Smooth optimal synthesis for infinite horizon variational problems

Andrei A. Agrachev; Francesca C. Chittaro

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 173-188
  • ISSN: 1292-8119

Abstract

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We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.

How to cite

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Agrachev, Andrei A., and Chittaro, Francesca C.. "Smooth optimal synthesis for infinite horizon variational problems." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 173-188. <http://eudml.org/doc/250545>.

@article{Agrachev2009,
abstract = { We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems. },
author = {Agrachev, Andrei A., Chittaro, Francesca C.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Infinite-horizon; optimal synthesis; Hamiltonian dynamics; infinite-horizon},
language = {eng},
month = {1},
number = {1},
pages = {173-188},
publisher = {EDP Sciences},
title = {Smooth optimal synthesis for infinite horizon variational problems},
url = {http://eudml.org/doc/250545},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Agrachev, Andrei A.
AU - Chittaro, Francesca C.
TI - Smooth optimal synthesis for infinite horizon variational problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 173
EP - 188
AB - We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.
LA - eng
KW - Infinite-horizon; optimal synthesis; Hamiltonian dynamics; infinite-horizon
UR - http://eudml.org/doc/250545
ER -

References

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  1. A.A. Agrachev, Geometry of Optimal Control Problem and Hamiltonian Systems, in Nonlinear and Optimal Control Theory, Lecture Notes in Mathematics1932, Fondazione C.I.M.E., Firenze, Springer-Verlag (2008).  
  2. A.A. Agrachev and R.V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals. J. Dyn. Contr. Syst.3 (1997) 343–389.  Zbl0952.49019
  3. A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004).  Zbl1062.93001
  4. A. Bressan and Y. Hong, Optimal control problems on stratified domains. Netw. Heterog. Media2 (2007) 313–331.  Zbl1123.49028
  5. G.M. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems: an introduction. Oxford University Press (1998).  Zbl0915.49001
  6. L. Cesari, Optimization theory and applications. Springer-Verlag (1983).  Zbl0506.49001
  7. R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978).  Zbl0401.49001
  8. A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995).  Zbl0878.58020
  9. A.V. Sarychev and D.F.M. Torres, Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. Appl. Math. Optim.41 (2000) 237–254.  Zbl0961.49021
  10. M.P. Wojtkovski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math.163 (2000) 177–191.  Zbl0997.37011

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