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Interior sphere property for level sets of the value function of an exit time problem. (English) Zbl 1155.49024
Summary: We consider an optimal control problem for a system of the form $$\dot{x} = f(x,u)$$, 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.

##### MSC:
 49N60 Regularity of solutions in optimal control 93B03 Attainable sets, reachability 49L20 Dynamic programming in optimal control and differential games 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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##### References:
 [1] O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Bound.7 (2005) 415-434. Zbl1099.35148 · Zbl 1099.35148 · doi:10.4171/IFB/131 [2] P. Cannarsa and P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex Anal.13 (2006) 253-267. · Zbl 1114.93018 · www.heldermann.de [3] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems. ESAIM: COCV12 (2006) 350-370. · Zbl 1105.93007 · doi:10.1051/cocv:2006002 · numdam:COCV_2006__12_2_350_0 · eudml:90800 [4] P. Cannarsa and C. Sinestrari, Convexity properties of the minimun time function. Calc. Var. Partial Differential Equations3 (1995) 273-298. Zbl0836.49013 · Zbl 0836.49013 · doi:10.1007/BF01189393 [5] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhauser, Boston (2004). · Zbl 1095.49003 [6] P. Cannarsa, C. Pignotti and C. Sinestrari, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dynam. Systems6 (2000) 975-997. Zbl1009.49024 · Zbl 1009.49024 · doi:10.3934/dcds.2000.6.975 [7] L.C. Evans and F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. Boca Raton (1992). · 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. · Zbl 1064.49024 · doi:10.3934/cpaa.2004.3.757
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