## 5 Main Results
### Learning-Based Control Law  
Assume that the acceleration of the target is available to each UAV, we design the distributed learning control law as follows  
\[\mathbf{u}_{i}=-\mathbf{\mu}_{i}+\mathbf{G}_{i}^{-1}\left[-k_{1}\left(\sum_{j=1}^{n}a_{ij}(\bm {e}_{i}-\mathbf{e}_{j})+b_{ii}\mathbf{e}_{i}\right)-k_{2}\left(\sum_{j=1}^{n}a_{ij}(\dot {\mathbf{e}}_{i}-\dot{\mathbf{e}}_{j})+b_{ii}\dot{\mathbf{e}}_{i}\right)+\mathbf{f}_{0}\right], \tag{21}\]  
where \(k_{1},k_{2}>0\) are constants, \(\mathbf{\mu}_{i}=(\mu_{i1},\mu_{i2})^{\top}\) is the posterior mean function of unknown dynamic function \(\mathbf{f}_{i}=(f_{i1},f_{i2})^{\top}\) based on GP regression. Then theorem 1 will show the main result in this paper. **Theorem 1**. : _Consider a group of \(n\) UAVs with dynamics Equation (14) and the target vehicle Equation (16), a distributed control law Equation (21) employing predictions based on GP, will make the consensus tracking error Equation (19) of each UAV converge to certain bound with probability._  
Proof of Theorem 1.: The second time derivative of the consensus tracking error \(\varepsilon_{i}\) along Equation (18) is  
\[ \begin{array}{ll}\dot{\mathbf{\varepsilon}}_{i}&=\sum_{j=1}^{n}a_{ij}(\dot{\mathbf{ \varepsilon}}_{i}-\dot{\mathbf{e}}_{j})+b_{ii}\dot{\mathbf{e}}_{i}\\ &=\sum_{j=1}^{n}a_{ij}[\mathbf{G}_{i}(\mathbf{u}_{i}+\mathbf{f}_{i}+\mathbf{d}_{i})-\mathbf{G}_{j} (\mathbf{u}_{j}+\mathbf{f}_{j}+\mathbf{d}_{j})]+b_{ii}[\mathbf{G}_{i}(\mathbf{u}_{i}+\mathbf{f}_{i}+\bm {d}_{i})-\mathbf{f}_{0}].\end{array}  \tag{22}\]  
Figure 2: The flowchart of GP regression and its application. Define \(\mathbf{\eta}_{1}=\left(\mathbf{\varepsilon}_{1}^{\top},\mathbf{\varepsilon}_{2}^{\top},\cdots \mathbf{\varepsilon}_{n}^{\top}\right)^{\top}\in\mathbb{R}^{2n}\), \(\mathbf{\eta}_{2}=\left(\mathbf{\varepsilon}_{1}^{\top},\mathbf{\varepsilon}_{2}^{\top}, \cdots\mathbf{\varepsilon}_{n}^{\top}\right)^{\top}\in\mathbb{R}^{2n}\), and substitute Equation (21) into Equation (22), we can rewrite Equation (22) as  
\[ \begin{split}\dot{\mathbf{\varepsilon}}_{i}=&\sum_{j=1}^ {n}a_{ij}\big{[}-k_{1}\big{(}\mathbf{\varepsilon}_{i}-\mathbf{\varepsilon}_{j}\big{)} -k_{2}\big{(}\dot{\mathbf{\varepsilon}}_{i}-\dot{\mathbf{\varepsilon}}_{j}\big{)} \big{]}+b_{ii}(-k_{1}\mathbf{\varepsilon}_{i}-k_{2}\dot{\mathbf{\varepsilon}}_{i})\\ &+\sum_{j=1}^{n}a_{ij}\big{[}\big{(}\mathbf{G}_{i}(f_{i}-\mathbf{\mu}_{i} )-\mathbf{G}_{j}\big{(}f_{j}-\mathbf{\mu}_{j}\big{)}\big{)}+\big{(}\mathbf{G}_{i}\mathbf{d}_{i }-\mathbf{G}_{j}\mathbf{d}_{j}\big{)}\big{]}\\ &+b_{ii}\big{[}\big{(}\mathbf{G}_{i}(f_{i}-\mathbf{\mu}_{i})-\mathbf{G}_{j} \big{(}f_{j}-\mathbf{\mu}_{j}\big{)}\big{)}+\big{(}\mathbf{G}_{i}\mathbf{d}_{i}-\mathbf{G}_{j }\mathbf{d}_{j}\big{)}\big{]}\\ \triangleq&-k_{1}\big{(}\tilde{\mathbf{L}}_{i}\otimes\mathbf{I }_{2}\big{)}\mathbf{\eta}_{1}-k_{2}\big{(}\tilde{\mathbf{L}}_{i}\otimes\mathbf{I}_{2}\big{)} \mathbf{\eta}_{2}+\big{(}\tilde{\mathbf{L}}_{i}\otimes\mathbf{I}_{2}\big{)}\mathbf{\Delta}+ \big{(}\tilde{\mathbf{L}}_{i}\otimes\mathbf{I}_{2}\big{)}\mathbf{H},\end{split}  \tag{23}\]  
where \(\tilde{\mathbf{L}}_{i}\) is the \(i\)-th row of \(\mathbf{L}\), and  
\[ \begin{split}\mathbf{\Delta}&=\Big{(}(\mathbf{G}_{1}\mathbf{f}_ {1}-\mathbf{G}_{1}\mathbf{\mu}_{1})^{\top},(\mathbf{G}_{2}\mathbf{f}_{2}-\mathbf{G}_{2}\mathbf{\mu}_{2 })^{\top},\cdots(\mathbf{G}_{n}\mathbf{f}_{n}-\mathbf{G}_{n}\mathbf{\mu}_{n})^{\top}\Big{)}^{ \top},\\ \mathbf{H}&=\Big{(}(\mathbf{G}_{1}\mathbf{d}_{1})^{\top},(\mathbf{G }_{2}\mathbf{d}_{2})^{\top},\cdots(\mathbf{G}_{n}\mathbf{d}_{n})^{\top}\Big{)}^{\top}. \end{split} \]  
Then we can obtain the differential function of the consensus tracking error:  
\[