Threshold Absolute Value of a Relay Control when Time-Optimally Bringing a Satellite to a Gravitationally Stable Position

The time-optimal problem is considered for the orientation of a satellite in the plane of a circular orbit and bringing it to the desirable angular position. The constraint on the control torque module is considered as the main parameter of the problem. It turned out that even in the case when the admissible control torque exceeds the gravitational moment in magnitude, additional switching curves corresponding to the relay control with two switchings are possible in the phase plane. As a result, a simple numerical algorithm is proposed, and with its help for each terminal angular position the threshold absolute control value is found at which the indicated switching curves are infinitesimal. Their coordinates in the phase plane are found in the most well-known case of the gravitationally stable terminal angular position. A suboptimal control law, which does not need a numerical calculation of the switching curve, is offered.