# Compensation Control Strategy for Orbital Pursuit-Evasion Problem with Imperfect Information
## 3 Construction of Guaranteed Cost Strategy with Perfect Information
### The Guaranteed Cost Strategy  
For the sake of brevity, we define the zero-effort-miss (ZEM) variables as follows:  
\[\mathbf{Z}(t)=\mathbf{D\Phi}\Big{(}t_{f}-t\Big{)}\mathbf{X}_{PE} \tag{7}\]  
where \(\mathbf{\Phi}\Big{(}t_{f}-t\Big{)}\) is the zero-input state transfer matrix of Equation (5) and satisfies \(\mathbf{\Phi}\Big{(}t_{f}-t\Big{)}=e^{\mathbf{A}(t_{f}-t)}\). Substituting Equation (7) into Equations (5) and (6), the system is reduced to  
\[\left\{\begin{array}{c}\dot{\mathbf{Z}}(t)=\mathbf{  
Define the Lyapunov-like function \(V(t)=\|\mathbf{Z}(t)\|\), \(t\in\left[0,t_{f}\right]\). Thus, the change rate of \(V(t)\) over time satisfies  
\[\frac{\mathrm{d}V(t)}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\|\mathbf{Z}(t)\|= \mathbf{\xi}^{T}\mathbf{B}_{P}\Big{(}t_{f}-t\Big{)}\mathbf{U}_{P}(t)+\mathbf{\xi}^{T}\mathbf{C}_{E} \Big{(}t_{f}-t\Big{)}\mathbf{U}_{E}(t) \tag{9}\]  
where \(\mathbf{\xi}=\frac{\mathbf{Z}(t)}{\|\mathbf{Z}(t)\|}\). The feedback control strategies of the pursuuer and the evader can be obtained by minimizing or maximizing the change rate:  
\[\left\{ \begin{array}{l}\mathbf{U}_{P}^{*}=-\rho_{P}\frac{\mathbf{B}_{P}^{T}(t_{f}-t) \mathbf{\xi}}{\|\mathbf{B}_{P}^{T}(t_{f}-t)\mathbf{\xi}\|}\\ \mathbf{U}_{E}^{*}=\rho_{E}\frac{\mathbf{C}_{E}^{T}(t_{f}-t)\mathbf{\xi}}{\|\mathbf{C}_{E}^{T} (t_{f}-t)\mathbf{\xi}\|}\end{array} \right. \tag{10}\]  
where \(\rho_{P}\) and \(\rho_{E}\) are the upper boundaries of the control of the pursuuer and the evader, respectively. The time-to-go \(t_{go}=t_{f}-t\) is an unknown variable and needs to be calculated in real time. The feedback control strategies can guarantee the cost (i.e., miss distance) for the pursuuer and the evader when the time-to-go is appropriate. The feedback control strategies with appropriate time-to-go are called guaranteed cost strategies. When the pursuuer adopts \(\mathbf{U}_{P}^{*}\), its corresponding time-to-go can be obtained by substituting \(\mathbf{U}_{P}^{*}\) into Equation (9) to obtain  
\[\frac{\mathrm{d}V(t)}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\|\mathbf{Z}(t) \|\ =-\rho_{P}\|\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|+\mathbf{\xi}^{T}\mathbf{C}_{ E}\Big{(}t_{f}-t\Big{)}\mathbf{U}_{E}(t) \tag{11}\]  
As \(\mathbf{U}_{E}^{*}\) is derived by maximizing the change rate (9), we get  
\[\frac{\mathrm{d}}{\mathrm{d}t}\|\mathbf{Z}(t)\|=-\rho_{P}\|\mathbf{B}_{P}^{T}\Big{(}t _{f}-t\Big{)}\mathbf{\xi}\|+\mathbf{\xi}^{T}\mathbf{C}_{E}\Big{(}t_{f}-t\Big{)}\mathbf{U}_{E}( t)\ \leq-\rho_{P}\|\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|+\rho_{E}\|\mathbf{C}_{E}^{T} \Big{(}t_{f}-t\Big{)}\mathbf{\xi}\| \tag{12}\]  
As \(\|\mathbf{\xi}\|=1\), according to the matrix analysis theory, we get  
\[\lambda_{\min}^{1/2}\leq\|\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|\ \leq \lambda_{\max}^{1/2} \tag{13}\]  
where \(\lambda_{\min}\) and \(\lambda_{\max}\), respectively, represent the maximum and minimum eigenvalues of the matrix \(\mathbf{B}_{P}\Big{(}t_{f}-t\Big{)}\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\). Thus, Equation (12) can be further expressed as  
\[ \begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\|\mathbf{Z}(t)\|&=-\rho_{P}\|\bm {B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|+\mathbf{\xi}^{T}\mathbf{C}_{E}\Big{(}t_{f}- t\Big{)}\mathbf{U}_{E}(t)\leq-\rho_{P}\|\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|+ \rho_{E}\|\mathbf{C}_{E}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|\\ &=-\rho_{P}\|\mathbf{B}_{P}^{T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|+\rho_{E}\|\mathbf{B}_{P}^ {T}\Big{(}t_{f}-t\Big{)}\mathbf{\xi}\|\leq-\rho_{P}\lambda_{\min}^{1/2}+\rho_{E} \lambda_{\max}^{1/2}\end{array}