Motion planning and fastly oscillating controls. (English) Zbl 1206.53035
Given a control system \(\Delta\) and a parametrized curve \(\Gamma\) in the phase space, the authors study the problem of approximating \(\Gamma\) optimally (or suboptimally) by a trajectory of the system. The control system is
\[ x'(t) = \sum_{j=1}^p F_j(x(t)) u_j(t) \]
with phase space an open set in \(n\)-dimensional space and \(\Gamma\) satisfies (generic) conditions of transversality type with respect to \(\Delta\). The approximation is done in the framework of sub-Riemannian geometry, and the suboptimal approximation is achieved using fastly oscillating sinusoidal controls.
\[ x'(t) = \sum_{j=1}^p F_j(x(t)) u_j(t) \]
with phase space an open set in \(n\)-dimensional space and \(\Gamma\) satisfies (generic) conditions of transversality type with respect to \(\Delta\). The approximation is done in the framework of sub-Riemannian geometry, and the suboptimal approximation is achieved using fastly oscillating sinusoidal controls.
Reviewer: Hector O. Fattorini (Los Angeles)
MSC:
| 53C17 | Sub-Riemannian geometry |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
| 34H05 | Control problems involving ordinary differential equations |
