The existence of smooth solutions in a problem of the optimal control of the rotation of an axisymmetric rigid body

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The problem of the existence of a solution in the problem of the optimal control of the rotation of an axisymmetric rigid body for the arbitrary case of angular velocity boundary conditions is studied. A square integrable functional, which is consistent with the symmetry of the rotating body and characterizes the power consumption, is chosen as the criterion. The principal moment of the applied external forces serves as the control and the time of termination of a manoeuvre can be both specified as well as free. In the case of a specified termination time, it is shown that the solution (control) belongs to the class of infinitely-differentiable functions of time. The reasoning is based on the use of the singularities of the structure of the differential equations and the possibility of reducing the initial problem to two successive variational problems. The existence of a solution of the first of these problems in the class of square integrable functions is proved using the Cauchy–Bunyakovskii inequality. The second problem reduces to a search for the minimum of a functional which is weakly lower semi-continuous on a weakly compact set and the existence of its solution in the same class of functions follows from the Weierstrass theorem. The required conclusion concerning the smoothness of the solution of the optimal control problem is obtained from the necessary conditions of Pontryagin's maximum principle. In the case of a free termination time, one of the minimizing sequence can be constructed and it can be shown that, in the general case, there is no solution in the class of measurable controls.