## 2 Theoretical Basis
### The Terminal Trajectory of Anti-Ship Missiles
Similar to Phase 1, the missile maintains consistent altitude and cruising speed, with adjustments confined to speed adjustments within the \(x_{m}\text{O}y_{m}\) plane. The sole distinction between the two phases lies in the following directional guidance mechanism: Phase 5 employs \(\hat{\theta}^{\prime}\) rather than \(\hat{\theta}\) as the directional reference:  
\[f_{5}(\mathbf{X})|_{f_{4}}=\left\{ \begin{array}{l}\tilde{\alpha}=sat\Big{(}-K_ {1}\hat{\theta}^{\prime}-K_{2}\hat{\alpha},\hat{\alpha}_{\text{max}}-\hat{ \alpha}\Big{)}\\ \tilde{\beta}=sat\Big{(}K_{3}(H_{1}-H),\hat{\beta}_{\text{max}}-\hat{\beta} \Big{)}\end{array} \right., \tag{8}\]  
where \(\hat{\theta}^{\prime}\) is derived from the predicted value of \(\theta^{\prime}\). (6) Phase 6 represents a critical operational phase. Upon achieving sufficient proximity to the target, the anti-ship missile executes a rapid ascent to designated elevation angles, subsequently redirecting its speed vector to a downward trajectory perpendicular to the sea surface in anticipation of a diving motor-driven maneuver. The corresponding state equation for this phase is expressed as follows:  
\[f_{6}(\mathbf{X})|_{f_{5}}=\left\{ \begin{array}{l}\dot{\alpha}=0\\ \tilde{\beta}=\left\{\begin{array}{l}sat\Big{(}K_{7}(H_{3}-H),\frac{n_{ \text{max}}\mathcal{S}}{V}-\hat{\beta}\Big{)},\text{if }H<H_{1}+c_{2}(H_{3}-H_{1})\\ sat\Big{(}K_{8}(-\frac{\pi}{2}-\hat{\beta})-K_{9}\hat{\beta}-K_{10}\text{sgn}( \beta+\frac{\pi}{2}),\frac{n_{\text{max}}\mathcal{S}}{V}-\hat{\beta}\Big{)} \end{array}\right.,\end{array} \right. \tag{9}\]  
where \(\text{sgn}(\cdot)\) represents the signum function. (7) Phase 7 represents the most crucial phase of the terminal trajectory. Leveraging its superior aerodynamic configuration, the anti-ship missile executes a terminal dive with a trajectory perpendicular to the sea surface and is guided by the proportional navigation law. During this phase, the radar seeker is activated promptly to acquire enhanced target information. The corresponding state equation for this final stage is as follows:  
\[f_{\mathcal{T}}(\mathbf{X})|_{f_{6}}=\left\{ \begin{array}{l}\dot{\alpha}=\text{ sat}\big{(}-K_{11}\frac{\hat{\beta}}{\hat{\beta}}\Delta\theta^{\prime}\Delta d^{ \prime}\cos\beta,\dot{\alpha}_{\text{max}}\big{)}\\ \hat{\beta}=\text{sat}\big{(}-K_{11}\frac{\hat{\beta}}{\hat{\beta}}\Delta\phi^ {\prime}\Delta d^{\prime}\cos\beta,\dot{\beta}_{\text{max}}\big{)}\end{array} \right., \tag{10}\]  
where \(\Delta\theta^{\prime}\), \(\Delta\phi^{\prime}\), and \(\Delta d^{\prime}\) denote the variations in target angle and distance information as perceived by the radar seeker over successive temporal intervals. Through the comprehensive integration of all switching conditions associated with the final trajectory segments, the following result can be derived:  
\[f(\mathbf{X},t)=\left\{ \begin{array}{l}f_{1}(\mathbf{X}),\text{ if }d^{\prime}>R_{1}\\ f_{2}(\mathbf{X}),\text{if }R_{1}\geq d^{\prime}>R_{2}\\ \vdots\\ f_{k}(\mathbf{X}),\text{if }R_{k-1}\geq d^{\prime}>R_{k}\\ \vdots\\ f_{\mathcal{T}}(\mathbf{X}),\text{if }R_{6}\geq d^{\prime}\end{array} \right.,k=1,2, \ldots,6. \tag{11}\]  
Based on \(f(\mathbf{X})\), the termination criteria for the final phase trajectory include \(t\geq N_{\text{max}}/F\), \(d_{\text{end}}\leq D\), and \(\left\|\mathbf{p}_{m}\right\|_{\text{end}}\leq R\), which correspond to missile fuel exhaustion, sea impact, and target engagement, respectively. Upon completing the seven-stage sequence, it is imperative that the anti-ship missile satisfies condition \(\beta_{\text{end}}\rightarrow-\frac{\pi}{2}\) simultaneously with the target impact. Consequently, the objective function and constraints for the terminal trajectory optimization of anti-ship missiles are established as follows:  
\[