On the simplest time-optimal problem with phase constraints in the control of the spatial orientation of a body

Various problems of control of the spatial orientation of a rigid body arise in many applied problems of control of this body in the atmosphere under gravitational and reactive forces. Here, one of such control problems is investigated in an extremely simplified statement. The rotational motion of a rod around its mass center on a plane under the action of a force which is constant in absolute value and applied to one of the ends of the rod is considered. The rate of change of the angle between the rod and the vector that specifies the direction of the force is used as a control parameter. There are constraints on the control and the current phase state of the linear dynamic system describing the rod motion. The desired control must satisfy the constraints and take the system from its initial state to a given terminal state in the minimum time provided the phase constraints are fulfilled. The structure of the optimal control is designed in the time-optimal problem in the presence of phase constraints. The questions of existence and uniqueness of a solution of this optimal control problem are discussed. The obtained results are illustrated by examples of numerical solution of several model problems.