Controllability of Partial Differential Equations Governed by Multiplicative Controls
The goal of this monograph is to address the issue of the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The mathematical models we examine
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1995
Alexander Y. Khapalov
Controllability of Partial Differential Equations Governed by Multiplicative Controls
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Alexander Y. Khapalov Washington State University Department of Mathematics Pullman, WA 99163 USA [email protected]
ISBN: 978-3-642-12412-9 e-ISBN: 978-3-642-12413-6 DOI: 10.1007/978-3-642-12413-6 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010927757 Mathematics Subject Classification (2000): 35, 93, 76, 49, 92 c Springer-Verlag Berlin Heidelberg 2010 ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
To Irina, Elena and Dasha
Foreword
In a typical mathematical model of a controlled distributed parameter process one usually finds either boundary or internal locally distributed controls to serve as the means to describe the effect of external actuators on the process at hand. However, these classical controls, entering the model equations as additive terms, are not suitable to deal with a vast array of processes that can change their principal intrinsic properties due to the control actions. Important examples here include (but not limited to) the chain reaction-type processes in biomedical, nuclear, chemical and financial applications, which can change their (reaction) rate when certain “catalysts” are applied, and the so-called “smart materials”, which can, for instance, alter their frequency response. The goal of this monograph is to address the issue of global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The mathematical models of our interest include the linear and nonlinear parabolic and hyperbolic PDE’s, the Schr¨odinger equation, and coupled hybrid nonlinear distributed parameter systems associated with the swimming phenomenon. Pullman, WA, USA January 2010
Alexander Khapalov
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Preface
This monograph developed from the research conducted in 2001–2009 in the area of controllability theory of partial differential equations. The concept of controllability is a principal co
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