4.1 Optimal feedback control problems of population diffusion models
Let Ω be a bounded domain in with boundary of class and . We consider the following feedback control problem:
with , where and .
We assume the following hypotheses:
a and b are globally measurable in , and there exist two functions such that, for a.a. and every ,
the map satisfies the following conditions:
for all , is measurable;
for a.a. and , is continuous;
there exists such that, for a.a. and every and ,
Let , , , and be the maps defined by
We can write the feedback control problem (4.1) as a first-order inclusion in the Hilbert space
where , is the linear operator defined as and .
The operators and are defined as and . Trivially, L is a linear and bounded operator, and M is sequentially continuous with respect to the weak topology, and the operator K of condition (L2) is the identity operator, in particular it is invertible. Moreover, it is known that A generates a semigroup of contractions on E with the constant D of (2.1) equal to 1 (see, e.g.,  p.209-210). Hence hypotheses (A),(L1),(L2) and (M) are satisfied with .
We prove that there exists a mild solution
where , .
We show now that all the hypotheses of Theorem 3.1 are satisfied.
The map defined as
is a selection of . Moreover, by the separability of the space , conditions (i), (ii), (iii), and by the Pettis measurability theorem (see Theorem 2.4), we obtain that g is a measurable map, and so we have obtained for every the existence of a measurable selection of .
We prove now that the map F verifies condition (F2), i.e., that the map is weakly sequentially closed for a.a. .
We start proving the sequential closedness with respect to the weak topology of the multimap V.
Let , in , , , . According to Mazur’s convexity lemma, for each j there exist and positive numbers , such that and the sequence in . Then we can extract a subsequence, denoted as the sequence, satisfying for a.a. . We have that the convex combination converges weakly to ω. Moreover, by the definition of V, we have that
Applying again Mazur’s convexity lemma, for each j there exist and positive numbers , such that and the sequence in . Then we can extract a subsequence, denoted as the sequence, satisfying for a.a. . As before,
Then passing to the limit, we obtain
Let now be fixed, let , weakly convergent to , and let with for every , weakly convergent to . By the definition of the multimap F, we have
where , , .
By the definition of the multimap V and the weak convergence of , we have that the sequence is norm bounded. Hence, by the reflexivity of the space , up to a subsequence, weakly converges to and the weak closure of the multimap V implies . Moreover, by the continuity of the map p and condition (iv), we have that converges weakly to and it is easy to see that converges weakly to . In conclusion, we have obtained
Furthermore, easily, V has convex and closed values; thus, by the linearity of the map and following the same reasonings as above, F is convex closed-valued as well.
Finally, the multimap verifies all the hypotheses of Theorem 3.1. Indeed from (i), (iv) and again from the definition of V, we have that
Denoting with , we have
obtaining both that for every and the set is bounded (hence relatively compact by the reflexivity of ) and that condition (F3′′′) is satisfied. Then applying Theorem 3.1 (see also Remark 3.2), we obtain the existence of a function such that
Finally, applying the implicit function theorem of Filippov type (see [, Theorem 7.2]), we have that there exists such that and , .
Theorem 4.1 Let be a lower semicontinuous (l.s.c. for short) functional with respect to the weak topology. Then, under conditions (i)-(iv), there exists a mild solution of problem (4.1) such that
where is the set of all mild trajectories of the system (4.1) with the initial value .
Proof Under the hypotheses (i)-(iv), the set ; see the proof above. Moreover, according to [, Theorem 4.2] (see also Remark 3.2), the solution set is weakly compact in . Let be the minimizer of j on , and let be the corresponding control, then the pair is the required optimal solution. □
4.2 Age-structure population model
We consider the nonlinear hyperbolic integro-differential problem (1.2) which was already introduced in Section 1. This model arises in population biology and describes the time evolution of the age-structure of a population. Here we consider the case of a nonlinear equation with linear boundary conditions. The independent variables t and a denote respectively time and age, and represents the density of individuals of age a at time t. The death rate f is a nonnegative term depending on the time, the age and the total size of the population . The boundary condition accounts for the birth in the population. The weight function b measures the fertility at age a.
We consider problem (1.2) under the following hypothesis:
for all , is measurable;
for a.a. and , is continuous;
there exists such that, for every , and , ;
Problem (1.2) can be written as the following Cauchy problem in the Banach space :
where is defined as , , is the single-valued map and , with
is the linear operator . A is the generator of the translation semigroup, which satisfies the identity
This semigroup is intensely studied in [29, 30], thus we do not give any details here. We just recall that the translation semigroup is not compact.
