(1,1.5)\). **Theorem 1. **_For system (22), with the fixed-time distributed observer (24), assuming that \(u_{ri}\) and \(u_{rj}\) are known and the disturbance caused by the target satisfies the boundary condition, \(|w_{ri}|\leq w_{1}\), where \(w_{1}\) is finite and unknown, the observation error of \(\left|\sum\limits_{j=1}^{N}a_{ij}(t_{goi}-t_{goj})-Z_{1}\right|\) and \(|w_{ri}-Z_{2i}|\) will converge to a neighborhood of the origin in fixed time. The result of the contrast is shown in Section 5._  
**Proof of Theorem 1. ** The proof is provided in Appendix A. \(\square\)  
**Remark 2. **_Because of the special distributed model (23) in the followers' cooperative-guidance law, the traditional fixed-time disturbance observer (FxTDO) is no longer effective. To solve this problem, we proposed the DDOB. As is shown in (24), the state variables that \(Z_{1}\) tracks are designed as a distributed form (\(\sum\limits_{j=1}^{N}a_{ij}(t_{goi}-t_{goj})\)) by the communication graph. Compared with the other single-vehicle disturbance observer methods [23,32], our DDOB is more suitable for multivehicle-cooperative-combat environments._  
**Theorem 2. **_Under the undirected graph, consider a one-order system as follows:_  
\[\dot{x}=u;\ x(t_{0})=x_{0} \tag{26}\]  
_The system can converge to zero within a finite time by the guidance law as:_  
\[u_{con}=-k_{r1}sig^{\alpha}\left(\sum\limits_{j=1}^{N}a_{ij}(x_{i}-x_{j})\right) \tag{27}\]  
_where \(k_{ri}>0\) and \(0<\alpha<1\)._  
**Proof of Theorem 2. ** The proof is provided in Appendix B. \(\square\)  
**Theorem 3. **_If the undirected graph of the multivehicle system is connected, the impact time of all the missiles will converge to the same value within a finite time with the guidance law:_  
\[u_{ri}=-\frac{\dot{r_{i}}^{2}}{\sum\limits_{j=1}^{N}a_{ij}r_{i}}\left(-\sum \limits_{j=1}^{N}a_{ij}(\frac{r_{i}^{2}\dot{q_{i}}^{2}}{\dot{r_{i}}^{2}}-\frac {r_{j}^{2}\dot{q_{j}}^{2}}{\dot{r_{j}}^{2}})+Z_{2}-k_{r1}sig^{\gamma}(\sum \limits_{j=1}^{N}a_{ij}(t_{goi}-t_{goj}))-\sum\limits_{j=1}^{N}a_{ij}\frac{r_ {j}}{\dot{r_{j}}}u_{rj}\right) \tag{28}\]  
**Proof of Theorem 3. ** By combining (28) with (23), one can obtain:  
\[\dot{e}_{ti}=-k_{r1}sig^{\gamma}\left(\sum\limits_{j=1}^{N}a_{ij}(t_{goi}-t_{ goj})\right)=-k_{r1}sig^{\gamma}\left(\sum\limits_{j=1}^{N}a_{ij}(t_{fi}-t_{ fj})\right) \tag{29}\]  
By utilizing Theorem 2, \(e_{ti}\) can converge to zero in finite time. This indicates that the total flight time of the followers can remain consistent, and furthermore, that it can be consistent with that of the leader. \(\square\)  
**Remark 3**.