On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities

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.