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Beginning with the works of N.N.Krasovskii [81, 82, 83], which
clari fied the functional nature of systems with delays, the
functional approach provides a foundation for a complete theory of
differential equations with delays. Based on the functional
approach, different aspects of time-delay system theory have been
developed with almost the same completeness as the corresponding
field of ODE (ordinary differential equations) the ory. The term
functional differential equations (FDE) is used as a syn onym for
systems with delays 1. The systematic presentation of these re
sults and further references can be found in a number of excellent
books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102,
128]. In this monograph we present basic facts of i-smooth calculus
~ a new differential calculus of nonlinear functionals, based on
the notion of the invariant derivative, and some of its
applications to the qualitative theory of functional differential
equations. Utilization of the new calculus is the main distinction
of this book from other books devoted to FDE theory. Two other
distinguishing features of the volume are the following: - the
central concept that we use is the separation of finite dimensional
and infinite dimensional components in the structures of FDE and
functionals; - we use the conditional representation of functional
differential equa tions, which is convenient for application of
methods and constructions of i~smooth calculus to FDE theory.
Beginning with the works of N.N.Krasovskii [81, 82, 83], which
clari fied the functional nature of systems with delays, the
functional approach provides a foundation for a complete theory of
differential equations with delays. Based on the functional
approach, different aspects of time-delay system theory have been
developed with almost the same completeness as the corresponding
field of ODE (ordinary differential equations) the ory. The term
functional differential equations (FDE) is used as a syn onym for
systems with delays 1. The systematic presentation of these re
sults and further references can be found in a number of excellent
books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102,
128]. In this monograph we present basic facts of i-smooth calculus
~ a new differential calculus of nonlinear functionals, based on
the notion of the invariant derivative, and some of its
applications to the qualitative theory of functional differential
equations. Utilization of the new calculus is the main distinction
of this book from other books devoted to FDE theory. Two other
distinguishing features of the volume are the following: - the
central concept that we use is the separation of finite dimensional
and infinite dimensional components in the structures of FDE and
functionals; - we use the conditional representation of functional
differential equa tions, which is convenient for application of
methods and constructions of i~smooth calculus to FDE theory.
i-SMOOTH ANALYSIS A totally new direction in mathematics, this
revolutionary new study introduces a new class of invariant
derivatives of functions and establishes relations with other
derivatives, such as the Sobolev generalized derivative and the
generalized derivative of the distribution theory. i-smooth
analysis is the branch of functional analysis that considers the
theory and applications of the invariant derivatives of functions
and functionals. The important direction of i-smooth analysis is
the investigation of the relation of invariant derivatives with the
Sobolev generalized derivative and the generalized derivative of
distribution theory. Until now, i-smooth analysis has been
developed mainly to apply to the theory of functional differential
equations, and the goal of this book is to present i-smooth
analysis as a branch of functional analysis. The notion of the
invariant derivative (i-derivative) of nonlinear functionals has
been introduced in mathematics, and this in turn developed the
corresponding i-smooth calculus of functionals and showed that for
linear continuous functionals the invariant derivative coincides
with the generalized derivative of the distribution theory. This
book intends to introduce this theory to the general mathematics,
engineering, and physicist communities. i-Smooth Analysis: Theory
and Applications Introduces a new class of derivatives of functions
and functionals, a revolutionary new approach Establishes a
relationship with the generalized Sobolev derivative and the
generalized derivative of the distribution theory Presents the
complete theory of i-smooth analysis Contains the theory of FDE
numerical method, based on i-smooth analysis Explores a new
direction of i-smooth analysis, the theory of the invariant
derivative of functions Is of interest to all mathematicians,
engineers studying processes with delays, and physicists who study
hereditary phenomena in nature. AUDIENCE Mathematicians, applied
mathematicians, engineers, physicists, students in mathematics
The main aim of the book is to present new constructive methods of
delay differential equation (DDE) theory and to give readers
practical tools for analysis, control design and simulating of
linear systems with delays. Referred to as "systems with delays" in
this volume, this class of differential equations is also called
delay differential equations (DDE), time-delay systems, hereditary
systems, and functional differential equations. Delay differential
equations are widely used for describing and modeling various
processes and systems in different applied problems At present
there are effective control and numerical methods and corresponding
software for analysis and simulating different classes of ordinary
differential equations (ODE) and partial differential equations
(PDE). There are many applications for these types of equations,
because of this progress, but there are not as many methodologies
in systems with delays that are easily applicable for the engineer
or applied mathematician. there are no methods of finding solutions
in explicit forms, and there is an absence of generally available
general-purpose software packages for simulating such systems.
Systems with Delays fills this void and provides easily applicable
methods for engineers, mathematicians, and scientists to work with
delay differential equations in their operations and research.
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