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2. Analytical solution
This section establishes a state feedback formula for the optimal thrust direction and shows that the optimal injection point is an apside of the target orbit. These properties allow us to reduce the optimal control problem to a non-linear one.
2.1 Thrust direction
Figure 1 shows the vectors and angles used for the calculations. The trajectory takes place in a plane (Oxy), with origin O at the center of the Earth and arbitrary inertial axes Ox and Oy. φ is the longitude (or polar angle) from Ox. θ and γ are the local pitch of the launcher and the local slope of the velocity, respectively. By convention, angles θ and γ measured from the local horizontal are positive upwards....
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