# Optimal Interplanetary Transfer of Solar Wind Ion Focusing Thruster-Based Spacecraft
## 3 Trajectory Optimization  
Consider a circle-to-circle (coplanar) orbit transfer, where the SWIFT-based spacecraft, which initially (at \(t_{0}=0\)) tracks a heliocentric circular orbit of radius \(r_{0}\triangleq r(t_{0})=r_{\oplus}\), reaches a target circular orbit of given radius \(r_{f}\triangleq r(t_{f})\) at time \(t_{f}\). It is assumed that at time \(t_{0}\) the spacecraft leaves the Earth along a parabolic (escape) orbit. In addition, without loss of generality, the spacecraft's initial polar angle is set equal to zero, while the final value of  
Figure 7: Comparison between the force bubble of a SWIFT and an ideal (photonic) flat solar sail when \(k=k^{\star}\), \(\delta=60\,\deg\) and \(\Delta\alpha=0\). \(\theta\) is left free, that is, it is an output of the optimization process. Accordingly, the initial and final constraints on the spacecraft state variables \(\{r,\,\theta,\,v_{r},\,v_{\theta}\}\) are  
\[r(t_{0})=r_{\oplus}\quad,\quad\theta(t_{0})=0\quad,\quad v_{r}(t_{0})=0\quad, \quad v_{\theta}(t_{0})=\sqrt{\frac{\mu_{\odot}}{r_{\oplus}}} \tag{40}\]  
\[r(t_{f})=r_{f}\quad,\quad v_{r}(t_{f})=0\quad,\quad v_{\theta}(t_{f})=\sqrt{ \frac{\mu_{\odot}}{r_{f}}}. \tag{41}\]  
The orbit transfer is analyzed in an optimal framework [44; 45; 46], in which the trajectory is found by minimizing the flight time necessary for the spacecraft to reach the target orbit. This amounts to maximizing the performance index  
\[J\triangleq-t_{f}, \tag{42}\]  
where \(t_{f}\) coincides with the total flight time to be calculated. The optimal control problem is faced with an indirect approach [47; 48], in which the Hamiltonian \(\mathcal{H}\) is obtained from Equations (3), (4), (38) and (39) as  
\[\mathcal{H}=\lambda_{r}\,v_{r}+\lambda_{\theta}\,v_{\theta}+ \lambda_{v_{r}}\,\left[\frac{v_{\theta}^{2}}{r}-\frac{\mu_{\odot}}{r^{2}}+a_ {D_{\oplus}}\Big{(}\frac{r_{\oplus}}{r}\Big{)}^{2}(1+k\,\cos\alpha)\right]+\\ +\lambda_{v_{\theta}}\,\left[-\frac{v_{r}\,v_{\theta}}{r}+a_{D_{ \oplus}}\Big{(}\frac{r_{\oplus}}{r}\Big{)}^{2}k\,\sin\alpha\right], \tag{43}\]  
where \(\{\lambda_{r},\,\lambda_{\theta},\,\lambda_{v_{f}},\,\lambda_{v_{\theta}}\}\) are the variables adjoint to the spacecraft states. The time variation of the adjoint variables is described by the Euler-Lagrange equations [47; 48],  
\[\lambda_{r}=-\frac{\partial\mathcal{H}}{\partial r}\,\ \lambda_{\theta}=-\frac{ \partial\mathcal{H}}{\partial\theta}\,\ \lambda_{v_{r}}=-\frac{\partial\mathcal{H}}{ \partial v_{r}}\,\ \lambda_{v_{\theta}}=-\frac{\partial\mathcal{H}}{\partial v_{\theta}}, \tag{44}\]  
whose explicit expressions are here omitted for the sake of brevity. It can be easily verified that \(\lambda_{\theta}=0\) (in fact \(\mathcal{H}\) does not explicitly depend on the polar angle), so that \(\lambda_{\theta}\) is a constant of motion. Its value is found from the transversality condition that, in this case, reads  
\[\lambda_{\theta}(t_{f})=0\quad,\quad\mathcal{H}(t_{f})=1, \tag{45}\]  
from which \(\lambda_{\theta}=0\) during the whole flight. The optimal control law \(\alpha=\alpha^{\star}(t)\), that is, the optimal time variation of the thrust angle, comes from the [PERSON]'s maximum principle [48], according to which \(\alpha\) must maximize the Hamiltonian at any time instant. Observing from Equation (43) that  
\[\mathcal{H}^{\prime}\triangleq a_{D_{\oplus}}\,k\,\Big{(}\frac{r_{\oplus}}{r} \Big{)}^{2}\left[\lambda_{v_{r}}\,\cos\alpha+\lambda_{v_{\theta}}\,\sin\alpha\right] \tag{46}\]  
coincides with that part of \(\mathcal{H}\) that explicitly depends on \(\alpha\), and recalling that the thrust angle modulus is upper constrained by \(\alpha_{\max}\) of Equation (8), the optimal control law is found to be  
\[\alpha^{\star}= \begin{cases}\alpha_{\max}&\text{if }\sigma>\alpha_{\max}\\ -\alpha_{\max}&\text{if }\sigma<-\alpha_{\max}\\ \sigma&\text{otherwise},\end{cases}