zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.