On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities: A Guide to Theory, Applications, and Some Open Problems ... Equations and Their Applications, 104)

This monograph presents the most recent developments in the study of Hamilton-Jacobi Equations and control problems with discontinuities, mainly from the viewpoint of partial differential equations. Two main cases are investigated in detail: the case of codimension 1 discontinuities and the stratified case in which the discontinuities can be of any codimensions. In both, connections with deterministic control problems are carefully studied, and numerous examples and applications are illustrated throughout the text. After an initial section that provides a “toolbox” containing key results which will be used throughout the text, Parts II and III completely describe several recently introduced approaches to treat problems involving either codimension 1 discontinuities or networks. The remaining sections are concerned with stratified problems either in the whole space R^N or in bounded or unbounded domains with state-constraints. In particular, the use of stratified solutions to treat problems with boundary conditions, where both the boundary may be non-smooth and the data may present discontinuities, is developed. Many applications to concrete problems are explored throughout the text – such as Kolmogorov-Petrovsky-Piskunov (KPP) type problems, large deviations, level-sets approach, large time behavior, and homogenization – and several key open problems are presented. This monograph will be of interest to graduate students and researchers working in deterministic control problems and Hamilton-Jacobi Equations, network problems, or scalar conservation laws. Preface Acknowledgements: Survival kit for the potential reader: how can this book be useful to you? Additional information: Notations and Terminology Contents Introduction Viscosity solutions and discontinuities A simple, universal, and efficient notion of solution Discontinuities, a potential weakness of viscosity solutions The end of universality? Toward more general discontinuities Networks Key considerations related to discontinuities Overview of the content Part I A Toolbox for Discontinuous Hamilton–Jacobi Equations and Control Problems Chapter 1 The Basic Continuous Framework Revisited 1.1 The value function and the associated PDE 1.2 Important remarks on the comparison proof 1.3 Basic assumptions Chapter 2 PDE Tools 2.1 Discontinuous viscosity solutions for equations with discontinuities 2.1.1 Discontinuous viscosity solutions 2.1.2 The half-relaxed limits method 2.2 Strong comparison results: how to cook them? 2.2.1 Stationary equations 2.2.2 The evolution case 2.2.3 Viscosity inequalities at t = Tf in the evolution case 2.2.4 The simplest examples of comparison results: the continuous case 2.3 Whitney stratifications 2.3.1 General and admissible flat stratifications 2.3.2 Locally flattenable stratifications 2.3.3 Limits of the (LFS) approach 2.3.4 Tangentially flattenable stratifications 2.4 Partial regularity, partial regularization 2.4.1 Regular discontinuous functions 2.4.2 Regularity of subsolutions 2.4.3 Regularization of subsolutions 2.4.4 What about regularization for supersolutions? 2.5 Sub- and superdifferentials, inequalities at the boundary Chapter 3 Control Tools 3.1 Introduction: how to define deterministic control problems with discontinuities? The two half-spaces problem 3.2 A general framework for deterministic control problems 3.2.1 Dynamics, discounts, and costs 3.2.2 The control problem 3.2.3 The value function 3.3 Ishii solutions for the Bellman Equation 3.3.1 Discontinuous viscosity solutions 3.3.2 The dynamic programming principle 3.3.3 The value function is an Ishii solution 3.4 Supersolutions of the Bellman Equation 3.4.1 The super-dynamic programming principle 3.4.2 The value function is the minimal supersolution Chapter 4 Mixed Tools 4.1 Initial conditions for suband supersolutions of the Bellman Equation 4.1.1 The general result 4.1.2 A relevant example involving unbounded control 4.2 The sub-dynamic programming principle for subsolutions 4.3 Local comparison for discontinuous HJB Equations 4.4 The “good framework for HJ Equations with discontinuities” 4.4.1 General definition at the PDE level 4.4.2 The stratified case, “good assumptions” on the control problem 4.4.3 Ishii solutions for a codimension-1 discontinuous Hamilton–Jacobi Equation Chapter 5 Other Tools 5.1 Semiconvex and semiconcave functions: the main properties 5.2 Quasiconvexity: definition and main properties 5.2.1 Quasiconvex functions on the real line 5.2.2 On the maximum of two quasiconvex functions 5.2.3 Application to quasiconvex Hamiltonians 5.3 A strange, Kirchhoff-related lemma 5.4 A