By Sebastian Aniţa (auth.)

The fabric of the current ebook is an extension of a graduate direction given through the writer on the collage "Al.I. Cuza" Iasi and is meant for stu dents and researchers attracted to the purposes of optimum regulate and in mathematical biology. Age is likely one of the most vital parameters within the evolution of a bi ological inhabitants. no matter if for a really lengthy interval age constitution has been thought of in basic terms in demography, these days it really is primary in epidemiology and ecology too. this is often the 1st ebook dedicated to the keep an eye on of constant age based populationdynamics.It specializes in the fundamental homes ofthe recommendations and at the regulate of age dependent inhabitants dynamics without or with diffusion. the most aim of this paintings is to familiarize the reader with an important difficulties, ways and leads to the mathematical thought of age-dependent types. certain realization is given to optimum harvesting and to precise controllability difficulties, that are vitally important from the econom ical or ecological issues of view. We use a few new suggestions and strategies in smooth regulate concept equivalent to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and strategies we use could be utilized to different regulate problems.

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**Sample text**

17) . ) sign w(a, t + a)da, tER .

Denote by * the set of all sp with the above mentioned properties. e . e, a E (O,at). 4. 1) has a unique weak solution. Proof. 1). 16) io io for any sp E . 5. Let 9 E Loo(QT). e. e. in QT. e. e, in (0, at) . 5. 10) . e, t E (0,T), n E N* . 18) . 15) is verified. e. e, in (0, T) . e. in QT. 4 - continued. 23) cp(at> t) = 0 t E (O,T). 23) belongs to *

*Then for t large enough we have = (0'* - M (h o (zo + 'Yo))) (zo + 'Yo). = 0 and t-+oo lim y (t) = 'Yo . then 'Yo = 0, otherwise, for t large enough we have > 0 and so Zo - 'Yo > O. We conclude that 3 lim y' (t) > 0, t-+oo which is absurd. 2) that t-+oo lim t-++oo where P (a) = 'UZo exp { -O'*a and _ p(a) ='Uzoexp { * -0' a- l lip (t) - pllu"'(o a -l J-L (s) ds } , a E (0, at). 1). In the same manner follows the conclusion for 0'* [] Va E 0, at , = O. 4. 1) (a,t) E Q, where Q = [0, at) x R. l are T-periodic with respect to the time t . *