## 2 Thrust Vector Description and Mathematical Model
### Notes on the Control Law and the sailcraft's Trajectory Design  
For an assigned value of the dimensionless pair \(\{\widetilde{a}_{c},\Delta t\}\), the control law \(\tau=\tau(\widetilde{t})\) is selected to minimize the flight time \(\Delta\widetilde{t}\) required to reach the final boundary conditions reported in Equation (7). In this regard, the minimization of the time of flight (or, equivalently, the maximization of the dimensionless performance index \(J\triangleq-\widetilde{t}_{f}\)) has been obtained using an indirect method [52] and following the general approach illustrated by [PERSON] and [PERSON] [44], to which the interested reader is referred for useful comments on the procedure illustrated below. In particular, the method used in this part of the section is general, in the sense that it can be applied to any elliptical orbit with a prescribed eccentricity \(e_{0}\). Within this framework, the special case of a circular reference orbit (i.e., \(e_{0}=0\)) allows for a simplified form of the control law, as discussed later in this paper.  
The employed method, based on the calculus of variations, introduces the dimensionless variables \(\{\lambda_{\widetilde{r}},\,\lambda_{\theta},\,\lambda_{\widetilde{r}_{f}}, \,\lambda_{\widetilde{v}_{t}}\}\) adjoint to the states \(\{\widetilde{r},\,\theta,\,\widetilde{v}_{r},\,\widetilde{v}_{t}\}\) to write the dimensionless Hamiltonian function \(\mathcal{H}\) defined as  
\[\mathcal{H}=\lambda_{\widetilde{r}}\,\widetilde{v}_{r}+\frac{\lambda_{\theta} \,\widetilde{v}_{t}}{\widetilde{r}}+\frac{\lambda_{\widetilde{v}_{t}}\, \widetilde{v}_{t}^{2}}{\widetilde{r}}+\frac{\lambda_{\widetilde{v}_{r}}\,( \widetilde{a}_{c}\,\widetilde{r}_{\widetilde{v}_{\widetilde{v}}}^{2}-\sqrt{ 2})}{\sqrt{2}\,\widetilde{r}^{2}}-\frac{\lambda_{\widetilde{v}_{t}}\, \widetilde{v}_{r}\,\widetilde{v}_{t}}{\widetilde{r}}-\frac{\tau\,\lambda_{ \widetilde{v}_{t}}\,\widetilde{a}_{c}\,\widetilde{r}_{\widetilde{v}_{ \widetilde{v}}}^{2}}{\sqrt{2}\,\widetilde{r}^{2}} \tag{8}\]  
which gives the Euler-Lagrange equations  
\[\lambda_{\widetilde{r}}^{\prime}=-\frac{\partial\mathcal{H}}{\partial \widetilde{r}},\quad\lambda_{\theta}^{\prime}=-\frac{\partial\mathcal{H}}{ \partial\theta},\quad\lambda_{\widetilde{v}_{r}}^{\prime}=-\frac{\partial \mathcal{H}}{\partial\widetilde{v}_{r}},\quad\lambda_{\widetilde{v}_{t}}^{ \prime}=-\frac{\partial\mathcal{H}}{\partial\widetilde{v}_{t}} \tag{9}\]  
whose explicit form is  
\[\lambda_{\widetilde{r}}^{\prime  
where \(\text{sign}(\square)\) is the signum function. The latter equation, in fact, gives the value of the binary control term that maximizes the local value of the Hamiltonian function. Note that Equation (11) indicates that \(\lambda_{\theta}\) is a constant of motion, so keeping in mind that the Euler-Lagrange equations are invariant under a scaling factor, the constant value of \(\lambda_{\theta}\) is taken as a scaling factor, i.e., \(|\lambda_{\theta}|=1\). The minimum flight time \(\tilde{t}_{f}\) and the initial value of the remaining three adjoint variables \(\{\lambda_{\tilde{\tau}}(\tilde{t}_{0})\), \(\lambda_{\tilde{\omega}_{\tilde{\tau}}}(\tilde{t}_{0})\), \(\lambda_{\tilde{\omega}_{\tilde{\tau}}}(\tilde{t}_{0})\}\) are obtained by enforcing the four final boundary conditions summarized in Equation (7). In this regard, a typical single-shooting procedure has been employed with a trial-and-error procedure to estimate the first guess solution, because the analytical method recently proposed in [54] fails in this specific scenario.