Interior sphere property for level sets of the value function of an exit time problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 102-116.

We consider an optimal control problem for a system of the form x ˙ = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.

DOI : 10.1051/cocv:2008018
Classification : 93B03, 49L20, 49L25
Mots-clés : control theory, interior sphere property, value function, semiconcavity, perimeter 
@article{COCV_2009__15_1_102_0,
     author = {Castelpietra, Marco},
     title = {Interior sphere property for level sets of the value function of an exit time problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {102--116},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {1},
     year = {2009},
     doi = {10.1051/cocv:2008018},
     mrnumber = {2488570},
     zbl = {1155.49024},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2008018/}
}
TY  - JOUR
AU  - Castelpietra, Marco
TI  - Interior sphere property for level sets of the value function of an exit time problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 102
EP  - 116
VL  - 15
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2008018/
DO  - 10.1051/cocv:2008018
LA  - en
ID  - COCV_2009__15_1_102_0
ER  - 
%0 Journal Article
%A Castelpietra, Marco
%T Interior sphere property for level sets of the value function of an exit time problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 102-116
%V 15
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2008018/
%R 10.1051/cocv:2008018
%G en
%F COCV_2009__15_1_102_0
Castelpietra, Marco. Interior sphere property for level sets of the value function of an exit time problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 102-116. doi : 10.1051/cocv:2008018. http://archive.numdam.org/articles/10.1051/cocv:2008018/

[1] O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Bound. 7 (2005) 415-434. | MR | Zbl

[2] P. Cannarsa and P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex Anal. 13 (2006) 253-267. | MR | Zbl

[3] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems. ESAIM: COCV 12 (2006) 350-370. | Numdam | MR | Zbl

[4] P. Cannarsa and C. Sinestrari, Convexity properties of the minimun time function. Calc. Var. Partial Differential Equations 3 (1995) 273-298. | MR | Zbl

[5] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhauser, Boston (2004). | MR | Zbl

[6] P. Cannarsa, C. Pignotti and C. Sinestrari, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dynam. Systems 6 (2000) 975-997. | MR | Zbl

[7] L.C. Evans and F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. Boca Raton (1992). | MR | Zbl

[8] C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target. Commun. Pure Appl. Anal. 3 (2004) 757-774. | MR | Zbl

Cité par Sources :