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This book is aimed primarily towards physicists and mechanical
engineers specializing in modeling, analysis, and control of
discontinuous systems with friction and impacts. It fills a gap in
the existing literature by offering an original contribution to the
field of discontinuous mechanical systems based on mathematical and
numerical modeling as well as the control of such systems. Each
chapter provides the reader with both the theoretical background
and results of verified and useful computations, including
solutions of the problems of modeling and application of friction
laws in numerical computations, results from finding and analyzing
impact solutions, the analysis and control of dynamical systems
with discontinuities, etc. The contents offer a smooth
correspondence between science and engineering and will allow the
reader to discover new ideas. Also emphasized is the unity of
diverse branches of physics and mathematics towards understanding
complex piecewise-smooth dynamical systems. Mathematical models
presented will be important in numerical experiments, experimental
measurements, and optimization problems found in applied mechanics.
Stability and Controls Analysis for Delay Systems is devoted to
stability, controllability and iterative learning control (ILC) to
delay systems, including first order system, oscillating systems,
impulsive systems, fractional systems, difference systems and
stochastic systems raised from physics, biology, population
dynamics, ecology and economics, currently not presented in other
books on conventional fields. Delayed exponential matrix function
approach is widely used to derive the representation and stability
of the solutions and the controllability. ILC design are also
established, which can be regarded as a way to find the control
function. The broad variety of achieved results with rigorous
proofs and many numerical examples make this book unique.
This book extends classical Hermite-Hadamard type inequalities to
the fractional case via establishing fractional integral
identities, and discusses Riemann-Liouville and Hadamard integrals,
respectively, by various convex functions. Illustrating theoretical
results via applications in special means of real numbers, it is an
essential reference for applied mathematicians and engineers
working with fractional calculus. Contents Introduction
Preliminaries Fractional integral identities Hermite-Hadamard
inequalities involving Riemann-Liouville fractional integrals
Hermite-Hadamard inequalities involving Hadamard fractional
integrals
This book presents fractional difference, integral, differential,
evolution equations and inclusions, and discusses existence and
asymptotic behavior of their solutions. Controllability and relaxed
control results are obtained. Combining rigorous deduction with
abundant examples, it is of interest to nonlinear science
researchers using fractional equations as a tool, and physicists,
mechanics researchers and engineers studying relevant topics.
Contents Fractional Difference Equations Fractional Integral
Equations Fractional Differential Equations Fractional Evolution
Equations: Continued Fractional Differential Inclusions
1. 1 Preface Many phenomena from physics, biology, chemistry and
economics are modeled by di?erential equations with parameters.
When a nonlinear equation is est- lished, its behavior/dynamics
should be understood. In general, it is impossible to ?nd a
complete dynamics of a nonlinear di?erential equation. Hence at
least, either periodic or irregular/chaotic solutions are tried to
be shown. So a pr- erty of a desired solution of a nonlinear
equation is given as a parameterized boundary value problem.
Consequently, the task is transformed to a solvability of an
abstract nonlinear equation with parameters on a certain functional
space. When a family of solutions of the abstract equation is known
for some para- ters, the persistence or bifurcations of solutions
from that family is studied as parameters are changing. There are
several approaches to handle such nonl- ear bifurcation problems.
One of them is a topological degree method, which is rather
powerful in cases when nonlinearities are not enough smooth. The
aim of this book is to present several original bifurcation results
achieved by the author using the topological degree theory. The
scope of the results is rather broad from showing periodic and
chaotic behavior of non-smooth mechanical systems through the
existence of traveling waves for ordinary di?erential eq- tions on
in?nite lattices up to study periodic oscillations of undamped
abstract
waveequationsonHilbertspaceswithapplicationstononlinearbeamandstring
partial di?erential equations. 1.
This handbook is the fourth volume in a series of volumes devoted
to self-contained and up-to-date surveys in the theory of ordinary
differential equations, with an additional effort to achieve
readability for mathematicians and scientists from other related
fields so that the chapters have been made accessible to a wider
audience.