As showed in Section 4.1, the initial condition satisfies (L1), (L2) and (M).
According to (iii), for every , the function belongs to , hence F is nonempty-valued. Moreover, the Pettis measurability theorem (see Theorem 2.4), the separability of and conditions (i) and (ii) imply that F is globally measurable (see [, Corollary 1.3.1]), and hence, according to Remark 3.1, since F is single-valued, condition (F1) is also satisfied.
We now prove that is weakly sequentially continuous for a.a. t, so we take in . Then , thus (ii) implies that for a.a. . Moreover, it is possible to show that in obtaining condition (F2).
Finally, according to (iii), we have, for a.a. and every ,
and so the growth condition (F3′′′) is satisfied. Recalling Remark 3.2, this condition is sufficient for the existence of at least a solution for the Cauchy problem.
Hence we find a mild solution . We stress that by (4.4) the solution is a mild solution of (1.2) satisfying the required boundary conditions. Indeed,
With our techniques, we can attach also problems of the following general form:
where the Laplacian operator is understood only with respect to the space variable x and is a bounded closed domain with Lipschitz boundary. This model represents the age-structure of a population with spatial diffusion which takes into account the consequences of the environment changes on populations dynamics (see [31, 32]). The same arguments as above show that Theorem 3.1 applies to problem (4.5), taking again into account Remark 3.2. In fact, in this case, given the Banach space , we define the operator on
This operator generates a strongly continuous semigroup (see ), and all the other conditions can be shown as in the previous example.
4.3 Multipoint boundary value problem
Let and be a bounded set with a sufficiently regular boundary. Consider the multipoint boundary value problem
under the following hypotheses:
is measurable with and for all ;
, , satisfy the following conditions:
for every and , is measurable;
for a.a. and all , is l.s.c. and is u.s.c.;
there exist and increasing such that, for a.a. and every , and , and
for every , and , ;
is measurable with for a.a. and ;
, , satisfy and .
We search for solutions , and we can transform problem (4.6) into the nonlocal boundary value problem
in the reflexive Hilbert space , where , A and are as in Section 4.1. For , the function defined by is well defined and measurable according to (i). Moreover, for every . The multimap is defined by if and only if there is a measurable function satisfying and for a.a. . This definition is well posed, i.e., F is nonempty-valued according to (ii) and (iii), and it is easy to see that F is also convex-valued.
Given , from (ii) we have that
for every . Hence F has weakly relatively compact values. Moreover, according to (4.9) and (ii)(c), we have that
and therefore (F3′) follows from (4.7).
Now we investigate (F2), and hence we fix a value t in for which (ii)(b) is satisfied and consider the sequences such that , in and for all . Notice that the weak convergence of implies both the existence of such that and for all . Moreover, for all n and some measurable satisfying for a.a. . According to (ii)(c), a.e. in Ω for all n. Since Ω is bounded in , the set and it is weakly relatively compact. Hence we can find a subsequence, again denoted as the sequence, satisfying . Consequently, applying Mazur’s convexity lemma, for each n there exist and positive numbers , , such that and the sequence . Up to a subsequence, denoted again as the sequence, for a.a. . Since
passing to the limit as and according to (ii)(b), we obtain that a.e. in Ω. Let . It is possible to see that . However, , and hence for a.a. and the convergence is dominated since for a.a. . Therefore in , and the uniqueness of the weak limit implies that . We have showed that is weakly sequentially closed, i.e., that (F2) is satisfied, for a.a. . Then by (4.9) F has weakly sequentially compact values since is reflexive, and according to Corollary 2.1, it is weakly compact-valued. Furthermore, according to the Pettis measurability theorem (see Theorem 2.4), it is possible to show that the maps , , are measurable selection of for every ; hence condition (F1) is satisfied.
For we have that and then, as showed in Section 3, it satisfies both (L1) and (L2). Whereas is such that and it satisfies condition (M) (see Introduction); estimate (3.1), in particular, depends on (v). All the assumptions of Theorem 3.1 are then satisfied, and hence problem (4.8) is solvable. It implies that the multipoint boundary value problem (4.6) has a solution .
Remark 4.2 Similarly as before, it is possible to show that also the nonlocal boundary value problem given by the differential inclusion in (4.6) associated with the boundary condition
is solvable, provided that is Lipschitzian of some constant .
Remark 4.3 In (4.1) and (4.6) it is possible to substitute the Laplacian operator with a more general operator in divergence form of the following type:
with , and
, , ;
there exists such that, for a.a. and every , .