few results for penalized problems 5.4.1 The compact case 5.4.2 Penalization at infinity Part II Deterministic Control Problems and Hamilton–Jacobi Equations for Codimension-1 Discontinuities Chapter 6 Introduction: Ishii Solutions for the Hyperplane Case 6.1 The PDE viewpoint 6.2 The control viewpoint 6.3 The uniqueness question Chapter 7 The Control Problem and the “Natural” Value Function 7.1 Finding trajectories by differential inclusions 7.2 The Uvalue function 7.3 The complementary equation 7.4 A characterization of U- Chapter 8 A Less Natural Value Function, Regular Dynamics 8.1 Introducing U+ 8.2 More on regular trajectories 8.3 A Magical Lemma for U+ 8.4 Maximality of U+ 8.5 Appendix: stability of regular trajectories Chapter 9 Uniqueness and Non-Uniqueness Features 9.1 A typical example where U+ U− 9.2 Equivalent definitions for HT and Hreg T 9.3 A sufficient condition to get uniqueness 9.4 More examples of uniqueness and non-uniqueness Chapter 10 Adding a Specific Problem to the Interface 10.1 The control problem 10.2 The minimal solution 10.3 The maximal solution Chapter 11 Remarks on the Uniqueness Proofs, Problems Without Controllability 11.1 The main steps of the uniqueness proofs and the role of the normal controllability 11.2 Some problems without controllability Chapter 12 Further Discussions and Open Problems 12.1 The Ishii subsolution inequality: natural or unnatural from the control point-of-view? 12.2 Infinite horizon control problems and stationary equations 12.3 Towards more general discontinuities: a bunch of open problems. 12.3.1 Non-uniqueness in the case of codimension N discontinuities 12.3.2 Puzzling examples Part III Hamilton–Jacobi Equations with Codimension-1 Discontinuities: the “Network Point-of-View” Chapter 13 Introduction 13.1 The “network approach”: a different point-of-view 13.1.1 A larger space of test-functions 13.1.2 Different types of junction conditions 13.2 The “good assumptions” used in Part III 13.2.1 Good assumptions on H1, H2 13.2.2 Good assumptions on the junction condition 13.3 What do we do in this part? Chapter 14 Flux-Limited Solutions for Control Problems and Quasiconvex Hamiltonians 14.1 Definition and first properties 14.2 Stability of flux-limited solutions 14.3 Comparison results for flux-limited solutions and applications 14.3.1 The convex case 14.3.2 The quasiconvex case 14.4 Flux-limited solutions and control problems 14.5 Vanishing viscosity approximation (I): convergence via flux-limited solutions 14.6 Classical viscosity solutions as flux-limited solutions 14.7 Extension to second-order equations (I) Chapter 15 Junction Viscosity Solutions 15.1 Definition and first properties 15.1.1 Lack of regularity of subsolutions 15.1.2 The case of Kirchhoff-type conditions 15.2 Stability of junction viscosity solutions 15.3 Comparison results for junction viscosity solutions: the Lions–Souganidis approach 15.3.1 Preliminary lemmas 15.3.2 A comparison result for the Kirchhoff condition 15.3.3 Remarks on the comparison proof and some possible variations 15.3.4 Comparison results for more general junction conditions 15.3.5 Extension to second-order problems (II) 15.4 Vanishing viscosity approximation (II): convergence via junction viscosity solutions Chapter 16 From One Notion of Solution to the Others 16.1 Ishii and flux-limited solutions 16.2 Flux-limited and junction viscosity solutions for flux-limited conditions 16.3 The Kirchhoff condition and flux limiters 16.4 General Kirchhoff conditions and flux limiters 16.5 Vanishing viscosity approximation (III) 16.6 A few words about existence 16.7 Where the equivalence helps to pass to the limit Chapter 17 Emblematic Examples 17.1 HJ analog of a discontinuous one-dimensional scalar conservation law 17.1.1 On the condition at the interface 17.1.2 Network viscosity solutions 17.1.3 Main results 17.2 Traffic flow models with a fixed or moving flow constraint 17.2.1 The LWR model 17.2.2 Constraints on the flux Chapter 18 Further Discussions and Open Problems Part IV General Discontinuities: Stratified Problems Chapter 19 Stratified Solutions 19.1 Introduction 19.2 Definition of weak and strong stratified solutions 19.3 The regularity of strong stratified subsolutions 19.4 The comparison result 19.5 Regular weak stratified subsolutions are strong stratified subsolutions Chapter 20 Connections with Control Problems and Ishii Solutions 20.1 Value functions as stratified solutions 20.2 Stratified solutions and classical Ishii viscosity solutions 20.2.1 The stratified solution as the minimal Ishii solution 20.2.2 Ishii subsolutions as stratified subsolutions 20.3 Concrete situations that fit into the stratified framework 20.3.1 A general control-oriented framework 20.3.2 A general PDE-oriented framework Chapter 