* Covers a variety of problems in ordinary differential equations
* Pure mathematical and real-world applications
* Written for mathematicians and scientists of many related fields
1. 1 Preface Many phenomena from physics, biology, chemistry and
economics are modeled by di?erential equations with parameters.
When a nonlinear equation is est- lished, its behavior/dynamics
should be understood. In general, it is impossible to ?nd a
complete dynamics of a nonlinear di?erential equation. Hence at
least, either periodic or irregular/chaotic solutions are tried to
be shown. So a pr- erty of a desired solution of a nonlinear
equation is given as a parameterized boundary value problem.
Consequently, the task is transformed to a solvability of an
abstract nonlinear equation with parameters on a certain functional
space. When a family of solutions of the abstract equation is known
for some para- ters, the persistence or bifurcations of solutions
from that family is studied as parameters are changing. There are
several approaches to handle such nonl- ear bifurcation problems.
One of them is a topological degree method, which is rather
powerful in cases when nonlinearities are not enough smooth. The
aim of this book is to present several original bifurcation results
achieved by the author using the topological degree theory. The
scope of the results is rather broad from showing periodic and
chaotic behavior of non-smooth mechanical systems through the
existence of traveling waves for ordinary di?erential eq- tions on
in?nite lattices up to study periodic oscillations of undamped
abstract
waveequationsonHilbertspaceswithapplicationstononlinearbeamandstring
partial di?erential equations. 1.
This book introduces iterative learning control (ILC) and its
applications to the new equations such as fractional order
equations, impulsive equations, delay equations, and multi-agent
systems, which have not been presented in other books on
conventional fields. ILC is an important branch of intelligent
control, which is applicable to robotics, process control, and
biological systems. The fractional version of ILC updating laws and
formation control are presented in this book. ILC design for
impulsive equations and inclusions are also established. The broad
variety of achieved results with rigorous proofs and many numerical
examples make this book unique. This book is useful for graduate
students studying ILC involving fractional derivatives and
impulsive conditions as well as for researchers working in pure and
applied mathematics, physics, mechanics, engineering, biology, and
related disciplines.
This book introduces iterative learning control (ILC) and its
applications to the new equations such as fractional order
equations, impulsive equations, delay equations, and multi-agent
systems, which have not been presented in other books on
conventional fields. ILC is an important branch of intelligent
control, which is applicable to robotics, process control, and
biological systems. The fractional version of ILC updating laws and
formation control are presented in this book. ILC design for
impulsive equations and inclusions are also established. The broad
variety of achieved results with rigorous proofs and many numerical
examples make this book unique. This book is useful for graduate
students studying ILC involving fractional derivatives and
impulsive conditions as well as for researchers working in pure and
applied mathematics, physics, mechanics, engineering, biology, and
related disciplines.
Poincare-Andronov-Melnikov Analysis for Non-Smooth Systems is
devoted to the study of bifurcations of periodic solutions for
general n-dimensional discontinuous systems. The authors study
these systems under assumptions of transversal intersections with
discontinuity-switching boundaries. Furthermore, bifurcations of
periodic sliding solutions are studied from sliding periodic
solutions of unperturbed discontinuous equations, and bifurcations
of forced periodic solutions are also investigated for impact
systems from single periodic solutions of unperturbed impact
equations. In addition, the book presents studies for weakly
coupled discontinuous systems, and also the local asymptotic
properties of derived perturbed periodic solutions. The
relationship between non-smooth systems and their continuous
approximations is investigated as well. Examples of 2-, 3- and
4-dimensional discontinuous ordinary differential equations and
impact systems are given to illustrate the theoretical results. The
authors use so-called discontinuous Poincare mapping which maps a
point to its position after one period of the periodic solution.
This approach is rather technical, but it does produce results for
general dimensions of spatial variables and parameters as well as
the asymptotical results such as stability, instability, and
hyperbolicity.
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