Finding the principal bifurcation value of the maximum control torque in the problem of optimal control synthesis for a pendulum

The principal bifurcation value in the problem of synthesis of a time-optimal control for damping the oscillations of a nonlinear pendulum is found. A one-link pendulum controlled by a torque of forces produced at the suspension point is considered. The magnitude that bounds the modulus of the control torque is the only essential parameter of the investigated system. It is supposed that an admissible control may exceed the moment of the gravitational force. By the principal bifurcation value, we mean the value of the maximum admissible control torque for which a trajectory with two switches of relay optimal control arises in the synthesis pattern. This publication is a logical complement to papers [1-3], where the mentioned synthesis patterns were obtained numerically with the use of the maximum principle. In contrast to [1-3], we formulate a system of explicit relations of integral type for the numerical construction of the boundaries of domains of the phase plane inside which the trajectories with two switches start. Based on these relations, an analytical upper estimate of the principal bifurcation value is obtained. This value is also found numerically. In a special coordinate system, the synthesis patterns are constructed that illustrate the specific features of the optimal control for large angular velocities of pendulum rotations.