Complex calculus of variations

Michel Gondran; Rita Hoblos Saade

Kybernetika (2003)

  • Volume: 39, Issue: 2, page [249]-263
  • ISSN: 0023-5954

Abstract

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In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to functions in . It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, in particular by generalizing the Hopf-Lax formula.

How to cite

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Gondran, Michel, and Saade, Rita Hoblos. "Complex calculus of variations." Kybernetika 39.2 (2003): [249]-263. <http://eudml.org/doc/33638>.

@article{Gondran2003,
abstract = {In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to $\{\mathbf \{C\}\}^n$ functions in $\{\mathbf \{C\}\}$. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, in particular by generalizing the Hopf-Lax formula.},
author = {Gondran, Michel, Saade, Rita Hoblos},
journal = {Kybernetika},
keywords = {complex calculus of variation; Hamilton-Jacobi equations; complex calculus of variation; Hamilton-Jacobi equations},
language = {eng},
number = {2},
pages = {[249]-263},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Complex calculus of variations},
url = {http://eudml.org/doc/33638},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Gondran, Michel
AU - Saade, Rita Hoblos
TI - Complex calculus of variations
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 2
SP - [249]
EP - 263
AB - In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to ${\mathbf {C}}^n$ functions in ${\mathbf {C}}$. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, in particular by generalizing the Hopf-Lax formula.
LA - eng
KW - complex calculus of variation; Hamilton-Jacobi equations; complex calculus of variation; Hamilton-Jacobi equations
UR - http://eudml.org/doc/33638
ER -

References

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  1. Balian R., Bloch C., Solution of the Schrödinger Equation in Terms of Classical Paths, Academic Press, New York 1974 Zbl0281.35029MR0438937
  2. Evans L. C., Partial Differential Equations, (Graduate Studies in Mathematics 19.) American Mathematical Society, 1998 (19.)) MR1625845
  3. Gondran M., 10.1016/S0764-4442(99)90007-1, C.R. Acad. Sci., Paris 1999, t. 329, série I, pp. 783–777 (1999) MR1724540DOI10.1016/S0764-4442(99)90007-1
  4. Gondran M., 10.1016/S0764-4442(01)01901-2, C.R. Acad. Sci., Paris 2001, t. 332, série I, pp. 677–680 Zbl1007.49014MR1842467DOI10.1016/S0764-4442(01)01901-2
  5. Lions P. L., Generalized Solutions of Hamilton–Jacobi Equations, (Research Notes in Mathematics 69.) Pitman, London 1982 Zbl0497.35001MR0667669
  6. Voros A., The return of the quadratic oscillator, The complex WKB method. Ann. Inst. H. Poincaré Phys. Théor. 39 (1983), 3, 211–338 (1983) MR0729194

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