Analysis and Control of Age-Dependent Population Dynamics

Analysis and Control of Age-Dependent Population Dynamics

MATHEMATICAL MODELLING: Theory and Applications VOLUME I I This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the under standing of the systems being modelled are particularly welcomed. Manuscripts to be considered for publication lie within the following, non-exhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programmering, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural science s, mathematical biology, mathematical ecology, image proce ssing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractal s. Original monographs, comprehensive surveys as well as edited collections will be considered for publication. Editors: R. Lowen (Antwerp. Belgium) Editorial Board: E. Jouini (University ofparis 1 and ENSAE, France) GJ. Klir (New York, U.S.A.) J.-L. Lions (Paris, France) P.G. Mezey (Saskatchewan, Canada) F. Pfeiffer (Miinchen. Germany) H.-J. Zimmerman (Aachen. Germany) The titles published in this series are listed at the end of this volume.

Analysis and Control of Age-Dependent Population Dynamics by Sebastian Anita Faculty ofmathematics, University 'AI. I. Cuza' Iasi and Institute ofmathematics, Romanian Academy, lasi, Romania Springer-Science+Business Media, B.V.

Library ofcongress Cataloging-in-Publication Data Anita, Sebastian. Analysis and control ofage-dependent population dynamics / by Sebastian Anita. p. em. -- (Mathematical modelling-theory and applications; v, II) Includes bibliographical references (p. ). 1. Age-structured populations--mathematical models. 2. Population biology--mathematical models. 1. Title. II. Series. QH352 A54 2000 577.8'8'0151 I8--dc21 ISBN 978-90-481-5590-3 ISBN 978-94-015-9436-3 (ebook) 001 10.1007/978-94-015-9436-3 00-064699 Printed on acid-free paper All Rights Reserved 2000 Springer Science-Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint of the hardcover Ist edition 200() No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

CONTENTS Preface vii 1 INTRODUCTION 1 1.1 Overview of the problems................ 1 1.2 General models of population dynamics with diffusion 6 1.3 Age-dependent epidemic models............. 10 2 ANALYSIS OF AGE-DEPENDENT POPULATION DYNAMICS 15 2.1 The linear age-dependent model.... 15 2.2 A general nonlinear model....... 29 2.3 Asymptotic behaviour of the solution. 42 2.4 A linear periodic age-dependent population dynamics 51 2.5 Exercises 60 3 OPTIMAL CONTROL OF POPULATION DYNAMICS 65 3.1 Optimal harvesting for linear age-dependent population dynamics............................... 67 3.2 Optimal harvesting for a nonlinear model........... 75 3.3 Optimal harvesting for a linear periodic population dynamics 88 3.4 A nonlinear optimal control problem 95 3.5 Exercises 104 4 ANALYSIS OF POPULATION DYNAMICS WITH DIFFUSION 109 4.1 Basic properties of the solution. The linear model. 109 4.2 A general nonlinear model....... 123 4.3 Asymptotic behaviour of the solution. 128 4.4 Exercises 132 v

VI CONTENTS 5 CONTROL OF POPULATION DYNAMICS WITH DIFFUSION 137 5.1 Optimal harvesting for a general nonlinear model.... 137 5.2 A null controllability problem with distributed parameter 147 5.3 A local exact controllability result 154 5.4 Exercises 167 APPENDIX 1: ELEMENTS OF NONLINEAR ANALYSIS 171 Al.1 Convex functions and sub differentials....... 171 Al.2 Generalized gradients of locally Lipschitz functions 177 Al.3 The Ekeland variational principle.......... 180 APPENDIX 2: THE LINEAR HEAT EQUATION 183 REFERENCES 193 INDEX 199

PREFACE The material of the present book is an extension of a graduate course given by the author at the University "Al.I. Cuza" Iasi and is intended for students and researchers interested in the applications of optimal control and in mathematical biology. Age is one of the most important parameters in the evolution of a biological population. Even if for a very long period age structure has been considered only in demography, nowadays it is fundamental in epidemiology and ecology too. This is the first book devoted to the control of continuous age structured population dynamics. It focuses on the basic properties of the solutions and on the control of age structured population dynamics with or without diffusion. The main goal of this work is to familiarize the reader with the most important problems, approaches and results in the mathematical theory of age-dependent models. Special attention is given to optimal harvesting and to exact controllability problems, which are very important from the economical or ecological points of view. We use some new concepts and techniques in modern control theory such as Clarke's generalized gradient, Ekeland's variational principle, and Carleman estimates. The methods and techniques we use can be applied to other control problems. The book is intended to be accesible to graduate students whose mathematical background includes basic courses in functional analysis, ordinary and partial differential equations. Some technical details of the proofs are omitted but the reader can always fill the gaps. Several exercises are proposed at the end of the chapters. For most of them we have introduced substantial hints. The references are not complete and refer only to the works closely related or used in this book. I would like to express my gratitude to Professor Viorel Barbu for his support and interest in the development of the present work. I enjoyed the fruitful cooperation and discussions with Professor Mimmo Iannelli. I extend to him my warmest thanks. Thanks are also due to Professor Michel Langlais for stimulating discussions which helped me to improve this material. VB

NOTATIONS R = (-00, +00) - the set of real numbers R + = [0,+00) r+ = max{r, a},r" = - min{r, o} = r+ - r R" - the n-dimensional Euclidean space C - the set of complex numbers o ern - an open subset of R" j ao is the boundary of 0 o - the closure of the set 0 wee 0 means that w is a compact subset of 0 au au au) " V'u = ( -a' -a'...,-a - the gradient of u Xl X2 X n X " Y = L~=l XiYi - n a2 b.. = tr ax~ the scalar product of X = (Xl,..., Xn), Y = (YI'"""' Yn) - the Laplace operator 1I"llx - the norm of space X D u ( a,t ) = 1 " u(a+c,t+c)-u(a,t) im ~---~--'-----'- c-to c " u(a+c,t+c,x) -u(a,t,x) D u( a,t, X ) = 1 im ---..:...---..:.-----:----:..----:...:...-:-...:.. c-to c ~~ - the outward normal derivative V(O) = {u measurable on OJ In lulpdx < +oo}, 1 s p < +00 Hm(O) - the Sobolev space {u E 2(0); DQu E 2(0), lal::s; m} Ck([O, T]; X) - the space of continuously differentiable functions u : [0, T] --+ X, of order up to and including k V(O,TiX) = {u measurable from (O,T) to Xi 1T liu(t)li~dt < +oo} IX