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.
Mots-clés : infinite-horizon, optimal synthesis, hamiltonian dynamics
@article{COCV_2009__15_1_173_0,
author = {Agrachev, Andrei A. and Chittaro, Francesca C.},
title = {Smooth optimal synthesis for infinite horizon variational problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {173--188},
publisher = {EDP-Sciences},
volume = {15},
number = {1},
year = {2009},
doi = {10.1051/cocv:2008029},
mrnumber = {2488574},
zbl = {1158.49039},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008029/}
}
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 PY - 2009 SP - 173 EP - 188 VL - 15 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008029/ DO - 10.1051/cocv:2008029 LA - en ID - COCV_2009__15_1_173_0 ER -
%0 Journal Article %A Agrachev, Andrei A. %A Chittaro, Francesca C. %T Smooth optimal synthesis for infinite horizon variational problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 173-188 %V 15 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008029/ %R 10.1051/cocv:2008029 %G en %F COCV_2009__15_1_173_0
Agrachev, Andrei A.; Chittaro, Francesca C. Smooth optimal synthesis for infinite horizon variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 173-188. doi : 10.1051/cocv:2008029. http://www.numdam.org/articles/10.1051/cocv:2008029/
[1] , 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). | MR | Zbl
[2] and , Feedback-invariant optimal control theory and differential geometry, I. Regular extremals. J. Dyn. Contr. Syst. 3 (1997) 343-389. | MR | Zbl
[3] and , Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004). | MR | Zbl
[4] and , Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313-331. | MR | Zbl
[5] , and , One-dimensional variational problems: an introduction. Oxford University Press (1998). | MR | Zbl
[6] , Optimization theory and applications. Springer-Verlag (1983). | MR | Zbl
[7] , Principles of Optimal Control Theory. Plenum Press, New York (1978). | MR | Zbl
[8] and , Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995). | MR | Zbl
[9] and , Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. Appl. Math. Optim. 41 (2000) 237-254. | MR | Zbl
[10] , Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math. 163 (2000) 177-191. | MR | Zbl
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