Interior sphere property of attainable sets and time optimal control problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 350-370.

This paper studies the attainable set at time T>0 for the control system

( ˙t)=f((t),u(t))u(t)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 C 1,1 -regularity for boundaries of attainable sets.

DOI: 10.1051/cocv:2006002
Classification: 26B25, 49K15, 93B03
Keywords: control theory, attainable sets, minimum time function, semiconcave functions
@article{COCV_2006__12_2_350_0,
     author = {Cannarsa, Piermarco and Frankowska, H\'el\`ene},
     title = {Interior sphere property of attainable sets and time optimal control problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {350--370},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {2},
     year = {2006},
     doi = {10.1051/cocv:2006002},
     mrnumber = {2209357},
     zbl = {1105.93007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2006002/}
}
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
PY  - 2006
SP  - 350
EP  - 370
VL  - 12
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2006002/
DO  - 10.1051/cocv:2006002
LA  - en
ID  - COCV_2006__12_2_350_0
ER  - 
%0 Journal Article
%A Cannarsa, Piermarco
%A Frankowska, Hélène
%T Interior sphere property of attainable sets and time optimal control problems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2006
%P 350-370
%V 12
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2006002/
%R 10.1051/cocv:2006002
%G en
%F COCV_2006__12_2_350_0
Cannarsa, Piermarco; Frankowska, Hélène. Interior sphere property of attainable sets and time optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 350-370. doi : 10.1051/cocv:2006002. http://www.numdam.org/articles/10.1051/cocv:2006002/

[1] J.-P. Aubin, A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984). | MR | Zbl

[2] J.-P. Aubin, H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990). | MR | Zbl

[3] M. Bardi, I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi equations. Birkhäuser, Boston (1997). | Zbl

[4] M. Bardi, M. Falcone, An approximation scheme for the minimum time function. SIAM J. Control Optim. 28 (1990) 950-965. | Zbl

[5] A. Bressan, On two conjectures by Hájek. Funkcial. Ekvac. 23 (1980) 221-227. | Zbl

[6] P. Cannarsa, P. Cardaliaguet, Perimeter estimates for the reachable set of control problems. J. Convex. Anal. (to appear). | MR | Zbl

[7] P. Cannarsa, C. Pignotti, C. Sinestrari, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dynam. Syst. 6 (2000) 975-997. | Zbl

[8] P. Cannarsa, C. Sinestrari, Convexity properties of the minimum time function. Calc. Var. 3 (1995) 273-298. | Zbl

[9] P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control. Birkhäuser, Boston (2004). | MR | Zbl

[10] F.H. Clarke, Optimization and nonsmooth analysis. Wiley, New York (1983). | MR | Zbl

[11] R. Conti, Processi di controllo lineari in n . Quad. Unione Mat. Italiana 30, Pitagora, Bologna (1985). | Zbl

[12] M.C. Delfour, J.-P. Zolésio, Shape analysis via oriented distance functions. J. Funct. Anal. 123 (1994) 129-201. | Zbl

[13] H. Frankowska, B. Kaskosz, Linearization and boundary trajectories of nonsmooth control systems. Canad. J. Math. 40 (1988) 589-609. | Zbl

[14] H. Hermes, J.P. Lasalle, Functional analysis and time optimal control. Academic Press, New York (1969). | MR | Zbl

[15] E.B. Lee, L. Markus, Foundations of optimal control theory. John Wiley & Sons Inc., New York (1967). | MR | Zbl

[16] S. Lojasiewicz Jr., A. Pliś, R. Suarez, Necessary conditions for a nonlinear control system. J. Differ. Equ., 59, 257-265. | Zbl

[17] N.N. Petrov, On the Bellman function for the time-optimal process problem. J. Appl. Math. Mech. 34 (1970) 785-791. | Zbl

[18] A. Pliś, Accessible sets in control theory. Int. Conf. on Diff. Eqs., Academic Press (1975) 646-650. | Zbl

[19] R.T. Rockafellar, R.J.-B. Wets, Variational analysis. Springer-Verlag, Berlin (1998). | MR | Zbl

[20] 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. | Zbl

[21] V.M. Veliov, Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335-363. | Zbl

[22] P. Wolenski, Y. Zhuang, Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048-1072. | Zbl

Cited by Sources: