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

Guy Barles (1) , Emmanuel Chasseigne (1)

1 IDP - Institut Denis Poisson

This manuscrit is a project of book on Hamilton-Jacobi Equations and Control Problems with discontinuities. For the time being, it is composed of 4 parts: -- Part I is a toolbox with key results which are used in all the other parts. -- Part II describes several approaches which were introduced recently to treat problems involving co-dimension 1 discontinuities or networks. -- Part III concerns stratified problems in $R^N$, i.e. problems with discontinuities of any co-dimensions. -- Part IV addresses the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. This second version is still imperfect and we will welcome any (fair) comment on it. In this second version, besides of improving few points in the first one, we have added applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and started to develop the use of stratified solutions to treat problems with boundary conditions (Dirichlet, Neumann and mixed boundary conditions), where both the boundary may be non-smooth and the data may present discontinuities. Of course, for all these questions, we only provide examples of what can be done since developing a whole theory would be too long, out of the scope of this book and probably a little bit beyond what we are able to do up to now. In these two directions, we address, in particular, a new question which is interesting when considering applications: under which conditions can one prove that Ishii's viscosity solutions are stratified solutions? The interest of this question is clear: when these conditions hold, one can take advantages of both the nice stability properties of classical viscosity solutions and the general comparison result for stratified solutions. A priori the third version (next release planned in june 2020) will probably be the last one: besides of improving the second version, we plan to address Large Deviations and homogenization problems. This will be the occasion to test the results we have presented so far and to improve them.

Mots clés: Discontinuous Hamiltonians; Comparison principles; Stratification problems; Hamilton-Jacobi-Bellman Equations; Deterministic control problems; Viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Guy Barles, Emmanuel Chasseigne. An Illustrated Guide of the Modern Approches of Hamilton-Jacobi Equations and Control Problems with Discontinuities. 2018. ⟨hal-01962955v2⟩