Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T01:29:22.485Z Has data issue: false hasContentIssue false
  • English
  • Français

Interior sphere property of attainable sets and time optimal control problems

Published online by Cambridge University Press:  22 March 2006

Piermarco Cannarsa  and
Hélène Frankowska
Piermarco Cannarsa
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy; cannarsa@mat.uniroma2.it
Hélène Frankowska
Affiliation:
CREA, École Polytechnique, 1 Rue Descartes, 75005 Paris, France; franko@shs.polytechnique.fr
Get access

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 valuefunction of time optimal control problems with a general target, and todeduce C1,1-regularity for boundaries of attainable sets.

Keywords

Control theoryattainable setsminimum time functionsemiconcavefunctions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 2 , April 2006 , pp. 350 - 370
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Bardi, M., Falcone, M., An approximation scheme for the minimum time function. SIAM J. Control Optim. 28 (1990) 950–965. CrossRef
Bressan, A., 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).
Cannarsa, P., Pignotti, C., Sinestrari, C., Semiconcavity for optimal control problems with exit time. Discrete Contin. Dynam. Syst. 6 (2000) 975–997.
Cannarsa, P., Sinestrari, C., Convexity properties of the minimum time function. Calc. Var. 3 (1995) 273–298. CrossRef
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 $\mathbb R^n$ . Quad. Unione Mat. Italiana 30, Pitagora, Bologna (1985).
Delfour, M.C., Zolésio, J.-P., Shape analysis via oriented distance functions. J. Funct. Anal. 123 (1994) 129–201. CrossRef
Frankowska, H., Kaskosz, B., Linearization and boundary trajectories of nonsmooth control systems. Canad. J. Math. 40 (1988) 589–609. CrossRef
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.
Petrov, N.N., On the Bellman function for the time-optimal process problem. J. Appl. Math. Mech. 34 (1970) 785–791. CrossRef
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).
Sinestrari, C., 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. CrossRef
Veliov, V.M., Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335–363. CrossRef
Wolenski, P., Zhuang, Y., Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048–1072. CrossRef