Geodesic
Smooth optimal synthesis for infinite horizon variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 173-188.

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1051/cocv:2008029
Classification : 93B50, 49K99
Keywords: 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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.1051/cocv:2008029/

[1] 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). | Zbl | MR

[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. | Zbl | MR

[3] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004). | Zbl | MR

[4] A. Bressan and Y. Hong, Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313-331. | Zbl | MR

[5] G.M. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems: an introduction. Oxford University Press (1998). | Zbl | MR

[6] L. Cesari, Optimization theory and applications. Springer-Verlag (1983). | Zbl | MR

[7] R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978). | Zbl | MR

[8] A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995). | Zbl | MR

[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. | Zbl | MR

[10] M.P. Wojtkovski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math. 163 (2000) 177-191. | Zbl | MR

Cité par Sources :