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

Beschreibung

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.
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.
Presents a unified viewpoint of the study of Hamilton-Jacobi equations and control problems with discontinuities Considers results and methods useful for researchers working in network problems Illustrates key concepts with numerous examples and concrete applications

Autor*in

Guy Barles

Themen in »On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities«

Hamilton-Jacobi Equations with discontinuities Initial-boundary value problems Deterministic control problems Viscosity solutions Stratified problems Comparison principles Stability Vanishing viscosity method

Stimmen zu »On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities«

“The monograph is ambitious in scope, spanning around 600 pages, and is structured around six central parts. Each part plays a distinct role, gradually leading the reader from fundamental tools to advanced topics. ... This book succeeds in its ambitious mission: to create a comprehensive, evolving framework for Hamilton-Jacobi equations with discontinuities. ... The text is mathematically demanding, but it rewards the reader with deep insights, careful proofs, and a wealth of applications.” (Alpár R. Mészáros, zbMATH 1573.35001, 2026) 

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