243154 Introduction to Hamilton-Jacobi-Bellman Equations and Optimal Control (V) (SoSe 2023)

Overview: This collection of 15 lectures will introduce the students to the main re-
sults 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 you to 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,II, Maß- und Integrationstheorie und Funktionalanalysis
helpfuL: PDE 1 und WTheorie 1+2

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]