## 2 Preliminaries and Problem Formulation
### Problem Formulation  
Considering the Multiple Autonomous Unmanned linear systems (MAUS) constructed by the directed communication graph \(G(V,E,A)\) having N agents, the dynamics of each agent is described in the following:  
\[\dot{x}_{i}=Ax_{i}+Bu_{i} \tag{1}\]  
where \(x_{i}\in R^{n},\;u_{i}\in R^{m_{i}}\) are states and control inputs of node \(i\), respectively. The cooperative control of homogeneous systems is investigated in this paper and the leader node \(x_{0}\in R^{n}\) is set to satisfy the following dynamic  
\[\dot{x}_{0}=Ax_{0} \tag{2}\]  
The problem of MAUS synchronization is designing control protocols \(u_{i}\) for each agent so that states of each node track the leader node, i.e., \(\|x_{0}-x_{i}\|\to 0,\;\forall i\).  
The neighborhood error for each node is defined as [26]  
\[\delta_{i}=\sum_{j\in N_{i}}a_{ij}(x_{i}-x_{j})+g_{i}(x_{i}-x_{0}) \tag{3}\]where \(g_{i}\geq 0\) denotes the pinning gain and there is at least one node that has a link to the leader node.  
For the neighborhood error (3), the overall neighborhood error vector of graph \(G(V,E,A)\) is given by  
\[\delta=\left((L+\overline{G})\otimes I_{n}\right)(x_{i}-\underline{x}_{0}) \tag{4}\]  
where \(x=[x_{1}^{T}x_{2}^{T}\ \cdots x_{N}^{T}]^{T}\) and \(\delta=[\delta_{1}^{T}\delta_{2}^{T}\ \cdots\delta_{N}^{T}]^{T}\) denote the global state vector and global error vector, respectively. Moreover, for \(\underline{x}_{0}=Lx_{0}\in R^{nN}\) with \(\underline{I}=1_{N}\otimes I_{n}\in R^{nN\times n}\), \(I_{n}\) denotes the \(n\) dimensional identity matrix and \(1_{N}\) denotes the \(N\)-vector of ones. The symbol \(\otimes\) is the Kronecker product [27]. \(\overline{G}=diag(g_{1},g_{2},\cdots,g_{N})\) as a diagonal matrix represents the connection between all agents and the leader node.  
The overall synchronization error is  
\[\varepsilon=(x-\underline{x}_{0})\in R^{nN} \tag{5}\]  
**Assumption 1**.: _The communication graph is strongly connected, i.e., there is a directed path for any a pair of distinct nodes._  
On the basis of Assumption 1, if \(\overline{G}\
eq 0\), then \(g_{i}\
eq 0\) for at least one. In this case, the matrix \(L+\overline{G}\) is non-singular and the real parts of all eigenvalues are positive [26]. The following lemma can be obtained, which shows that the overall neighborhood error vector \(\delta\) is positively correlated with the overall synchronization error \(\varepsilon\).  
**Lemma 1**.: _If the communication graph is strongly connected and \(\overline{G}\
eq 0\), the synchronization errors are bounded, as follows_  
\[\delta/\overline{\lambda}(L+\overline{G})\leq\|\varepsilon\|\leq\delta/ \underline{\lambda}(L+\overline{G}) \tag{6}\]  
Furthermore, \(\delta\equiv 0\) if and only if all nodes are synchronized, i.e.,  
\[\varepsilon=(x-\underline{x}_{0})=0 \tag{7}\]  
The dynamics of the local neighborhood tracking errors are given as  
\[\dot{\delta}_{i}=\sum_{j\in N_{i}}a_{ij}(\dot{x}_{i}-\dot{x}_{j})+g_{i}(\dot{ x}_{i}-\dot{x}_{0}) \tag{8}\]  
Substituting (1) and (2) into the above equation, it can be obtained that  
\[\dot{\delta}_{i}=A\delta_{i}+(d_{i}+g_{i})B_{i}u_{i}-\sum_{j\in N_{i}}a_{ij}B _{j}u_{j} \tag{9}\]  
It can be seen that the dynamics of the local neighborhood error of each agent \(i\) is affected by multiple control inputs from node \(i\) and its adjacent nodes. The whole MAUS with the communication graph \(G(V,E,A)\) presents a complex coupling relationship. It is quite intricate to solve the optimal control problem of dynamic (9) affected by multi-coupling.