Differential equations can bring mathematics to life, describing phenomena originating in physics, chemistry, biology, economics, and more. Used by scientists and engineers alike, differential equations are also the starting point of much purely mathematical activity. They also play a role in the formulation and resolution of problems in harmonic analysis, differential geometry, and probability calculus. A large part of functional analysis has therefore been motivated by the need to solve questions in the analysis of differential systems, as with numerical analysis.Differential equations are doubly relevant, then: as significant in many areas of mathematics, and as important machinery for applying mathematics to real-world problems. This book therefore aims to provide a rigorous introduction to the theoretical study of differential equations, and to demonstrate their utility with applications in many fields.Ordinary Differential Equations and Applications originates from several courses given by the author for decades at the University of Seville. It aims to bring together rigorous mathematical theory and the rich variety of applications for differential equations. The book examines many aspects of differential equations: their existence, uniqueness, and regularity, alongside their continuous dependence on data and parameters. Delving into permanent interpretation of the laws of differential equations, we also look at the role of data and how their solutions behave. Each chapter finishes with a collection of exercises, many of which also contain useful hints.
The term “control theory” refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010. Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a friendly introduction to, and an updated account of, some of the most active trends in current research.
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