Differential Systems Involving Impulses
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954
Sudhakar G. Pandit Sadashiv G.Deo
Differential Systems Involving Impulses
Springer-Verlag Berlin Heidelberg New York 1982
Authors
Sudhakar G. Pandit Sadashiv G. Deo Department of Mathematics Centre of Post-graduate Instruction and Research University of Bombay Panaji, Goa - 403 001, India
AMS Subject Classifications (1980): 34 A XX, 34 C 11, 34 D XX, 34 H 05, 45FXX, 93DXX
ISBN 3-540-11606-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11606-0 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982' Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2t 46/3140-54321 0
PREFACE When a system described by an ordinary differential equation is sUbjected to perturbations, the perturbed system is again an ordinary differential equation in which the perturbation function is assumed to be continuous or integrable, and as such, the state of the system changes continuously with respect to time. However, in many physical problems (optimal control theory in particular), one can not expect perturbations to be well behaved. Biological systems such as heart beats, blOod flows, pulse frequency modulated systems and models for biological neural nets exhibit an impulsive behaviour. Therefore, perturbations of impulsive type are more realistic. This gives rise to Measure Differential Equations. The derivative involved in these equations is the distributional derivative. The fact that their solutions are discontinuous (they are functions of bounded variation), renders most of the classical, methods ineffective, thereby making their study interesting. The systems involving impUlsive behaviour are in abundance. We mention beloW some problems of this kind. (i) Growth Problem: A fish breeding pond maintained scientifi cally is an example of this kind. Here the natural growth of fish population is disturbed by making catches at certain time intervals and by adding fresh breed. The natural growth of fish population is disturbed at some time intervals. This problem therefore involves impulses. We study such a model in some details in Chapter 1. (ii) Case and Blaquiere Problem [2, 4] : The profit of a roadside inn on some prescribed interval of time t T is a function of the number of strangers who pass by on the road each day and of the number of times the inn is repainted during that period. The ability to attract new customers into the inn depends on its appearance which is supposed to be indexed by a number Xl' During time intervals between paint Jobs, Xl decays according to the law
r
xi
= k
xl'
k
= positive
The total profit in
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