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

This version is the last version of our book project on Hamilton-Jacobi Equations and Control Problems with discontinuities. Compared to the third version (online in december 2022), we have improved Part V (Stratified solutions for state-constraints problems) and Part VI on the applications but also the stability results for stratified solutions; we have rewritten a large part of the introduction and added guidelines for the reader. As in the previous versions, we have incorporated new results and examples, changed some points-of-view, detailed some proofs and corrected several mistakes. Version 3 had 550 pages, this one 630.As the third version, it is composed of six parts: Part I is still a toolbox with key results which are used in all the other parts. The study of the simplest case, i.e. the case of a co-dimension 1 discontinuity, is now split in two parts: in Part II, we only consider control problems and the associated Bellman Equations are treated by using only the classical notion of viscosity solutions. In this part, the methods are a combinations of control and pdes techniques. On the contrary, Part III describes purely pdes approaches which are inspired by the literature on Hamilton Jacobi Equations on networks and which can handle the case of non-convex Hamiltonians. In this part, we present two notions of solutions, namely flux-limited and junction viscosity solutions, and we study in detail their properties by providing comparison and stability results. We also show that they are ``almost'' equivalent when both make sense, i.e. for quasi-convex Hamiltonians. Part IV concerns stratified problems in $\R^N$, i.e. problems with discontinuities of any co-dimensions: the main change compared to the previous version is the introduction of a notion of ``weak'' stratified (sub)solution. In Part V, we address the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. Finally, in Part VI we describe some applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and we discuss possible extensions to problems with jumps and to ``stratified networks''.Even if we consider this version as being the final one, all comments are welcome!