By Sebastian Aniţa (auth.)

The fabric of the current booklet is an extension of a graduate path given via the writer on the college "Al.I. Cuza" Iasi and is meant for stu dents and researchers drawn 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. whether for a truly lengthy interval age constitution has been thought of in simple terms in demography, these days it truly is primary in epidemiology and ecology too. this can be the 1st e-book dedicated to the keep watch over of continuing age based populationdynamics.It makes a speciality of the fundamental houses ofthe suggestions and at the keep watch over of age dependent inhabitants dynamics without or with diffusion. the most objective of this paintings is to familiarize the reader with crucial difficulties, methods and leads to the mathematical thought of age-dependent versions. specific cognizance is given to optimum harvesting and to certain controllability difficulties, that are extremely important from the econom ical or ecological issues of view. We use a few new thoughts and strategies in smooth keep an eye on thought equivalent to Clarke's generalized gradient, Ekeland's variational precept, and Carleman estimates. The tools and methods we use might be utilized to different keep watch over problems.

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E. in QT x (0, +00). e. e, (a, t) E QT. e. (a, t) E QT. e. e. e. in (0, T). 1) for any T E (0,+00) . 1) will be proved via the Banach fixed point theorem. e. 1. e , s E (0, min{ a, t}) and b(·;P) E Loo(O ,T) is the solution of the Volterra integral equation b(t;P) = F(t; P) + lot K(t , s;P)b(t - s;P)ds, t E (0, T) . 4) Here we have set K(t , a;P) and 1 00 F(t; P) = (00 (3(a = (3(a, t, P(t))II(a , t, a; P) + t , t , P(t))po(a)II(a + t, t, t; P)da (min{a ,t} + Jo (3(a , t, P(t)) Jo f(a - s, t - s)II(a, t, s;P)ds da, where the functions Po, (3 and II are extended by zero outside their definition sets.

TK(t - s)b(s)dt ds = £(F)('x) + £(b)('x)£(K)('x) , £(F)('x) = £(F)('x) and in conclusion £(b)('x) = 1 - £(K)('x) + £(F)('x)£(K)('x) . 5) We shall use classical Laplace transform techniques in order to study the asymptotic behaviour of b, which is related to the singularities of £(b)('x). 6) £(K)(A) = 1. 1. 6) has a unique real solution o" , which is a simple rat root. This solution is negative if and only if Jo K(s )ds < 1. Ra < o". Proof. tK(t)dt < 0 44 CHAPTER 2 for any a E R and satisfies lim £(K)(>') = + 00, >''''''- 00 lim £(K)(>' ) = O.

E , s E (0, min{ a, t}) and b(·;P) E Loo(O ,T) is the solution of the Volterra integral equation b(t;P) = F(t; P) + lot K(t , s;P)b(t - s;P)ds, t E (0, T) . 4) Here we have set K(t , a;P) and 1 00 F(t; P) = (00 (3(a = (3(a, t, P(t))II(a , t, a; P) + t , t , P(t))po(a)II(a + t, t, t; P)da (min{a ,t} + Jo (3(a , t, P(t)) Jo f(a - s, t - s)II(a, t, s;P)ds da, where the functions Po, (3 and II are extended by zero outside their definition sets. e. e. in (0, T) x (0, +00) . We shall give the proof only for the case T > at .