# American Institute of Mathematical Sciences

December  2015, 10(4): 809-836. doi: 10.3934/nhm.2015.10.809

## (Almost) Everything you always wanted to know about deterministic control problems in stratified domains

 1 Laboratoire de Mathématiques et Physique Théorique(UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France, France

Received  February 2015 Revised  August 2015 Published  October 2015

We revisit the pioneering work of Bressan & Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of $\mathbb{R}^N$. By using slightly different methods, involving more partial differential equations arguments, we $(i)$ slightly improve the assumptions on the dynamic and the cost; $(ii)$ obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); $(iii)$ provide a general framework in which a stability result holds.
Citation: Guy Barles, Emmanuel Chasseigne. (Almost) Everything you always wanted to know about deterministic control problems in stratified domains. Networks and Heterogeneous Media, 2015, 10 (4) : 809-836. doi: 10.3934/nhm.2015.10.809
##### References:
 [1] Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDea Nonlinear Differential Equations Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1. [2] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients, J. Differential Equations, 241 (2007), 1-31. doi: 10.1016/j.jde.2007.05.039. [3] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. [4] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations, Systems & Control: Foundations & Applications, Birkhauser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [5] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994. [6] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbb{R}^N2$, ESAIM COCV, 19 (2013), 710-739. doi: 10.1051/cocv/2012030. [7] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^N2$, SIAM J. Control Optim., 52 (2014), 1712-1744. doi: 10.1137/130922288. [8] G. Barles, A. Briani, E. Chasseigne and N. Tchou, Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, preprint, arXiv:1405.0661. [9] G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN, 36 (2002), 33-54. doi: 10.1051/m2an:2002002. [10] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. in Control and Optimisation, 26 (1988), 1133-1148. doi: 10.1137/0326063. [11] R. Barnard and P. Wolenski, Flow invariance on stratified domains, Set-Valued and Variational Analysis, 21 (2013), 377-403. doi: 10.1007/s11228-013-0230-y. [12] A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media., 2 (2007), 313-331 (electronic) and Errata corrige: Optimal control problems on stratified domains. Netw. Heterog. Media., 8 (2013), p625. doi: 10.3934/nhm.2007.2.313. [13] F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z. [14] F. Camilli, C. Marchi and D. Schieborn, Eikonal equations on ramified spaces, Interfaces Free Bound, 15 (2013), 121-140. doi: 10.4171/IFB/297. [15] F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations, Comm. in Par. Diff. Eq., 30 (2005), 813-847. doi: 10.1081/PDE-200059292. [16] G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 4 (2007), 771-795. doi: 10.1142/S0219891607001355. [17] C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368. doi: 10.4171/IFB/238. [18] K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities, Interfaces Free Bound, 6 (2004), 329-349. doi: 10.4171/IFB/103. [19] P. Dupuis, A numerical method for a calculus of variations problem with discontinuous integrand, Applied stochastic analysis (New Brunswick, NJ, 1991), 90-107, Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992. doi: 10.1007/BFb0007050. [20] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics, Springer-Verlag, New York, 1993. [21] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 271-298. doi: 10.1007/s00030-004-1058-9. [22] 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. doi: 10.1007/s10957-006-9099-3. [23] 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. doi: 10.1090/S0002-9939-2010-10630-5. [24] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: COCV, 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [25] C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, preprint, arXiv:1410.3056. [26] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, preprint , arXiv:1306.2428. [27] H. Ishii, Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ., 28 (1985), 33-77. [28] Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman Equations on Multi-Domains, Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164, Birkhäuser Basel, 2013. doi: 10.1007/978-3-0348-0631-2_6. [29] Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014. doi: 10.1016/j.jde.2014.07.015. [30] P. Soravia, Degenerate eikonal equations with discontinuous refraction index, ESAIM COCV, 12 (2006), 216-230. doi: 10.1051/cocv:2005033. [31] H. Whitney, Tangents to an analytic variety, Annals of Mathematics, 81 (1965), 496-549. doi: 10.2307/1970400. [32] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205-244, Princeton Univ. Press, Princeton, N. J., 1965.

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##### References:
 [1] Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDea Nonlinear Differential Equations Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1. [2] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients, J. Differential Equations, 241 (2007), 1-31. doi: 10.1016/j.jde.2007.05.039. [3] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. [4] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations, Systems & Control: Foundations & Applications, Birkhauser Boston Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [5] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994. [6] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbb{R}^N2$, ESAIM COCV, 19 (2013), 710-739. doi: 10.1051/cocv/2012030. [7] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^N2$, SIAM J. Control Optim., 52 (2014), 1712-1744. doi: 10.1137/130922288. [8] G. Barles, A. Briani, E. Chasseigne and N. Tchou, Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, preprint, arXiv:1405.0661. [9] G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN, 36 (2002), 33-54. doi: 10.1051/m2an:2002002. [10] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. in Control and Optimisation, 26 (1988), 1133-1148. doi: 10.1137/0326063. [11] R. Barnard and P. Wolenski, Flow invariance on stratified domains, Set-Valued and Variational Analysis, 21 (2013), 377-403. doi: 10.1007/s11228-013-0230-y. [12] A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media., 2 (2007), 313-331 (electronic) and Errata corrige: Optimal control problems on stratified domains. Netw. Heterog. Media., 8 (2013), p625. doi: 10.3934/nhm.2007.2.313. [13] F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z. [14] F. Camilli, C. Marchi and D. Schieborn, Eikonal equations on ramified spaces, Interfaces Free Bound, 15 (2013), 121-140. doi: 10.4171/IFB/297. [15] F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations, Comm. in Par. Diff. Eq., 30 (2005), 813-847. doi: 10.1081/PDE-200059292. [16] G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 4 (2007), 771-795. doi: 10.1142/S0219891607001355. [17] C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368. doi: 10.4171/IFB/238. [18] K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities, Interfaces Free Bound, 6 (2004), 329-349. doi: 10.4171/IFB/103. [19] P. Dupuis, A numerical method for a calculus of variations problem with discontinuous integrand, Applied stochastic analysis (New Brunswick, NJ, 1991), 90-107, Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992. doi: 10.1007/BFb0007050. [20] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics, Springer-Verlag, New York, 1993. [21] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 271-298. doi: 10.1007/s00030-004-1058-9. [22] 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. doi: 10.1007/s10957-006-9099-3. [23] 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. doi: 10.1090/S0002-9939-2010-10630-5. [24] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: COCV, 19 (2013), 129-166. doi: 10.1051/cocv/2012002. [25] C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, preprint, arXiv:1410.3056. [26] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, preprint , arXiv:1306.2428. [27] H. Ishii, Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ., 28 (1985), 33-77. [28] Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman Equations on Multi-Domains, Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164, Birkhäuser Basel, 2013. doi: 10.1007/978-3-0348-0631-2_6. [29] Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014. doi: 10.1016/j.jde.2014.07.015. [30] P. Soravia, Degenerate eikonal equations with discontinuous refraction index, ESAIM COCV, 12 (2006), 216-230. doi: 10.1051/cocv:2005033. [31] H. Whitney, Tangents to an analytic variety, Annals of Mathematics, 81 (1965), 496-549. doi: 10.2307/1970400. [32] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205-244, Princeton Univ. Press, Princeton, N. J., 1965.
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