337 research outputs found

    Optimality of broken extremals

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    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

    L1L^1-Minimization for Mechanical Systems

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    Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the L1L^1-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 L1L^1-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

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    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

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    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

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    The idea of the Gauss map is unified with the concept of branes as hypersurfaces embedded into DD-dimensional Minkowski space. The map introduces new generalized coordinates of branes alternative to their world vectors x\mathbf{x} and identified with the gauge and other massless fields. In these coordinates the Dirac pp-branes realize extremals of the Euler-Lagrange equations of motion of a (p+1)(p+1)-dimensional SO(Dp1)SO(D-p-1) 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

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    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

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    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 pp--Laplacian as an example.Comment: 17 pages, 3 figures, 3 table
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