Time-optimal trajectories construction in problem for the double-integrator system with the terminal constraint of a sloped line segment type

We consider a time-optimal problem for the double integrator system. Our problem generalizes the classical one by introducing terminal (endpoint) block from two ineqaulities and one equality that define the line segment with parametrized slope. Using Maximum pronciple, we get all types of optimal trajectories, construct optimal synthesis and provide its evolution analysis. This problem related also to motorized antenna movement either from current position to track segment or between two tracks (since we parametrize both of position and velocity)