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, p. 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 : https://doi.org/10.1051/cocv:2008018
Classification:  93B03,  49L20,  49L25
@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},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {1},
     year = {2009},
     pages = {102-116},
     doi = {10.1051/cocv:2008018},
     zbl = {1155.49024},
     mrnumber = {2488570},
     language = {en},
     url = {http://www.numdam.org/item/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. https://www.numdam.org/item/COCV_2009__15_1_102_0/

[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 2191694 | Zbl 1099.35148

[2] P. Cannarsa and P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex Anal. 13 (2006) 253-267. | MR 2252231 | Zbl 1114.93018

[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 2209357 | Zbl 1105.93007

[4] P. Cannarsa and C. Sinestrari, Convexity properties of the minimun time function. Calc. Var. Partial Differential Equations 3 (1995) 273-298. | MR 1385289 | Zbl 0836.49013

[5] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhauser, Boston (2004). | MR 2041617 | Zbl 1095.49003

[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 1788264 | Zbl 1009.49024

[7] L.C. Evans and F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. Boca Raton (1992). | MR 1158660 | Zbl 0804.28001

[8] C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target. Commun. Pure Appl. Anal. 3 (2004) 757-774. | MR 2106298 | Zbl 1064.49024