21 Stability Results 21.1 Strong convergence of stratifications when the local structure is unchanged 21.2 Weak convergence of stratifications and the associated stability result 21.2.1 A half-relaxed limits type result for weakly converging stratifications 21.2.2 Some problematic examples 21.2.3 Sufficient conditions for stability 21.3 Stability under structural modifications of the stratification 21.3.1 Introducing new parts of the stratification 21.3.2 Eliminable parts of the stratification 21.3.3 Sub- and super-stratified problems: a general stability result Chapter 22 Applications and Extensions 22.1 A crystal growth model—where the stratified formulation is needed 22.1.1 Ishii solutions 22.1.2 The stratified formulation 22.1.3 Generalization 22.2 Combustion—where the stratified formulation may unexpectedly help 22.2.1 The level set approach 22.2.2 The stratified formulation 22.2.3 Asymptotic analysis 22.3 Large time behavior 22.3.1 A short overview of the periodic case 22.3.2 The discontinuous framework 22.3.3 An example 22.4 Lower semicontinuous solutions à la Barron–Jensen 22.4.1 A typical lower semicontinuous eikonal example 22.4.2 Definition and regularity of subsolutions 22.4.3 The comparison result for stratified Barron–Jensen solutions Chapter 23 Further Discussions and Open Problems 23.1 More general dependence in time 23.2 Unbounded control problems 23.3 Large deviations-type problems 23.4 Homogenization 23.5 Convergence of numerical schemes and estimates 23.6 About Ishii inequalities and weak stratified solutions 23.7 Are value functions always regular? Part V State-Constrained Problems Chapter 24 Introduction to State-Constrained Problems 24.1 Why only state-constrained problems? 24.2 State-constraints and boundary conditions 24.2.1 The Tanker Problem mixing boundary conditions 24.2.2 A counter-example for the Tanker Problem 24.3 A first difficulty: boundary regularity of subsolutions 24.4 A second difficulty: initial and boundary data interaction Chapter 25 Stratified Solutions for State-Constrained Problems 25.1 Admissible stratifications for state-constrained problems 25.2 Stratified solutions and a basic comparison result 25.3 On the boundary regularity of subsolutions 25.3.1 An inward-pointing cone condition 25.3.2 Redefining the boundary values 25.3.3 Quasi-regular boundaries 25.4 Refined versions of the comparison result 25.5 Control problems, stratifications, and state-constraints conditions Chapter 26 Classical Boundary Conditions and Stratified Formulation 26.1 On the Dirichlet problem 26.1.1 Stratified formulation of the classical case 26.1.2 Continuous data with a stratified boundary 26.1.3 Discontinuous data well-adapted to a stratified boundary 26.1.4 The case of non well-adapted data 26.2 On the Neumann problem 26.2.1 Stratified formulation of the classical case 26.2.2 Codimension-1 discontinuities in the direction of reflection 26.2.3 The Dupuis–Ishii configurations 26.2.4 Applications to domains with corners 26.3 Mixing the Dirichlet and Neumann problems 26.3.1 The most standard case 26.3.2 The Tanker Problem Chapter 27 On the Stability for Singular Boundary Value Problems 27.1 Stability via classical stability results 27.2 Stability via stratified approximations 27.3 A concrete application: singularities in Dirichlet problems 27.3.1 Non-smooth domains 27.3.2 Non-smooth data Chapter 28 Further Discussions and Open Problems Part VI Investigating Other Applications Chapter 29 KPP-Type Problems with Discontinuities 29.1 Introduction to KPP Equations and front propagations 29.2 A simple discontinuous example 29.3 The codimension-1 case 29.4 The variational inequality in the codimension-1 case 29.5 Remarks on more general discontinuities 29.5.1 Using the standard notion of Ishii viscosity solution 29.5.2 Going further with stratified solutions Chapter 30 Dealing with jumps 30.1 A simple obstacle problem 30.2 Quasi-variational inequalities The case k > 0 The case k = 0 30.3 A Large Deviations problem involving jumps Analysis of the problem Sketching the comparison result Chapter 31 On Stratified Networks 31.1 Stratified networks by penalization 31.2 Some examples An easy one Ad augusta, per angusta Chapter 32 Further Discussions and Open Problems Final words Appendix: Main Assumptions and Notions of Solutions Main Assumptions Basic or fundamental assumptions Stratification assumptions Assumptions for the differential inclusion and the value function Normal controllability, tangential continuity, monotonicity Localization, convexity, subsolutions Comparison results “Good Assumptions” for the Network Approach “Good Assumptions” for Stratified Problems in RN × [0, Tf) “Good Assumptions” for Stratified Problems in the State-Constraints Case Notions of solutions References Index