Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T06:01:44.202Z Has data issue: false hasContentIssue false
  • English
  • Français

A Bellman approach for two-domains optimal control problems inℝN∗

Published online by Cambridge University Press:  03 June 2013

G. Barles,
A. Briani and
E. Chasseigne
G. Barles
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr
A. Briani
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr
E. Chasseigne
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

Keywords

Optimal controldiscontinuous dynamicBellman Equationviscosity solutions

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013, pp. 710 - 739
Copyright
© EDP Sciences, SMAI, 2013

References

J.-P. Aubin and H. Frankowska, Set-valued analysis. Systems AND Control: Foundations and Applications, vol. 2. Birkhuser Boston, Inc. Boston, MA (1990).
Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks, NDEA Nonlinear Differential Equation and Application, to appear (2012).
Mishra, A.S. and Veerappa Gowda, G.D., Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients. J. Diff. Eq. 241 (2007) 1–31. Google Scholar
M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations & Applications. Birkhauser Boston Inc., Boston, MA (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag, Paris (1994).
Barles, G. and Jakobsen, E.R., On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM: M2AN 36 (2002) 33–54. Google Scholar
Barles, G. and Perthame, B., Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148. Google Scholar
Barles, G. and Perthame, B., Comparison principle for Dirichlet type Hamilton-Jacobi Equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21–44. Google Scholar
Blanc, A.-P., Deterministic exit time control problems with discontinuous exit costs. SIAM J. Control Optim. 35 (1997) 399–434. Google Scholar
Blanc, A-P., Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data. Nonlinear Anal. Ser. A Theory Methods 45 (2001) 1015–1037. Google Scholar
Bressan, A. and Hong, Y., Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313–331 (electronic). Google Scholar
Camilli, F and Siconolfi, A., Time-dependent measurable Hamilton-Jacobi equations. Comm. Partial Differ. Equ. 30 (2005) 813–847. Google Scholar
Coclite, G. and Risebro, N., Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 4 (2007) 771–795. Google Scholar
De Zan, C. and Soravia, P., Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients. Interfaces Free Bound 12 (2010) 347–368. Google Scholar
Deckelnick, K. and Elliott, C., Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities. Interfaces Free Bound 6 (2004) 329–349. Google Scholar
Dupuis, P., A numerical method for a calculus of variations problem with discontinuous integrand. Applied stochastic analysis, New Brunswick, NJ 1991. Lect. Notes Control Inform. Sci., vol. 177. Springer, Berlin (1992) 90–107. Google Scholar
Filippov, A.F., Differential equations with discontinuous right-hand side. Matematicheskii Sbornik 51 (1960) 99–128. Amer. Math. Soc. Transl. 42 (1964) 199–231 (English translation Series 2). Google Scholar
Garavello, M. and Soravia, P., Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. NoDEA Nonlinear Differ. Equ. Appl. 11 (2004) 271–298. Google Scholar
Garavello, M. and Soravia, P., Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games. J. Optim. Theory Appl. 130 (2006) 209–229. Google Scholar
Giga, Y., Gòrka, P. and Rybka, P., A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians. Proc. Amer. Math. Soc. 139 (2011) 1777–1785. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second-Order. Springer, New-York (1983).
Lions P.L. Generalized Solutions of Hamilton-Jacobi Equations, Res. Notes Math., vol. 69. Pitman, Boston (1982).
R.T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J. (1970).
Imbert, C., Monneau, R. and Zidani, H., A Hamilton-Jacobi approach to junction problems and applications to traffic flows, ESAIM: COCV 19 (2013) 1–316. Google Scholar
Soner, H.M., Optimal control with state-space constraint I. SIAM J. Control Optim. 24 (1986) 552–561. Google Scholar
D. Schieborn and F. Camilli, Viscosity solutions of Eikonal equations on topological networks, to appear in Calc. Var. Partial Differential Equations.
P. Soravia, Degenerate eikonal equations with discontinuous refraction index. ESAIM: COCV 12 (2006).
Wasewski, T., Systèmes de commande et équation au contingent. Bull. Acad. Pol. Sci. 9 (1961) 151–155.Google Scholar