The geometry of the plate-ball problem
The author considers the plate-ball problem introduced recently by \textit{R. Brockett} and \textit{L. Dai} in an unpublished preprint. The author presents this as an optimal control problem on the five-dimensional Lie group \(G = {\mathbb{R}}^ 2 \times \text{SO}(3)\) described by the differential system \[ {dx_ 1\over dt} = u_ 1,\quad {dx_ 2 \over dt} = u_ 2, \quad {dR \over dt} = R \left( \begin{smallmatrix} 0 & 0 & -u_ 1 \\ 0 & 0 & -u_ 2\\ u_ 1 & u_ 2 & 0\end{smallmatrix} \right), \] where \(R\) is the rotation matrix in \(\text{SO}(3)\) and \(u_ 1\) and \(u_ 2\) are the control functions. One has to minimize \({1\over 2} \int_ 0 ^ T (u_ 1^ 2 + u_ 2^ 2) dt\) over all solution curves subject to the prescribed boundary condition in \(G\). The author shows that the problem is integrable in terms of elliptic functions. He shows also that the solution is related to the classical problem of the elastica described in a survey paper by \textit{T. Truesdell} [Bull. Am. Math. Soc. 9, 293-310 (1983; Zbl 0555.73030)].
scientific article; zbMATH DE number 484802