## 2 Mathematical Preliminaries
### Trajectory Optimization  
The minimum time circle-to-circle orbit transfer of a diffractive sail is obtained with an indirect method [62]. The Hamiltonian function \(\mathcal{H}\) is defined as  
\[\mathcal{H}=\lambda,u+\frac{\lambda_{\theta}\,v}{r}-\frac{\lambda_{u}\,\mu_{ \odot}}{r^{2}}+\frac{a_{\mathrm{c}}\,\lambda_{u}}{\sqrt{2}}\left(\frac{r_{ \oplus}}{r}\right)^{2}-\frac{\lambda_{v}\,u\,v}{r}+\mathcal{H}^{\prime} \tag{10}\]  
where \(\mathcal{H}^{\prime}\) is that part of \(\mathcal{H}\) that explicitly depends on the control variable \(\tau\), that is  
\[\mathcal{H}^{\prime}=-\frac{a_{\mathrm{c}}\,\lambda_{v}\,\tau}{\sqrt{2}}\left( \frac{r_{\oplus}}{r}\right)^{2} \tag{11}\]  
and \(\{\lambda_{r},\lambda_{\theta},\lambda_{u},\lambda_{v}\}\) are the variables adjoint to the states \(\{r,\theta,u,v\}\). The optimal control law, that is, the time variation of \(\tau\), is found by means of [PERSON]'s maximum principle, which requires the reduced Hamiltonian \(\mathcal{H}^{\prime}\) to be maximized at any time instant. Observing that the sign of Equation (11) only depends on the product \(\lambda_{v}\,\tau\), the control law that maximizes \(\mathcal{H}^{\prime}\) is found when \(\text{sign}(\tau\,\lambda_{v})=-1\,\,\forall t\geq t_{0}\), that is  
\[\tau=-\text{sign}(\lambda_{v}) \tag{12}\]  
where \(\text{sign}(\Box)\) is the signum function. Equation (12) states that the spacecraft optimal transfer trajectory is a sequence of patched arcs, where \(\tau\) is constant in each arc, and the diffractive film state change is determined by the sign of the adjoint variable \(\lambda_{v}\). This situation is illustrated in Figure 8 in the exemplary case of four arcs. Figure 8: Sequence of trajectory arcs related to the value of the adjoint variable \(\lambda_{v}\). The time variation of the adjoint variables is given by the Euler-Lagrange equations  
\[\dot{\lambda}_{r} =-\frac{\partial\mathcal{H}}{\partial r}=\frac{v(\lambda_{\theta}-u \lambda_{v}+v\lambda_{u})}{r^{2}}-\frac{2\lambda_{u}\mu_{\odot}+\sqrt{2}\,a_{c} \,r_{\oplus}^{2}(\tau\,\lambda_{v}-\lambda_{u})}{r^{3}} \tag{13}\] \[\dot{\lambda}_{\theta} =-\frac{\partial\mathcal{H}}{\partial\theta}=0\] (14) \[\dot{\lambda}_{u} =-\frac{\partial\mathcal{H}}{\partial u}=\frac{v\,\lambda_{v}}{ r}-\lambda_{r}\] (15) \[\dot{\lambda}_{v} =-\frac{\partial\mathcal{H}}{\partial v}=\frac{u\,\lambda_{v}-2\, v\,\lambda_{u}-\lambda_{\theta}}{r} \tag{16}\]  
which are completed by the four initial conditions  
\[\lambda_{r}(t_{0})=\lambda_{r_{0}}\quad,\quad\lambda_{\theta}(t_{0})=\lambda_ {\theta_{0}}\quad,\quad\lambda_{u}(t_{0})=\lambda_{u_{0}}\quad,\quad\lambda_{ v}(t_{0})=\lambda_{v_{0}} \tag{17}\]  
where \(\{\lambda_{\alpha},\lambda_{\theta_{0}},\lambda_{u_{0}},\lambda_{v_{0}}\}\) are four unknowns to be found by enforcing the transversality condition [62; 63] and the desired value of the spacecraft state variables at the end of the transfer. In a circle-to-circle transfer, it is required that at the final time \(t=t_{f}\), the spacecraft be inserted into the target circular orbit, from which  
\[r(t_{f})=r_{f}\quad,\quad u(t_{f})=0\quad,\quad v(t_{f})=\sqrt{\frac{\mu_{ \odot}}{r_{f}}} \tag{18}\]  
Note that the adjoint variable \(\lambda_{\theta}\) is a constant of motion, and the polar angle at the final time \(t_{f}\) is left free, that is, the spacecraft angular position on the target circular orbit is an output of the optimization process. In a minimum time orbit transfer, the performance index \(J\) to be maximized can be written as  
\[\mathcal{H}^{\prime}=-\frac{a_{\mathrm{c}}\,\lambda_{v}\,\tau}{\sqrt{2}}\left( \frac{r_{\oplus}}{r}\right)^{2} \tag{11}\]0  
so that the transversality condition [62; 63] gives  
\[\mathcal{H}^{\prime}=-\frac{a_{\mathrm{c}}\,\lambda_{v}\,\tau}{\sqrt{2}}\left( \frac{r_{\oplus}}{r}\right)^{2} \tag{11}\]1  
When Equation (14) is combined with the first of Equation (20), the result is that \(\lambda_{\theta}=0\) during the whole transfer, that is, the second of Equation (17) reads \(\lambda_{\theta_{0}}=0\).