## 3 Solution Method for Orbital Pursuit-Evasion-Defense Game  
### Necessary Conditions for Optimal Strategies  
The orbital pursuit-evasion-defense model given in the second section can be formulated as a non-cooperative N-person differential game. Necessary conditions for optimal strategies in this game are provided by [PERSON] [28] and applied to the system composed of (7) and (8) to obtain the form of optimal strategies.  
The Hamiltonian function is introduced as follows:  
\[\left\{\begin{array}{l}H_{\text{E}}=\mathbf{\Lambda}_{\text{E}}^{\text{T}}\dot{ \mathbf{\mathrm{Z}}}_{\text{PE}}=\mathbf{\Lambda}_{\text{E}}^{\text{T}}  
From Equations (20) and (21), we can find the following relationship:  
\[\left\{\begin{array}{l}A_{\rm PE}(t)=-k_{1}\Lambda_{\rm E}(t)\\ A_{\rm DP}(t)=-k_{2}\Lambda_{\rm D}(t)\end{array}\right. \tag{22}\]  
In addition, the optimal control strategies need to satisfy:  
\[\left\{\begin{array}{l}\mathbf{u}_{\rm D}^{*}=\underset{\|\mathbf{u}_{\rm D}\|\leq 1}{ \operatorname{argmin}}H_{\rm D}\\ \mathbf{u}_{\rm E}^{*}=\underset{\|\mathbf{u}_{\rm E}\|\leq 1}{\operatorname{argmin}}H_{\rm E }\\ \mathbf{u}_{\rm P}^{*}=\underset{\|\mathbf{u}_{\rm P}\|\leq 1}{\operatorname{ argmin}}H_{\rm P}\end{array}\right. \tag{23}\]  
For the sake of brevity, we define new variables \(\mathbf{M}_{i}\ (i=D,E,P)\) as follows:  
\[\left\{\begin{array}{l}\mathbf{M}_{\rm D}=\mathbf{\Lambda}_{\rm D}^{\rm T}\mathbf{\Phi}T _{\rm D}\\ \mathbf{M}_{\rm E}=-\mathbf{\Lambda}_{\rm E}^{\rm T}\mathbf{\Phi}\mathbf{\Phi}T_{\rm E}\\ \mathbf{M}_{\rm P}=-k_{1}\mathbf{\Lambda}_{\rm E}^{\rm T}\mathbf{\Phi}T_{\rm P}+k_{2}\mathbf{ \Lambda}_{\rm D}^{\rm T}\mathbf{D}\mathbf{\Phi}T_{\rm P}\end{array}\right. \tag{24}\]  
Combining Equations (19), (22), (23), and (24) yields:  
\[\left\{\begin{array}{l}\mathbf{u}_{\rm D}^{*}=[u_{\rm xD}^{*},u_{\rm yD}^{*},u_{ \rm zD}^{*}]^{\rm T}=-\frac{[\mathbf{M}_{\rm D}(4),\mathbf{M}_{\rm D}(5),\mathbf{M}_{\rm D }(6)]^{\rm T}}{\|\mathbf{M}_{\rm D}(4),\mathbf{M}_{\rm D}(5),\mathbf{M}_{\rm D}(6)\|^{\rm T }\|}\\ \mathbf{u}_{\rm E}^{*}=[u_{\rm xE}^{*},u_{\rm yE}^{*},u_{\rm zE}^{*}]^{\rm T}=- \frac{[\mathbf{M}_{\rm E}(4),\mathbf{M}_{\rm E}(5),\mathbf{M}_{\rm D}(6)]^{\rm T}}{\|\mathbf{ M}_{\rm D}(4),\mathbf{M}_{\rm E}(5),\mathbf{M}_{\rm D}(6)\|^{\rm T}\|}\\ \mathbf{u}_{\rm P}^{*}=[u_{\rm xP}^{*},u_{\rm yP}^{*},u_{\rm zP}^{*}]^{\rm T}=- \frac{[\mathbf{M}_{\rm D}(4),\mathbf{M}_{\rm P}(5),\mathbf{M}_{\rm D}(6)]^{\rm T}}{\|\mathbf{ M}_{\rm P}(4),\mathbf{M}_{\rm P}(5),\mathbf{M}_{\rm P}(6)\|^{\rm T}\|}\end{array}\right. \tag{25}\]  
Combining Equation (25) and the form of control vector, the optimal control variables are expressed as Equation (26), which satisfies Equation (27).  
\[\left\{\begin{array}{l}\mathbf{U}_{\rm D}^{*}=[0,0,0,u_{\rm xD}^{*},u_{\rm yD}^{* },u_{\rm zD}^{*}]^{\rm T}\\ \mathbf{U}_{\rm E}^{*}=[0,0,0,u_{\rm xE}^{*},u_{\rm yE}^{*},u_{\rm zE}^{*}]^{\rm T }\\ \mathbf{U}_{\rm P}^{*}=[0,0,0,u_{\rm xD}^{*},u_{\rm yP}^{*},u_{\rm zP}^{*}]^{\rm T }\end{array}\right. \tag{26}\]  
\[\left\{\begin{array}{l}J_{\rm P}\big{(}\mathbf{U}_{\rm P}^{*},\mathbf{U}_{\rm E}^{*},\mathbf{U}_{\rm D}^{*}\big{)}\leq J_{\rm P}\big{(}\mathbf{U}_{\rm P},\mathbf{U}_{\rm E}^{* },\mathbf{U}_{\rm D}^{*}\big{)}\\ J_{\rm E}\big{(}\mathbf{U}_{\rm P}^{*},\mathbf{U}_{\rm E}^{*}\big{)}\leq J_{\rm E}\big{(} \mathbf{U}_{\rm P},\mathbf{U}_{\rm E}\big{)}\\ J_{\rm D}\big{(}\mathbf{U}_{\rm P}^{*},\mathbf{U}_{\rm D}^{*}\big{)}\leq J_{\rm D}\big{(} \mathbf{U}_{\rm P}^{*},\mathbf{U}_{\rm D}\big{)}\end{array}\right. \tag{27}\]  
Equations (10), (20), (21), and (24)-(26) constitute a TPBVP.