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Intro -- 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 = 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 and reg -- 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 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 1, 2 -- 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.
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A Modern Introduction to Mathematical Analysis / / Alessandro Fonda
