Smooth optimal synthesis for infinite horizon variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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
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     title = {Smooth optimal synthesis for infinite horizon variational problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {173--188},
     publisher = {EDP-Sciences},
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Agrachev, Andrei A.; Chittaro, Francesca C. Smooth optimal synthesis for infinite horizon variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 1, pp. 173-188. doi : 10.1051/cocv:2008029. http://archive.numdam.org/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). | MR | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

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