We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.
Mots clés : optimal control, value function, semiconcavity
@article{COCV_2004__10_4_666_0,
author = {Sinestrari, Carlo},
title = {Regularity along optimal trajectories of the value function of a {Mayer} problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {666--676},
publisher = {EDP-Sciences},
volume = {10},
number = {4},
year = {2004},
doi = {10.1051/cocv:2004026},
mrnumber = {2111087},
zbl = {1068.49028},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2004026/}
}
TY - JOUR AU - Sinestrari, Carlo TI - Regularity along optimal trajectories of the value function of a Mayer problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 666 EP - 676 VL - 10 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2004026/ DO - 10.1051/cocv:2004026 LA - en ID - COCV_2004__10_4_666_0 ER -
%0 Journal Article %A Sinestrari, Carlo %T Regularity along optimal trajectories of the value function of a Mayer problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 666-676 %V 10 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2004026/ %R 10.1051/cocv:2004026 %G en %F COCV_2004__10_4_666_0
Sinestrari, Carlo. Regularity along optimal trajectories of the value function of a Mayer problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 666-676. doi : 10.1051/cocv:2004026. http://archive.numdam.org/articles/10.1051/cocv:2004026/
[1] and, Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch. Ration. Mech. Anal. 162 (2002) 1-23. | MR | Zbl
[2] and, Optimal control and viscosity solutions of Hamilton-Jacobi equations. Birkhäuser, Boston (1997). | MR | Zbl
[3] and, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322-1347. | MR | Zbl
[4] , and, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dyn. Syst. 6 (2000) 975-997. | MR | Zbl
[5] and, Convexity properties of the minimum time function. Calc. Var. 3 (1995) 273-298. | MR | Zbl
[6] and, On a class of nonlinear time optimal control problems. Discrete Contin. Dyn. Syst. 1 (1995) 285-300. | MR | Zbl
[7] and, Semiconcave functions, Hamilton-Jacobi equations and optimal control. Birkhäuser, Boston (2004). | MR | Zbl
[8] and, Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications. Nonlinear Anal. 13 (1989) 305-323. | MR | Zbl
[9] and, On the behaviour of the value function of a Mayer optimal control problem along optimal trajectories, in Control and estimation of distributed parameter systems (Vorau, 1996). Internat. Ser. Numer. Math. 126 81-88 (1998). | MR | Zbl
[10] and, The relationship between the maximum principle and dynamic programming. SIAM J. Control Optim. 25 (1987) 1291-1311. | MR | Zbl
[11] , The Cauchy problem for a nonlinear first order partial differential equation. J. Diff. Eq. 5 (1969) 515-530. | MR | Zbl
[12] and, On a many dimensional problem in the theory of quasilinear equations. Z. Vycisl. Mat. i Mat. Fiz. 4 (1964) 192-205. | MR
[13] , Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston (1982). | MR | Zbl
[14] , Convex Analysis. Princeton University Press, Princeton (1970). | MR | Zbl
[15] , Maximum principle, dynamic programming and their connection in deterministic control. J. Optim. Theory Appl. 65 (1990) 363-373. | MR | Zbl
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