General Introduction -- Basic Continuous Framework and Classical Assumptions Revisited -- Part I: A Toolbox for Discontinuous Hamilton-Jacobi Equations and Control Problems -- PDE Tools -- Control Tools -- Mixed Tools -- Other Tools -- Part II: Deterministic Control Problems and Hamilton-Jacobi Equations for Codimension One Discontinuities -- Introduction: Ishii Solutions for the Hyperplane Case -- The Control Problem and the "Natural" Value Function -- A Less Natural Value-Function, Regular and Singular Dynamics -- Uniqueness and Non-Uniqueness Features -- Adding a Specific Problem on the Interface -- Remarks on the Uniqueness Proofs, Problems without Controllability -- Further Discussions and Open Problems -- Part III: Hamilton-Jacobi Equations with Codimension One Discontinuities: The "Network" Point of View -- Introduction -- Flux-Limited Solutions for Control Problems and Quasi-Convex Hamiltonians -- Junction Viscosity Solutions -- From One Notion of Solution to the Others -- Applications and Emblematic Examples -- Further Discussions and Open Problems -- Part IV: General Discontinuities: Stratified Problems -- Stratified Solutions -- Connections with Control Problems and Ishii Solutions -- Stability Results -- Applications -- Further Discussions and Open Problems -- Part V: State-Constraint Problems -- Introduction to State-Constraint Problems -- Stratified Solutions for State-Constraint Problems -- Classical Boundary Conditions and Stratified Formulation -- Stability for Singular Boundary Value Problems -- Further Discussions and Open Problems -- Part VI: Investigating Other Applications -- KPP-Type Problems with Discontinuities -- And What about Jumps? And What about Networks -- Further Discussions and Open Problems -- Part VII: Appendices -- Notations and Terminology -- Assumptions, Hypotheses, Notions of Solutions.

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.