American Institute of Mathematical Sciences

Advanced
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

Guy Barles 1, and  Emmanuel Chasseigne 1,

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.
Keywords: stratified domains, viscosity solutions., Bellman equation, Optimal control, discontinuous dynamic.
Mathematics Subject Classification: Primary: 49L20, 49L25; Secondary: 35F2.
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.

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), 413-445. 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.

[1]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[2]

Alberto Bressan, Yunho Hong. Optimal control problems on stratified domains. Networks and Heterogeneous Media, 2007, 2 (2) : 313-331. doi: 10.3934/nhm.2007.2.313
[3]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[4]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046
[5]

Guy Barles, Emmanuel Chasseigne. Corrigendum to "(Almost) everything you always wanted to know about deterministic control problems in stratified domains". Networks and Heterogeneous Media, 2018, 13 (2) : 373-378. doi: 10.3934/nhm.2018016
[6]

Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
[7]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[8]

Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations and Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036
[9]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[10]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369
[11]

Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130
[12]

Roberta Ghezzi, Benedetto Piccoli. Optimal control of a multi-level dynamic model for biofuel production. Mathematical Control and Related Fields, 2017, 7 (2) : 235-257. doi: 10.3934/mcrf.2017008
[13]

B. M. Adams, H. T. Banks, Hee-Dae Kwon, Hien T. Tran. Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches. Mathematical Biosciences & Engineering, 2004, 1 (2) : 223-241. doi: 10.3934/mbe.2004.1.223
[14]

Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071
[15]

Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control and Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399
[16]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833
[17]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2237-2256. doi: 10.3934/cpaa.2021065
[18]

Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions. Networks and Heterogeneous Media, 2013, 8 (3) : 727-744. doi: 10.3934/nhm.2013.8.727
[19]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[20]

Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations and Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023

2021 Impact Factor: 1.41

Tools

Metrics

  • PDF downloads (173)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy