6,550 research outputs found
Singular trajectories of control-affine systems
When applying methods of optimal control to motion planning or stabilization
problems, some theoretical or numerical difficulties may arise, due to the
presence of specific trajectories, namely, singular minimizing trajectories of
the underlying optimal control problem. In this article, we provide
characterizations for singular trajectories of control-affine systems. We prove
that, under generic assumptions, such trajectories share nice properties,
related to computational aspects; more precisely, we show that, for a generic
system -- with respect to the Whitney topology --, all nontrivial singular
trajectories are of minimal order and of corank one. These results, established
both for driftless and for control-affine systems, extend previous results. As
a consequence, for generic systems having more than two vector fields, and for
a fixed cost, there do not exist minimizing singular trajectories. We also
prove that, given a control system satisfying the LARC, singular trajectories
are strictly abnormal, generically with respect to the cost. We then show how
these results can be used to derive regularity results for the value function
and in the theory of Hamilton-Jacobi equations, which in turn have applications
for stabilization and motion planning, both from the theoretical and
implementation issues
Singular trajectories of driftless and control-affine systems
We establish generic properties for singular trajectories, first for driftless, and then for control-affine systems. We show that, generically -- for the Whitney topology -- nontrivial singular trajectories are of minimal order and of corank one. As a consequence, if the number of vector fields of the system is greater than or equal to 3, then there exists generically no singular minimizing trajectory
Time-Optimal Trajectories of Generic Control-Affine Systems Have at Worst Iterated Fuller Singularities
We consider in this paper the regularity problem for time-optimal
trajectories of a single-input control-affine system on a n-dimensional
manifold. We prove that, under generic conditions on the drift and the
controlled vector field, any control u associated with an optimal trajectory is
smooth out of a countable set of times. More precisely, there exists an integer
K, only depending on the dimension n, such that the non-smoothness set of u is
made of isolated points, accumulations of isolated points, and so on up to K-th
order iterated accumulations
Optimality of broken extremals
In this paper we analyse the optimality of broken Pontryagin extremal for an
n-dimensional affine control system with a control parameter, taking values in
a k- dimensional closed ball. We prove the optimality of broken normal
extremals when n = 3 and the controllable vector fields form a contact
distribution, and when the Lie algebra of the controllable fields is locally
orthogonal to the singular locus and the drift does not belong to it. Moreover,
if k = 2, we show the optimality of any broken extremal even abnormal when the
controllable fields do not form a contact distribution in the point of
singularity.Comment: arXiv admin note: text overlap with arXiv:1610.0675
Geometric optimal control of the contrast imaging problem in Nuclear Magnetic Resonance
The objective of this article is to introduce the tools to analyze the
contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories
can be selected among extremal solutions of the Pontryagin Maximum Principle
applied to this Mayer type optimal problem. Such trajectories are associated to
the question of extremizing the transfer time. Hence the optimal problem is
reduced to the analysis of the Hamiltonian dynamics related to singular
extremals and their optimality status. This is illustrated by using the
examples of cerebrospinal fluid / water and grey / white matter of cerebrum.Comment: 30 pages, 13 figur
- …