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 : https://doi.org/10.1051/cocv:2008029
Classification : 93B50,  49K99
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},
zbl = {1158.49039},
mrnumber = {2488574},
language = {en},
url = {www.numdam.org/item/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/item/COCV_2009__15_1_173_0/

[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). | MR 2410710 | Zbl 1170.49035

[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. | MR 1472357 | Zbl 0952.49019

[3] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004). | MR 2062547 | Zbl 1062.93001

[4] A. Bressan and Y. Hong, Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313-331. | MR 2291823 | Zbl 1123.49028

[5] G.M. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems: an introduction. Oxford University Press (1998). | MR 1694383 | Zbl 0915.49001

[6] L. Cesari, Optimization theory and applications. Springer-Verlag (1983). | MR 688142 | Zbl 0506.49001

[7] R.V. Gamkrelidze, Principles of Optimal Control Theory. Plenum Press, New York (1978). | MR 686793 | Zbl 0401.49001

[8] A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995). | MR 1326374 | Zbl 0878.58020

[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. | MR 1731420 | Zbl 0961.49021

[10] M.P. Wojtkovski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math. 163 (2000) 177-191. | MR 1752103 | Zbl 0997.37011