983 research outputs found
Pontryagin Maximum Principle and Stokes Theorem
We present a new geometric unfolding of a prototype problem of optimal
control theory, the Mayer problem. This approach is crucially based on the
Stokes Theorem and yields to a necessary and sufficient condition that
characterizes the optimal solutions, from which the classical Pontryagin
Maximum Principle is derived in a new insightful way. It also suggests
generalizations in diverse directions of such famous principle.Comment: 21 pages, 7 figures; we corrected a few minor misprints, added a
couple of references and inserted a new section (Sect. 7); to appear in
Journal of Geometry and Physic
Invariant Carnot-Caratheodory metrics on , , and lens spaces
In this paper we study the invariant Carnot-Caratheodory metrics on
, and induced by their Cartan decomposition
and by the Killing form. Beside computing explicitly geodesics and conjugate
loci, we compute the cut loci (globally) and we give the expression of the
Carnot-Caratheodory distance as the inverse of an elementary function. We then
prove that the metric given on projects on the so called lens spaces
. Also for lens spaces, we compute the cut loci (globally).
For the cut locus is a maximal circle without one point. In all other
cases the cut locus is a stratified set. To our knowledge, this is the first
explicit computation of the whole cut locus in sub-Riemannian geometry, except
for the trivial case of the Heisenberg group
Low-Thrust Lyapunov to Lyapunov and Halo to Halo with -Minimization
In this work, we develop a new method to design energy minimum low-thrust
missions (L2-minimization). In the Circular Restricted Three Body Problem, the
knowledge of invariant manifolds helps us initialize an indirect method solving
a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the
optimal control problem is solved using Newton-like algorithms finding the zero
of a shooting function. To compute a Lyapunov to Lyapunov mission, we first
compute an admissible trajectory using a heteroclinic orbit between the two
periodic orbits. It is then used to initialize a multiple shooting method in
order to release the constraint. We finally optimize the terminal points on the
periodic orbits. Moreover, we use continuation methods on position and on
thrust, in order to gain robustness. A more general Halo to Halo mission, with
different energies, is computed in the last section without heteroclinic orbits
but using invariant manifolds to initialize shooting methods with a similar
approach
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