The case of time minimization for affine control systems with control on the disk is studied. After recalling the standard sufficient conditions for local optimality in the smooth case, the analysis focusses on the specific type of singularities encountered when the control is prescribed to the disk. Using a suitable stratification, the regularity of the flow is analyzed, which helps to devise verifiable sufficient conditions in terms of left and right limits of Jacobi fields at a switching point. Under the appropriate assumptions, piecewise regularity of the field of extremals is obtained.
Affine control problems arise naturally from controlled mechanical systems. Building on previous results [1,9] we prove that, in the case of time minimization with control on the disk, the extremal flow given by Pontrjagin's maximum principle is smooth along the strata of a well-chosen stratification. We also study this flow in terms of regular-singular transition and prove that the singularity along time-minimizing extremals crossing these strata is at most logarithmic. We then apply these results to mechanical systems, paying special attention to the case of the controlled three body problem.
We prove, using Moralès-Ramis theorem, that the minimum-time controlled Kepler problem is not meromorphically integrable in the Liouville sens on the Riemann surface of its Hamiltonian.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.