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 & 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. doi: 10.1007/s00030-012-0158-1. Google Scholar

[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. doi: 10.1016/j.jde.2007.05.039. Google Scholar

[3]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis,, Systems & Control: Foundations & Applications, (1990). Google Scholar

[4]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations,, Systems & Control: Foundations & Applications, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar

[5]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Springer-Verlag, (1994). Google Scholar

[6]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbbR^N$,, ESAIM COCV, 19 (2013), 710. doi: 10.1051/cocv/2012030. Google Scholar

[7]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbbR^N$,, SIAM J. Control Optim., 52 (2014), 1712. doi: 10.1137/130922288. Google Scholar

[8]

G. Barles, A. Briani, E. Chasseigne and N. Tchou, Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, preprint,, , (). Google Scholar

[9]

G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations,, M2AN, 36 (2002), 33. doi: 10.1051/m2an:2002002. Google Scholar

[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. doi: 10.1137/0326063. Google Scholar

[11]

R. Barnard and P. Wolenski, Flow invariance on stratified domains,, Set-Valued and Variational Analysis, 21 (2013), 377. doi: 10.1007/s11228-013-0230-y. Google Scholar

[12]

A. Bressan and Y. Hong, Optimal control problems on stratified domains,, Netw. Heterog. Media., 2 (2007), 313. doi: 10.3934/nhm.2007.2.313. Google Scholar

[13]

F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks,, Calc. Var. Partial Differential Equations, 46 (2013), 671. doi: 10.1007/s00526-012-0498-z. Google Scholar

[14]

F. Camilli, C. Marchi and D. Schieborn, Eikonal equations on ramified spaces,, Interfaces Free Bound, 15 (2013), 121. doi: 10.4171/IFB/297. Google Scholar

[15]

F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations,, Comm. in Par. Diff. Eq., 30 (2005), 813. doi: 10.1081/PDE-200059292. Google Scholar

[16]

G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients,, J. Hyperbolic Differ. Equ., 4 (2007), 771. doi: 10.1142/S0219891607001355. Google Scholar

[17]

C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients,, Interfaces Free Bound, 12 (2010), 347. doi: 10.4171/IFB/238. Google Scholar

[18]

K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities,, Interfaces Free Bound, 6 (2004), 329. doi: 10.4171/IFB/103. Google Scholar

[19]

P. Dupuis, A numerical method for a calculus of variations problem with discontinuous integrand,, Applied stochastic analysis (New Brunswick, (1991), 90. doi: 10.1007/BFb0007050. Google Scholar

[20]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Applications of Mathematics, (1993). Google Scholar

[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), 11 (2004), 271. doi: 10.1007/s00030-004-1058-9. Google Scholar

[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. doi: 10.1007/s10957-006-9099-3. Google Scholar

[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. doi: 10.1090/S0002-9939-2010-10630-5. Google Scholar

[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. doi: 10.1051/cocv/2012002. Google Scholar

[25]

C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, preprint,, , (). Google Scholar

[26]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, preprint ,, , (). Google Scholar

[27]

H. Ishii, Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open sets,, Bull. Fac. Sci. Eng. Chuo Univ., 28 (1985), 33. Google Scholar

[28]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman Equations on Multi-Domains,, Control and Optimization with PDE Constraints, 164 (2013). doi: 10.1007/978-3-0348-0631-2_6. Google Scholar

[29]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations,, J. Differential Equations, 257 (2014), 3978. doi: 10.1016/j.jde.2014.07.015. Google Scholar

[30]

P. Soravia, Degenerate eikonal equations with discontinuous refraction index,, ESAIM COCV, 12 (2006), 216. doi: 10.1051/cocv:2005033. Google Scholar

[31]

H. Whitney, Tangents to an analytic variety,, Annals of Mathematics, 81 (1965), 496. doi: 10.2307/1970400. Google Scholar

[32]

H. Whitney, Local properties of analytic varieties,, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), (1965), 205. Google Scholar

show all references

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. doi: 10.1007/s00030-012-0158-1. Google Scholar

