241206 Introduction to viscosity solutions for first order partial differential equations (VÜA) (SoSe 2024)

This collection of 15 lectures will introduce the main results involving what are called “viscosity solutions” for Hamilton-Jacobi-Bellman equations
(HJB), with an emphasis on the connection to optimal control. These are the natural class of equations that arise when looking at minimization problems for particle trajectories in a specified energy environment (among other origins), and they have played a fundamental role in physics, engineering, geometry, probability, and optimal control for over a century.

Despite having a well-known importance for so long, and because examples show that one expects solutions to not be classical, these equations lacked a notion of a (non-classical ) solution that could give existence and uniqueness until 1980s. The name granted to this type of non-classical solution is “viscosity solution”, and it has led to many mathematical advances since its inception. These lectures will introduce this class of problems, to viscosity solutions, and, time permitting, possibly some related topics involving random walks and or the structure of operators with the comparison principle. The pace and level of difficulty of the lectures will be adjusted based on the audience.

Requirements for participation, required level

Analysis I and II, Measure and integration theory, Linear Algebra 1

Knowledge of partial differential equations is not mandatory but very useful.

Bibliography

Guy Barles, “An introduction to the theory of viscosity solutions for first order
Hamilton-Jacobi equations and applications”, Lecture Notes in Math., 2074, Fond.
CIME/CIME Found. Subser., Springer, Heidelberg, 2013. [2].