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

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},
     year = {2009},
     publisher = {EDP Sciences},
     volume = {15},
     number = {1},
     doi = {10.1051/cocv:2008029},
     mrnumber = {2488574},
     zbl = {1158.49039},
     language = {en},
     url = {https://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  - https://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 https://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

[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 :