Angular velocity control of rigid body motions

The problem of finding a motion of a rigid body in three space with angular velocity close to a prescribed vector function ω is considered. In particular, if u is the angular velocity of the body, one seeks minimizers of |||u-ω|||_p on an admissible class consisting of C² rigid body motions on [0,1] with prescribed endpoints for 1⩽p⩽∞. It is shown that all of these problems are equivalent to autonomous problems that admit complete closed-form solutions when one works in an appropriate moving frame. While this moving frame must in general be obtained numerically, it can be obtained in advance, independently of the solutions; hence, a complete classification of the extremals for this problem is obtained along with existence and uniqueness results