[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. doi: 10.1016/j.jde.2007.05.039. Google Scholar

[3]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis,, Systems & Control: Foundations & Applications, (1990). Google Scholar

[4]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations,, Systems & Control: Foundations & Applications, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar

[5]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Springer-Verlag, (1994). Google Scholar

[6]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $\mathbbR^N$,, ESAIM COCV, 19 (2013), 710. doi: 10.1051/cocv/2012030. Google Scholar

[7]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbbR^N$,, SIAM J. Control Optim., 52 (2014), 1712. doi: 10.1137/130922288. Google Scholar

[8]

G. Barles, A. Briani, E. Chasseigne and N. Tchou, Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, preprint,, , (). Google Scholar

[9]

G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations,, M2AN, 36 (2002), 33. doi: 10.1051/m2an:2002002. Google Scholar

[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. doi: 10.1137/0326063. Google Scholar

[11]

R. Barnard and P. Wolenski, Flow invariance on stratified domains,, Set-Valued and Variational Analysis, 21 (2013), 377. doi: 10.1007/s11228-013-0230-y. Google Scholar

[12]

A. Bressan and Y. Hong, Optimal control problems on stratified domains,, Netw. Heterog. Media., 2 (2007), 313. doi: 10.3934/nhm.2007.2.313. Google Scholar

[13]

F. Camilli and D. Schieborn, Viscosity solutions of Eikonal equations on topological networks,, Calc. Var. Partial Differential Equations, 46 (2013), 671. doi: 10.1007/s00526-012-0498-z. Google Scholar

[14]

F. Camilli, C. Marchi and D. Schieborn, Eikonal equations on ramified spaces,, Interfaces Free Bound, 15 (2013), 121. doi: 10.4171/IFB/297. Google Scholar

[15]

F Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations,, Comm. in Par. Diff. Eq., 30 (2005), 813. doi: 10.1081/PDE-200059292. Google Scholar

[16]

G. Coclite and N. Risebro, Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients,, J. Hyperbolic Differ. Equ., 4 (2007), 771. doi: 10.1142/S0219891607001355. Google Scholar

[17]

C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients,, Interfaces Free Bound, 12 (2010), 347. doi: 10.4171/IFB/238. Google Scholar

[18]

K. Deckelnick and C. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities,, Interfaces Free Bound, 6 (2004), 329. doi: 10.4171/IFB/103. Google Scholar

[19]

P. Dupuis, A numerical method for a calculus of variations problem with discontinuous integrand,, Applied stochastic analysis (New Brunswick, (1991), 90. doi: 10.1007/BFb0007050. Google Scholar

[20]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Applications of Mathematics, (1993). Google Scholar

[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), 11 (2004), 271. doi: 10.1007/s00030-004-1058-9. Google Scholar

[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. doi: 10.1007/s10957-006-9099-3. Google Scholar

[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. doi: 10.1090/S0002-9939-2010-10630-5. Google Scholar

[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. doi: 10.1051/cocv/2012002. Google Scholar

[25]

C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, preprint,, , (). Google Scholar

[26]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, preprint ,, , (). Google Scholar

[27]

H. Ishii, Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open sets,, Bull. Fac. Sci. Eng. Chuo Univ., 28 (1985), 33. Google Scholar

[28]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman Equations on Multi-Domains,, Control and Optimization with PDE Constraints, 164 (2013). doi: 10.1007/978-3-0348-0631-2_6. Google Scholar

[29]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations,, J. Differential Equations, 257 (2014), 3978. doi: 10.1016/j.jde.2014.07.015. Google Scholar

[30]

P. Soravia, Degenerate eikonal equations with discontinuous refraction index,, ESAIM COCV, 12 (2006), 216. doi: 10.1051/cocv:2005033. Google Scholar

[31]

H. Whitney, Tangents to an analytic variety,, Annals of Mathematics, 81 (1965), 496. doi: 10.2307/1970400. Google Scholar

[32]

H. Whitney, Local properties of analytic varieties,, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), (1965), 205. Google Scholar

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