Interior sphere property of attainable sets and time optimal control problems
Piermarco Cannarsa; Hélène Frankowska
ESAIM: Control, Optimisation and Calculus of Variations (2006)
- Volume: 12, Issue: 2, page 350-370
- ISSN: 1292-8119
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topCannarsa, Piermarco, and Frankowska, Hélène. "Interior sphere property of attainable sets and time optimal control problems." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 350-370. <http://eudml.org/doc/249673>.
@article{Cannarsa2006,
abstract = {
This paper studies the attainable set at time T>0 for the control system $$\dot y(t)=f(y(t),u(t))\,\qquad u(t)\in U$$ showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere
condition. The interior sphere property is
then applied to recover a semiconcavity result for the value
function of time optimal control problems with a general target, and to
deduce C1,1-regularity for boundaries of attainable sets.
},
author = {Cannarsa, Piermarco, Frankowska, Hélène},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control theory; attainable sets; minimum time function; semiconcave
functions.; semiconcave functions},
language = {eng},
month = {3},
number = {2},
pages = {350-370},
publisher = {EDP Sciences},
title = {Interior sphere property of attainable sets and time optimal control problems},
url = {http://eudml.org/doc/249673},
volume = {12},
year = {2006},
}
TY - JOUR
AU - Cannarsa, Piermarco
AU - Frankowska, Hélène
TI - Interior sphere property of attainable sets and time optimal control problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 350
EP - 370
AB -
This paper studies the attainable set at time T>0 for the control system $$\dot y(t)=f(y(t),u(t))\,\qquad u(t)\in U$$ showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere
condition. The interior sphere property is
then applied to recover a semiconcavity result for the value
function of time optimal control problems with a general target, and to
deduce C1,1-regularity for boundaries of attainable sets.
LA - eng
KW - Control theory; attainable sets; minimum time function; semiconcave
functions.; semiconcave functions
UR - http://eudml.org/doc/249673
ER -
References
top- J.-P. Aubin, A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984).
- J.-P. Aubin, H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
- M. Bardi, I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton–Jacobi equations. Birkhäuser, Boston (1997).
- M. Bardi, M. Falcone, An approximation scheme for the minimum time function. SIAM J. Control Optim.28 (1990) 950–965.
- A. Bressan, On two conjectures by Hájek. Funkcial. Ekvac.23 (1980) 221–227.
- P. Cannarsa, P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex. Anal. (to appear).
- P. Cannarsa, C. Pignotti, C. Sinestrari, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dynam. Syst.6 (2000) 975–997.
- P. Cannarsa, C. Sinestrari, Convexity properties of the minimum time function. Calc. Var.3 (1995) 273–298.
- P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control. Birkhäuser, Boston (2004).
- F.H. Clarke, Optimization and nonsmooth analysis. Wiley, New York (1983).
- R. Conti, Processi di controllo lineari in . Quad. Unione Mat. Italiana 30, Pitagora, Bologna (1985).
- M.C. Delfour, J.-P. Zolésio, Shape analysis via oriented distance functions. J. Funct. Anal.123 (1994) 129–201.
- H. Frankowska, B. Kaskosz, Linearization and boundary trajectories of nonsmooth control systems. Canad. J. Math.40 (1988) 589–609.
- H. Hermes, J.P. LaSalle, Functional analysis and time optimal control. Academic Press, New York (1969).
- E.B. Lee, L. Markus, Foundations of optimal control theory. John Wiley & Sons Inc., New York (1967).
- S. Lojasiewicz Jr., A. Pliś, R. Suarez, Necessary conditions for a nonlinear control system. J. Differ. Equ., 59, 257–265.
- N.N. Petrov, On the Bellman function for the time-optimal process problem. J. Appl. Math. Mech.34 (1970) 785–791.
- A. Pliś, Accessible sets in control theory. Int. Conf. on Diff. Eqs., Academic Press (1975) 646–650.
- R.T. Rockafellar, R.J.-B. Wets, Variational analysis. Springer-Verlag, Berlin (1998).
- C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target. Communications on Pure and Applied Analysis. Commun. Pure Appl. Anal.3 (2004) 757–774.
- V.M. Veliov, Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl.94 (1997) 335–363.
- P. Wolenski, Y. Zhuang, Proximal analysis and the minimal time function. SIAM J. Control Optim.36 (1998) 1048–1072.
Citations in EuDML Documents
top- Marco Castelpietra, Interior sphere property for level sets of the value function of an exit time problem
- Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, Analysis of a time optimal control problem related to the management of a bioreactor
- Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, Analysis of a time optimal control problem related to the management of a bioreactor
- Thomas Lorenz, Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms
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