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This book presents cutting-edge contributions in the areas of
control theory and partial differential equations. Over the
decades, control theory has had deep and fruitful interactions with
the theory of partial differential equations (PDEs). Well-known
examples are the study of the generalized solutions of
Hamilton-Jacobi-Bellman equations arising in deterministic and
stochastic optimal control and the development of modern analytical
tools to study the controllability of infinite dimensional systems
governed by PDEs. In the present volume, leading experts provide an
up-to-date overview of the connections between these two vast
fields of mathematics. Topics addressed include regularity of the
value function associated to finite dimensional control systems,
controllability and observability for PDEs, and asymptotic analysis
of multiagent systems. The book will be of interest for both
researchers and graduate students working in these areas.
This volume covers selected topics addressed and discussed during
the workshop "PDE models for multi-agent phenomena," which was held
in Rome, Italy, from November 28th to December 2nd, 2016. The
content mainly focuses on kinetic equations and mean field games,
which provide a solid framework for the description of multi-agent
phenomena. The book includes original contributions on the
theoretical and numerical study of the MFG system: the uniqueness
issue and finite difference methods for the MFG system, MFG with
state constraints, and application of MFG to market competition.
The book also presents new contributions on the analysis and
numerical approximation of the Fokker-Planck-Kolmogorov equations,
the isotropic Landau model, the dynamical approach to the
quantization problem and the asymptotic methods for fully nonlinear
elliptic equations. Chiefly intended for researchers interested in
the mathematical modeling of collective phenomena, the book
provides an essential overview of recent advances in the field and
outlines future research directions.
This volume provides an introduction to the theory of Mean Field
Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a
mean-field model for Nash equilibria in the strategic interaction
of a large number of agents. Besides giving an accessible
presentation of the main features of mean-field game theory, the
volume offers an overview of recent developments which explore
several important directions: from partial differential equations
to stochastic analysis, from the calculus of variations to modeling
and aspects related to numerical methods. Arising from the CIME
Summer School "Mean Field Games" held in Cetraro in 2019, this book
collects together lecture notes prepared by Y. Achdou (with M.
Lauriere), P. Cardaliaguet, F. Delarue, A. Porretta and F.
Santambrogio. These notes will be valuable for researchers and
advanced graduate students who wish to approach this theory and
explore its connections with several different fields in
mathematics.
This book presents cutting-edge contributions in the areas of
control theory and partial differential equations. Over the
decades, control theory has had deep and fruitful interactions with
the theory of partial differential equations (PDEs). Well-known
examples are the study of the generalized solutions of
Hamilton-Jacobi-Bellman equations arising in deterministic and
stochastic optimal control and the development of modern analytical
tools to study the controllability of infinite dimensional systems
governed by PDEs. In the present volume, leading experts provide an
up-to-date overview of the connections between these two vast
fields of mathematics. Topics addressed include regularity of the
value function associated to finite dimensional control systems,
controllability and observability for PDEs, and asymptotic analysis
of multiagent systems. The book will be of interest for both
researchers and graduate students working in these areas.
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