Second-order tests in optimization theories View Full Text


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Article Info

DATE

1975-06

AUTHORS

C. Marchal

ABSTRACT

There are two main kinds of second-order tests in optimization theories. The simplest tests (Refs. 1–13), such as the generalized Legendre-Clebsch condition, require a local study around points of the trajectory of interest (generally, on a singular arc). Tests of the second kind (Refs. 14–22 and 51–53) are more difficult to use but also more efficient: they can be applied under various assumptions of convexity or linearity and generally require some integrations along the trajectory of interest (usually, these integrations can only be done by numerical methods). In favorable cases, the better tests of this second kind lead to either the conclusion of nonoptimality or the conclusion of local optimality.This survey paper begins with a broader question, the question of sufficient conditions for absolute optimality. Some results of this study are used to define anadjoint matrix (extension of the notion ofadjoint vector of Pontryagin). Then, the different second-order tests can be unified and generalized even if the trajectory of interest has switches and singular arcs.On a given trajectory, theconjugate points are easily related to the evolution of the adjoint matrix.Finally, this generalized second-order test is applied to a singular arc of astrodynamics, the reversible arc: this arc is globally optimal from end to end. More... »

PAGES

633-666

References to SciGraph publications

  • 1968-09. Sufficiency theorems for optimal control in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1971-09. A sufficiency theorem for optimal control in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1969-01. A note on a sufficiency theorem for optimal control in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1962. Optimal Intermediate-Thrust Arcs in a Gravitational Field in SPACE FLIGHT AND RE-ENTRY TRAJECTORIES
  • 1962. Etude théorique des trajectoires optimales dans un champ de gravitation. Application au cas d’un centre d’attraction unique in SPACE FLIGHT AND RE-ENTRY TRAJECTORIES
  • 1973-05. Chattering arcs and chattering controls in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1971-02. Sufficient conditions for optimal control with state and control constraints in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1970. Conditions d'optimalité pour les domaines de manoeuvrabilité à frontière semi-affine in COLLOQUIUM ON METHODS OF OPTIMIZATION
  • 1975-12. Minimum-fuel rocket trajectories involving intermediate-thrust arcs in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
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    http://scigraph.springernature.com/pub.10.1007/bf00935505

    DOI

    http://dx.doi.org/10.1007/bf00935505

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