A Bellman approach for two-domains optimal control problems in ℝ N
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 710-739.

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.

DOI : 10.1051/cocv/2012030
Classification : 49L20, 49L25, 35F21
Mots-clés : optimal control, discontinuous dynamic, Bellman equation, viscosity solutions 
@article{COCV_2013__19_3_710_0,
     author = {Barles, G. and Briani, A. and Chasseigne, E.},
     title = {A {Bellman} approach for two-domains optimal control problems in $\mathbb {R}^N$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {710--739},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     doi = {10.1051/cocv/2012030},
     zbl = {1287.49028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2012030/}
}
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Barles, G.; Briani, A.; Chasseigne, E. A Bellman approach for two-domains optimal control problems in $\mathbb {R}^N$. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 710-739. doi : 10.1051/cocv/2012030. http://archive.numdam.org/articles/10.1051/cocv/2012030/

[1] J.-P. Aubin and H. Frankowska, Set-valued analysis. Systems AND Control: Foundations and Applications, vol. 2. Birkhuser Boston, Inc. Boston, MA (1990). | MR | Zbl

[2] Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks, NDEA Nonlinear Differential Equation and Application, to appear (2012). | MR | Zbl

[3] A.S. Mishra and G.D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients. J. Diff. Eq. 241 (2007) 1-31. | MR | Zbl

[4] 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). | MR | Zbl

[5] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag, Paris (1994). | MR | Zbl

[6] G. Barles and E.R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM: M2AN 36 (2002) 33-54. | Numdam | MR | Zbl

[7] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR | Zbl

[8] G. Barles and B. Perthame, Comparison principle for Dirichlet type Hamilton-Jacobi Equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21-44. | MR | Zbl

[9] A.-P. Blanc, Deterministic exit time control problems with discontinuous exit costs. SIAM J. Control Optim. 35 (1997) 399-434. | MR | Zbl

[10] A-P. Blanc, Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data. Nonlinear Anal. Ser. A Theory Methods 45 (2001) 1015-1037. | MR | Zbl

[11] A. Bressan and Y. Hong, Optimal control problems on stratified domains. Netw. Heterog. Media 2 (2007) 313-331 (electronic). | MR | Zbl

[12] F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations. Comm. Partial Differ. Equ. 30 (2005) 813-847. | MR | Zbl

[13] G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 4 (2007) 771-795. | MR | Zbl

[14] C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients. Interfaces Free Bound 12 (2010) 347-368. | MR | Zbl

[15] K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities. Interfaces Free Bound 6 (2004) 329-349. | MR | Zbl

[16] P. Dupuis, 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. | MR

[17] A.F. Filippov, 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). | MR | Zbl

[18] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. NoDEA Nonlinear Differ. Equ. Appl. 11 (2004) 271-298. | MR | Zbl

[19] M. Garavello and P. Soravia, 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. | MR | Zbl

[20] Y. Giga, P. Gòrka and P. Rybka, A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians. Proc. Amer. Math. Soc. 139 (2011) 1777-1785. | MR | Zbl

[21] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second-Order. Springer, New-York (1983). | MR | Zbl

[22] Lions P.L. Generalized Solutions of Hamilton-Jacobi Equations, Res. Notes Math., vol. 69. Pitman, Boston (1982). | MR | Zbl

[23] R.T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J. (1970). | MR | Zbl

[24] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and applications to traffic flows, ESAIM: COCV 19 (2013) 1-316. | Numdam | MR | Zbl

[25] H.M. Soner, Optimal control with state-space constraint I. SIAM J. Control Optim. 24 (1986) 552-561. | MR | Zbl

[26] D. Schieborn and F. Camilli, Viscosity solutions of Eikonal equations on topological networks, to appear in Calc. Var. Partial Differential Equations. | MR | Zbl

[27] P. Soravia, Degenerate eikonal equations with discontinuous refraction index. ESAIM: COCV 12 (2006). | Numdam | MR | Zbl

[28] T. Wasewski, Systèmes de commande et équation au contingent. Bull. Acad. Pol. Sci. 9 (1961) 151-155. | Zbl

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