Interior sphere property for level sets of the value function of an exit time problem

Marco Castelpietra

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 102-116
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Castelpietra, Marco. "Interior sphere property for level sets of the value function of an exit time problem." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 102-116. <http://eudml.org/doc/250575>.

@article{Castelpietra2009,
abstract = { 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. },
author = {Castelpietra, Marco},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control theory; interior sphere property; value function; semiconcavity; perimeter; control theory; value function},
language = {eng},
month = {1},
number = {1},
pages = {102-116},
publisher = {EDP Sciences},
title = {Interior sphere property for level sets of the value function of an exit time problem},
url = {http://eudml.org/doc/250575},
volume = {15},
year = {2009},
}

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
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 102
EP - 116
AB - 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.
LA - eng
KW - Control theory; interior sphere property; value function; semiconcavity; perimeter; control theory; value function
UR - http://eudml.org/doc/250575
ER -

References

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  2. P. Cannarsa and P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex Anal.13 (2006) 253–267.  Zbl1114.93018
  3. P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems. ESAIM: COCV12 (2006) 350–370.  Zbl1105.93007
  4. P. Cannarsa and C. Sinestrari, Convexity properties of the minimun time function. Calc. Var. Partial Differential Equations3 (1995) 273–298.  Zbl0836.49013
  5. P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhauser, Boston (2004).  Zbl1095.49003
  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
  7. L.C. Evans and F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. Boca Raton (1992).  Zbl0804.28001
  8. C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target. Commun. Pure Appl. Anal.3 (2004) 757–774.  Zbl1064.49024

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