337 research outputs found
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
-Minimization for Mechanical Systems
Second order systems whose drift is defined by the gradient of a given
potential are considered, and minimization of the -norm of the control is
addressed. An analysis of the extremal flow emphasizes the role of singular
trajectories of order two [25,29]; the case of the two-body potential is
treated in detail. In -minimization, regular extremals are associated with
controls whose norm is bang-bang; in order to assess their optimality
properties, sufficient conditions are given for broken extremals and related to
the no-fold conditions of [20]. An example of numerical verification of these
conditions is proposed on a problem coming from space mechanics
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
Optimality conditions applied to free-time multi-burn optimal orbital transfers
While the Pontryagin Maximum Principle can be used to calculate candidate
extremals for optimal orbital transfer problems, these candidates cannot be
guaranteed to be at least locally optimal unless sufficient optimality
conditions are satisfied. In this paper, through constructing a parameterized
family of extremals around a reference extremal, some second-order necessary
and sufficient conditions for the strong-local optimality of the free-time
multi-burn fuel-optimal transfer are established under certain regularity
assumptions. Moreover, the numerical procedure for computing these optimality
conditions is presented. Finally, two medium-thrust fuel-optimal trajectories
with different number of burn arcs for a typical orbital transfer problem are
computed and the local optimality of the two computed trajectories are tested
thanks to the second-order optimality conditions established in this paper
Branes as solutions of gauge theories in gravitational field
The idea of the Gauss map is unified with the concept of branes as
hypersurfaces embedded into -dimensional Minkowski space. The map introduces
new generalized coordinates of branes alternative to their world vectors
and identified with the gauge and other massless fields. In these
coordinates the Dirac -branes realize extremals of the Euler-Lagrange
equations of motion of a -dimensional gauge-invariant action
in a gravitational backgroundComment: 21 pages. Published version: extended Introduction, additional
clarifications and comments, new references and improved styl
Minimum fuel horizontal flightpaths in the terminal area
The problem of minimum fuel airplane trajectories from arbitrary initial states to be fixed final state is considered. There are four state variables (two position coordinates, heading, and constrained velocity) and two constrained controls (thrust and bank angle). The fuel optimality of circular and straight line flightpaths is examined. Representative extremals (trajectories satisfying the necessary conditions of the minimum principle) of various types are computed and used to evaluate trajectories generated by an on line algorithm. Attention is paid to the existence of Darboux points (beyond which an extremal ceases to be globally optimal). One fuel flow rate model includes a term quadratic in thrust; hence, the optimal thrust is continuous and nonsingular. The other fuel flow rate model is linear in thrust, and consequently the optimal thrust is discontinuous and singular
Noether type discrete conserved quantities arising from a finite element approximation of a variational problem
In this work we prove a weak Noether type theorem for a class of variational
problems which include broken extremals. We then use this result to prove
discrete Noether type conservation laws for certain classes of finite element
discretisation of a model elliptic problem. In addition we study how well the
finite element scheme satisfies the continuous conservation laws arising from
the application of Noether's 1st Theorem (E. Noether 1918).
We summarise extensive numerical tests, illustrating the conservativity of
the discrete Noether law using the --Laplacian as an example.Comment: 17 pages, 3 figures, 3 table
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