## 3 Multi-Player Zero-Sum Differential Game for Decoupled Multi-Agent System
### Multi-Player Zero-Sum Differential Game  
The following equation is used to define the following performance index function for each agent. \[J_{i}(\delta_{i}(0)\,\ u_{i}\,\ \widetilde{u}_{-i})=\frac{1}{2}\int_{0}^{ \infty}\ \delta_{i}^{T}Q_{i}\delta_{i}+u_{i}^{T}R_{ii}u_{i}-\gamma_{i}^{2}\sum \limits_{j\in N_{i}}\widetilde{u}_{j}^{T}R_{ij}\widetilde{u}_{j}dt \tag{11}\]  
where \(\widetilde{u}_{-i}\) denotes the virtual coupling control inputs from neighboring nodes, i.e., \(\widetilde{u}_{-i}=\left\{\ \widetilde{u}_{j}|\ j\in N_{i}\ \right\}\). It should be noted that the main difference from [17] is that the coupling control inputs from neighboring nodes are regarded as the virtual external disturbances directly, which greatly simplifies the design of the control protocol \(u_{i}\). The solution for the bounded \(L_{2}\)-gain problem for coupling attenuation depicted in Section 3.1 can be equivalent to the Nash equilibrium solution of the multi-player zero-sum game-base on the performance index function (11). That is  
\[V_{i}(\delta_{i}(0))=\underset{u_{i}}{\text{minmax}}J_{i}(\delta_{i}(0)\,\ u_{i}\,\ \widetilde{u}_{-i}) \tag{12}\]  
In this multi-player zero-sum game, the goal of \(u_{i}\) is to minimize the value \(V_{i}(\delta_{i}(0))\). On the contrary, the virtual coupling inputs \(\widetilde{u}_{-i}\) are assumed to maximize the value. This game has a unique solution if a game-theoretic saddle point \((u_{i}^{*}\,\ \widetilde{u}_{-i}^{*})\) exists, i.e.,  
\[V_{i}^{*}(\delta_{i}(0))=\underset{u_{i}}{\text{minmax}}J_{i}(\delta_{i}(0)\,\ u_{i}\,\ \widetilde{u}_{-i})=\underset{ \widetilde{u}_{-i}}{\text{maxmin}}J_{i}(\delta_{i}(0)\,u_{i}\,\ \widetilde{u}_{-i}) \tag{13}\]  
Accordingly, the value \(V_{i}^{*}(\delta_{i}(0))\) in the above equation is the value of the zero-sum game and satisfies the following Nash equilibrium condition for all policies \(u_{i}\,\ \widetilde{u}_{-i}\)  
\[J_{i}(\delta_{i}(0)\,u_{i}^{*}\,\ \widetilde{u}_{-i})\leq J_{i}(\delta_{i}(0)\, u_{i}\,\ \widetilde{u}_{-i})\leq J_{i}(\delta_{i}(0)\,u_{i}\,\ \widetilde{u}_{-i}) \tag{14}\]  
When the policies \(u_{i}\,\ \widetilde{u}_{-i}\) are selected, the value function of node \(i\) can yield  
\[V_{i}(\delta_{i}(t)\,\ u_{i}\,\ \widetilde{u}_{-i})=\frac{1}{2}\int_{t}^{ \infty}\ \delta_{i}^{T}Q_{i}\delta_{i}+u_{i}^{T}R_{ii}u_{i}-\gamma_{i}^{2}\sum \limits_{j\in N_{i}}\widetilde{u}_{j}^{T}R_{ij}\widetilde{u}_{j}dt \tag{15}\]  
Differential equivalents to each value function are given as  
\[ \begin{array}{rl}0=&\frac{1}{2}\Bigg{(}\delta_{i}^{T}Q_{i}\delta_{i}+u_{i}^ {T}R_{ii}u_{i}-\gamma_{i}^{2}\sum\limits_{j\in N_{i}}\widetilde{u}_{j}^{T}R_{ ij}\widetilde{u}_{j}\Bigg{)}\\ &+\
abla V_{i}^{T}(A\delta_{i}+(d_{i}+g_{i})B_{i}u_{i}-\sum\limits_{j\in N_{i} }a_{ij}B_{j}\widetilde{u}_{j})\,\ V_{i}(0)=0\,\ i\in N\end{array}  \tag{16}\]where \(\
abla V_{i}=\frac{\partial V_{i}}{\partial\dot{x}_{i}}\in R^{n}\) denotes the gradient vector. The Hamiltonian functions are defined as follows,  
\[ \begin{array}{c}H_{i}(\delta_{i},\
abla V_{i},u_{i},\,\widehat{u}_{-i})\equiv \frac{1}{2}\left(\delta_{i}^{T}Q_{i}\delta_{i}+u_{i}^{T}R_{ii}u_{i}-\gamma_{i} ^{2}\sum\limits_{j\in N_{i}}\widehat{u}_{j}^{T}R_{ij}\widehat{u}_{j}\right)\\ \hskip 14.226378 pt+\
abla V_{i}^{T}(A\delta_{i}+(d_{i}+g_{i})B_{i}u_{i}-\sum \limits_{j\in N_{i}}a_{ij}B_{j}\widehat{u}_{j})\end{array}  \tag{17}\]  
Under certain policies \(u_{i}\), \(\widehat{u}_{-i}\), the partial differential equation \(H_{i}(\delta_{i},\
abla V_{i},u_{i},\,\widehat{u}_{-i})=0\) has a unique solution \(V_{i}(\delta_{i})\). The principle of optimality gives  
\[ \begin{array}{c}\frac{\partial H_{i}(\delta_{i},\
abla V_{i},u_{i},\widehat {u}_{-i})}{\partial u_{i}}=0\Rightarrow u_{i}=-(d_{i}+g_{i})R_{ii}^{-1}B_{i} ^{T}\
abla V_{i}\\ \frac{\partial H_{i}(\delta_{i},\
abla V_{i},u_{i},\widehat{u}_{-i})}{ \partial\widehat{u}_{j}}=0\Rightarrow\widehat{u}_{j}=-\frac{1}{\gamma_{i}^{2 }}a_{ij}R_{ij}^{-1}B_{j}^{T}\
abla V_{i},\,\,j\in N_{i}\end{array}