Infinite Dimensional Linear Control Systems

North-Holland Mathematics Studies

Book 201
Elsevier
Free sample

For more than forty years, the equation y’(t) = Ay(t) + u(t) in Banach spaces has been used as model for optimal control processes described by partial differential equations, in particular heat and diffusion processes. Many of the outstanding open problems, however, have remained open until recently, and some have never been solved. This book is a survey of all results know to the author, with emphasis on very recent results (1999 to date).



The book is restricted to linear equations and two particular problems (the time optimal problem, the norm optimal problem) which results in a more focused and concrete treatment. As experience shows, results on linear equations are the basis for the treatment of their semilinear counterparts, and techniques for the time and norm optimal problems can often be generalized to more general cost functionals.



The main object of this book is to be a state-of-the-art monograph on the theory of the time and norm optimal controls for y’(t) = Ay(t) + u(t) that ends at the very latest frontier of research, with open problems and indications for future research.



Key features:



· Applications to optimal diffusion processes.
· Applications to optimal heat propagation processes.
· Modelling of optimal processes governed by partial
differential equations.
· Complete bibliography.
· Includes the latest research on the subject.
· Does not assume anything from the reader except
basic functional analysis.
· Accessible to researchers and advanced graduate
students alike

· Applications to optimal diffusion processes.
· Applications to optimal heat propagation processes.
· Modelling of optimal processes governed by partial
differential equations.
· Complete bibliography.
· Includes the latest research on the subject.
· Does not assume anything from the reader except
basic functional analysis.
· Accessible to researchers and advanced graduate
students alike

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Additional Information

Publisher
Elsevier
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Published on
Jul 12, 2005
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Pages
332
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ISBN
9780080457345
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Language
English
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Genres
Mathematics / Applied
Mathematics / Functional Analysis
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Content Protection
This content is DRM protected.
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Available on Android devices
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Eligible for Family Library

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This volume introduces a unified, self-contained study of linear discrete parabolic problems through reducing the starting discrete problem to the Cauchy problem for an evolution equation in discrete time. Accessible to beginning graduate students, the book contains a general stability theory of discrete evolution equations in Banach space and gives applications of this theory to the analysis of various classes of modern discretization methods, among others, Runge-Kutta and linear multistep methods as well as operator splitting methods.



Key features:



* Presents a unified approach to examining discretization methods for parabolic equations.
* Highlights a stability theory of discrete evolution equations (discrete semigroups) in Banach space.
* Deals with both autonomous and non-autonomous equations as well as with equations with memory.
* Offers a series of numerous well-posedness and convergence results for various discretization methods as applied to abstract parabolic equations; among others, Runge-Kutta and linear multistep methods as well as certain operator splitting methods.
* Provides comments of results and historical remarks after each chapter.

· Presents a unified approach to examining discretization methods for parabolic equations.
· Highlights a stability theory of discrete evolution equations (discrete semigroups) in Banach space.
· Deals with both autonomous and non-autonomous equations as well as with equations with memory.
· Offers a series of numerous well-posedness and convergence results for various discretization methods as applied to abstract parabolic equations; among others, Runge-Kutta and linear multistep methods as well as certain operator splitting methods as well as certain operator splitting methods are studied in detail.
·Provides comments of results and historical remarks after each chapter.

The book is an almost self-contained presentation of the most important concepts and results in viability and invariance. The viability of a set K with respect to a given function (or multi-function) F, defined on it, describes the property that, for each initial data in K, the differential equation (or inclusion) driven by that function or multi-function) to have at least one solution. The invariance of a set K with respect to a function (or multi-function) F, defined on a larger set D, is that property which says that each solution of the differential equation (or inclusion) driven by F and issuing in K remains in K, at least for a short time.

The book includes the most important necessary and sufficient conditions for viability starting with Nagumo’s Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions. In the latter (i.e. multi-valued) cases, the results (based on two completely new tangency concepts), all due to the authors, are original and extend significantly, in several directions, their well-known classical counterparts.New concepts for multi-functions as the classical tangent vectors for functionsProvides the very general and necessary conditions for viability in the case of differential inclusions, semilinear and fully nonlinear evolution inclusions Clarifying examples, illustrations and numerous problems, completely and carefully solvedIllustrates the applications from theory into practice Very clear and elegant style
A caution to mathematics professors: Complex Variables does not follow conventional outlines of course material. One reviewer noting its originality wrote: "A standard text is often preferred [to a superior text like this] because the professor knows the order of topics and the problems, and doesn't really have to pay attention to the text. He can go to class without preparation." Not so here — Dr. Flanigan treats this most important field of contemporary mathematics in a most unusual way. While all the material for an advanced undergraduate or first-year graduate course is covered, discussion of complex algebra is delayed for 100 pages, until harmonic functions have been analyzed from a real variable viewpoint. Students who have forgotten or never dealt with this material will find it useful for the subsequent functions. In addition, analytic functions are defined in a way which simplifies the subsequent theory. Contents include: Calculus in the Plane, Harmonic Functions in the Plane, Complex Numbers and Complex Functions, Integrals of Analytic Functions, Analytic Functions and Power Series, Singular Points and Laurent Series, The Residue Theorem and the Argument Principle, and Analytic Functions as Conformal Mappings.
Those familiar with mathematics texts will note the fine illustrations throughout and large number of problems offered at the chapter ends. An answer section is provided. Students weary of plodding mathematical prose will find Professor Flanigan's style as refreshing and stimulating as his approach.
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