# 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

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topAgrachev, 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

top- 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).
- 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
- A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004). Zbl1062.93001
- A. Bressan and Y. Hong, Optimal control problems on stratified domains. Netw. Heterog. Media2 (2007) 313–331. Zbl1123.49028
- G.M. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems: an introduction. Oxford University Press (1998). Zbl0915.49001
- L. Cesari, Optimization theory and applications. Springer-Verlag (1983). Zbl0506.49001
- R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978). Zbl0401.49001
- A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995). Zbl0878.58020
- 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
- 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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