Abstract <p>The problem on optimal reorientation of a solid (spacecraft) from an initial position into a prescribed final angular position on the basis of quaternions is solved. A combined criteria of quality is used, combining in a given proportion the contribution of control forces and the duration of maneuver, as well as the integral of the rotational energy. The synthesis of optimal control is based on a differential equation relating the attitude quaternion and angular momentum of a spacecraft. Analytical solution of optimal control problem is obtained using the necessary conditions of optimality in the form of the Pontryagin’s maximum principle. The properties of optimal rotation are studied in detail. Formalized equations and computational formulas are written to construct the optimal rotation program. Analytical equations and relations for finding the optimal control are presented. Key relations that determine the optimal values of the parameters of rotation control algorithm are given. A constructive scheme for solving the boundary-value problem of the maximum principle for arbitrary turning conditions (initial and final positions and moments of inertia of a solid) is given also. The made numerical experiments confirm the analytical conclusions. In the case of a dynamically symmetric solid body, the problem of spatial reorientation with minimum energy and time consumption is completely solved (in closed form). An example and results of mathematical modeling that confirm the practical feasibility of the developed method for orientation control are given.</p>