Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T01:33:07.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Smooth optimal synthesis for infinite horizonvariational problems

Published online by Cambridge University Press:  23 January 2009

Andrei A. Agrachev  and
Francesca C. Chittaro
Andrei A. Agrachev
Affiliation:
SISSA, via Beirut 2-4, 34014 Trieste, Italy. agrachev@sissa.it
Francesca C. Chittaro
Affiliation:
Dipartimento di Matematica Applicata “G. Sansone”, via S. Marta 3, 50139 Firenze, Italy. chittaro@math.unifi.it
Get access

Abstract

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

Keywords

Infinite-horizonoptimal synthesisHamiltonian dynamics
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 173 - 188
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A.A. Agrachev, Geometry of Optimal Control Problem and Hamiltonian Systems, in Nonlinear and Optimal Control Theory, Lecture Notes in Mathematics 1932, Fondazione C.I.M.E., Firenze, Springer-Verlag (2008).
Agrachev, A.A. and Gamkrelidze, R.V., Feedback-invariant optimal control theory and differential geometry, I. Regular extremals. J. Dyn. Contr. Syst. 3 (1997) 343–389. CrossRef
A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004).
Bressan, A. and Hong, Y., Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313–331.
G.M. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems: an introduction. Oxford University Press (1998).
L. Cesari, Optimization theory and applications. Springer-Verlag (1983).
R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978).
A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995).
Sarychev, A.V. and Torres, D.F.M., Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. Appl. Math. Optim. 41 (2000) 237–254. CrossRef
Wojtkovski, M.P., Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math. 163 (2000) 177–191.