An Illustrated Guide of the Modern Approches of Hamilton-Jacobi Equations and Control Problems with Discontinuities

This manuscrit is a project of book on Hamilton-Jacobi Equations and Control Problems with discontinuities. For the time being, it is composed of 4 parts: -- Part I is a toolbox with key results which are used in all the other parts. -- Part II describes several approaches which were introduced recently to treat problems involving co-dimension 1 discontinuities or networks. -- Part III concerns stratified problems in $R^N$, i.e. problems with discontinuities of any co-dimensions. -- Part IV addresses the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. This second version is still imperfect and we will welcome any (fair) comment on it. In this second version, besides of improving few points in the first one, we have added applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and started to develop the use of stratified solutions to treat problems with boundary conditions (Dirichlet, Neumann and mixed boundary conditions), where both the boundary may be non-smooth and the data may present discontinuities. Of course, for all these questions, we only provide examples of what can be done since developing a whole theory would be too long, out of the scope of this book and probably a little bit beyond what we are able to do up to now. In these two directions, we address, in particular, a new question which is interesting when considering applications: under which conditions can one prove that Ishii's viscosity solutions are stratified solutions? The interest of this question is clear: when these conditions hold, one can take advantages of both the nice stability properties of classical viscosity solutions and the general comparison result for stratified solutions. A priori the third version (next release planned in june 2020) will probably be the last one: besides of improving the second version, we plan to address Large Deviations and homogenization problems. This will be the occasion to test the results we have presented so far and to improve them.

[1]  N. Forcadel,et al.  Specified homogenization of a discrete traffic model leading to an effective junction condition , 2018 .

[2]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[3]  G. Barles,et al.  Wavefront propagation for reaction-diffusion systems of PDE , 1990 .

[4]  François Hamel,et al.  Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN , 2001 .

[5]  H. Whitney.  Tangents to an Analytic Variety , 1965 .

[6]  Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinious data , 2001 .

[7]  Hasnaa Zidani,et al.  Hamilton-Jacobi-Bellman Equations on Multi-domains , 2013, Control and Optimization with PDE Constraints.

[8]  Guy Barles,et al.  Remarks on a flame propagation model , 1985 .

[9]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[10]  Flux-limited solutions and state constraints for quasi-convex Hamilton-Jacobi equations , 2016, 1607.03996.

[11]  Fabio Camilli,et al.  Hamilton-Jacobi equations with measurable dependence on the state variable , 2003, Advances in Differential Equations.

[12]  D. Nunziante,et al.  Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence , 1990, Differential and Integral Equations.

[13]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[14]  Ruth J. Williams.  Brownian motion in a wedge with oblique reflection at the boundary , 1985 .

[15]  Yves Achdou,et al.  Hamilton–Jacobi equations constrained on networks , 2012, Nonlinear Differential Equations and Applications NoDEA.

[16]  P. Souganidis,et al.  A PDE approach to geometric optics for certain semilinear parabolic equations , 1989 .

[17]  P. Dupuis,et al.  On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners , 1991 .

[18]  Z. Rao.  Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients , 2013 .

[19]  H. Zidani,et al.  Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions , 2017, ESAIM: Control, Optimisation and Calculus of Variations.

[20]  Alain-Philippe Blanc,et al.  Deterministic Exit Time Control Problems With Discontinuous Exit costs , 1997 .

[21]  E. Barron,et al.  Semicontinuous Viscosity Solutions For Hamilton–Jacobi Equations With Convex Hamiltonians , 1990 .

[22]  Fabio Camilli,et al.  A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks , 2013 .

[23]  E. Barron,et al.  Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable hamiltonians , 1987 .

[24]  H. Ishii.  Hamilton-Jacobi Equations with Discontinuous Hamiltonians on Arbitrary Open Sets , 1985 .

[25]  G. Barles,et al.  A STRONG COMPARISON RESULT FOR THE BELLMAN EQUATION ARISING IN STOCHASTIC EXIT TIME CONTROL PROBLEMS AND ITS APPLICATIONS , 1998 .

[26]  Fabio Camilli,et al.  The vanishing viscosity limit for Hamilton-Jacobi equations on networks , 2012, 1207.6535.

[27]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[28]  J. Harrison,et al.  Reflected Brownian Motion on an Orthant , 1981 .

[29]  L. Evans.  Periodic homogenisation of certain fully nonlinear partial differential equations , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[30]  Alberto Bressan,et al.  Optimal control problems on stratified domains , 2007, Networks Heterog. Media.

[31]  C. M. Elliott,et al.  Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities , 2004 .

[33]  N. Risebro,et al.  VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS , 2003, math/0303288.

[34]  B. Perthame,et al.  Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians , 1987 .

[35]  G. Barles,et al.  Exit Time Problems in Optimal Control and Vanishing Viscosity Method , 1988 .

[36]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[37]  Y. Giga,et al.  A Comparison Principle for Hamilton-Jacobi equations with discontinuous Hamiltonians , 2010 .

[38]  Pierre-Louis Lions,et al.  Well posedness for multi-dimensional junction problems with Kirchoff-type conditions , 2017 .

[39]  Mariane Bourgoing,et al.  Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach , 2008 .

[40]  Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients , 2010 .

[41]  Nicolas Forcadel,et al.  Singular Perturbation of Optimal Control Problems on MultiDomains , 2014, SIAM J. Control. Optim..

[42]  C. Imbert,et al.  Effective junction conditions for degenerate parabolic equations , 2016, 1601.01862.

