Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

We consider a sub-Riemannian problem on the group of motions of three-dimensional space. Such a problem is encountered in the analysis of 3D images as well as in describing the motion of a solid body in a fluid. Mathematically, this problem reduces to solving a Hamiltonian system the vertical part of which is a system of six differential equations with unknown functions $u_1, \ldots, u_6$. The optimality consideration arising from the Pontryagin maximum principle implies that the last component of the vector control $\bar{u}$, denoted by $u_6$, must be constant. In the problem of the motion of a solid body in a fluid, this means that the fluid flow has a unique velocity potential, i.e., is vortex-free. The case ($u_6 = 0$), which is the most important for applications and at the same time the simplest, was rigorously studied by the authors in 2017. There, a solution to the system was found in explicit form. Namely, the extremal controls $u_1, \ldots, u_5$ were expressed in terms of elliptic functions. Now we consider the general case: $u_6$ is an arbitrary constant. In this case, we obtain a solution to the system in an operator form. Although the explicit form of the extremal controls does not follow directly from these formulas (their calculation requires the inversion of some nontrivial operator), it allows us to construct an approximate analytical solution for a small parameter $u_6$. Computer simulation shows a good agreement between the constructed analytical approximations and the solutions computed via numerical integration of the system.