An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications. (English) Zbl 1269.49043
Loreti, Paola (ed.) et al., Hamilton-Jacobi equations: approximations, numerical analysis and applications. Based on the lectures of the CIME summer school, Cetraro, Italy, August 29–September 3, 2011. Berlin: Springer; Florence: Fondazione CIME (ISBN 978-3-642-36432-7/pbk; 978-3-642-36433-4/ebook). Lecture Notes in Mathematics 2074. CIME Foundation Subseries, 49-109 (2013).
Summary: In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton-Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron-Jensen solutions etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton-Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii [Lect. Notes Math. 2074, 111–249 (2013; Zbl 1269.49044)] which presents the dynamical system approach (“weak KAM approach”).
For the entire collection see [Zbl 1266.65002].
For the entire collection see [Zbl 1266.65002].
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