[43]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[44]  F. Camilli,et al.  Time-Dependent Measurable Hamilton–Jacobi Equations , 2005 .

[45]  Panagiotis E. Souganidis,et al.  Front propagation: Theory and applications , 1997 .

[46]  D. Nunziante.  Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence , 1992 .

[47]  H. Frankowska,et al.  Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints , 2013 .

[48]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[49]  Guy Barles,et al.  A Bellman approach for two-domains optimal control problems in R N , 2011, 1112.3727.

[50]  Degenerate Eikonal equations with discontinuous refraction index , 2006 .

[51]  Guy Barles,et al.  A Bellman Approach for Regional Optimal Control Problems in RN , 2013, SIAM J. Control. Optim..

[52]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[53]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[54]  G. Barles,et al.  A New Approach to Front Propagation Problems: Theory and Applications , 1998 .

[55]  Yves Achdou,et al.  HAMILTON-JACOBI EQUATIONS FOR OPTIMAL CONTROL ON JUNCTIONS AND NETWORKS ∗, ∗∗ , 2015 .

[56]  H. Whitney.  Local Properties of Analytic Varieties , 1992 .

[57]  Mauro Garavello,et al.  Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games , 2006 .

[58]  Cyril Imbert,et al.  Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks , 2013, 1306.2428.

[59]  P. E. Souganidis,et al.  Viscosity solutions for junctions: well posedness and stability , 2016 .

[61]  P. Souganidis,et al.  A PDE approach to certain large deviation problems for systems of parabolic equations , 1989 .

[62]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[63]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[64]  Richard B. Vinter,et al.  Hamilton-Jacobi-Bellman equations for optimal control processes with convex state constraints , 2017, Syst. Control. Lett..

[65]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[66]  Flux-limited and classical viscosity solutions for regional control problems , 2016, ESAIM: Control, Optimisation and Calculus of Variations.

[67]  G. Barles,et al.  Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations , 1990 .

[68]  H. Ishii.  On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's , 1989 .

[69]  H. Frankowska.  Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations , 1993 .

[70]  Guy Barles,et al.  Homogenization results for a deterministic multi-domains periodic control problem , 2014, Asymptot. Anal..

[71]  G. Barles,et al.  Large time behavior of fronts governed by eikonal equations , 2003 .

[72]  C. Imbert,et al.  Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case , 2014, 1410.3056.

[73]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[74]  Pierpaolo Soravia.  Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness , 1999 .

[75]  Wilfredo Salazar,et al.  Homogenization of second order discrete model and application to traffic flow , 2015 .

[76]  H. Zidani,et al.  Infinite horizon problems on stratifiable state-constraints sets ☆ , 2015 .

[77]  Erratum to the article Hamilton–Jacobi equations for optimal control on junctions and networks , 2016 .

[78]  A. F. Filippov.  On Certain Questions in the Theory of Optimal Control , 1962 .

[79]  H. Zidani,et al.  Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations , 2014 .

[80]  Y. Giga,et al.  Hamilton-Jacobi Equations with Discontinuous Source Terms , 2011 .

[81]  Emmanuel Chasseigne,et al.  (Almost) Everything you always wanted to know about deterministic control problems in stratified domains , 2015, Networks Heterog. Media.

[82]  R. Jensen.  The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations , 1988 .

[83]  Hasnaa Zidani,et al.  A Hamilton-Jacobi approach to junction problems and application to traffic flows , 2011, 1107.3250.

[84]  Michael G. Crandall,et al.  Viscosity Solutions of Fully Nonlinear Equations , 1994 .

[85]  P. Wolenski,et al.  The Mayer and Minimum Time Problems with Stratified State Constraints , 2017, Set-Valued and Variational Analysis.

[86]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[87]  Mariane Bourgoing,et al.  Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions , 2008 .

[88]  Dirk Schieborn.  Viscosity solutions of Hamilton-Jacobi equations of eikonal type on ramified spaces , 2006 .

[89]  Mauro Garavello,et al.  Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost , 2004 .

[90]  P. Lions,et al.  Viscosity solutions of fully nonlinear second-order elliptic partial differential equations , 1990 .

[91]  S. Kružkov,et al.  GENERALIZED SOLUTIONS OF NONLINEAR FIRST ORDER EQUATIONS WITH SEVERAL INDEPENDENT VARIABLES. II , 1967 .

[92]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[93]  Heinz Schättler,et al.  Optimal Control with State Space Constraints , 2015, Encyclopedia of Systems and Control.

[94]  Adimurthi,et al.  Explicit Hopf–Lax type formulas for Hamilton–Jacobi equations and conservation laws with discontinuous coefficients , 2007 .

[95]  E. Bouin.  A Hamilton-Jacobi approach for front propagation in kinetic equations , 2014, 1406.1898.

[96]  A numerical method for a calculus of variations problem with discontinuous integrand , 1992 .

[97]  H. Frankowska,et al.  Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations with Degenerate State Constraints , 2000 .

[98]  P. J. Graber,et al.  Discontinuous solutions of Hamilton–Jacobi equations on networks , 2017 .

[99]  H. Ishii,et al.  A New Formulation of State Constraint Problems for First-Order PDEs , 1996 .

[100]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[101]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[102]  P. Dupuis,et al.  On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains , 1990 .

[103]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[104]  G. Barles.  Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit , 1993 .

[105]  G. Barles,et al.  Discontinuous solutions of deterministic optimal stopping time problems , 1987 .

[106]  Fabio Camilli,et al.  Viscosity solutions of Eikonal equations on topological networks , 2011 .