On the Null Controllability of the Lotka-Mckendrick System

. In this work, we study null-controllability of the Lotka-McKendrick system of population dynamics. The control is acting on the individuals in a given age range. The main novelty we bring in this work is that the age interval in which the control is active does not necessarily contain a neighbourhood of 0 . The main result asserts the whole population can be steered into zero in large time. The proof is based on ﬁnal-state observability estimates of the adjoint system with the use of characteristics.


Introduction
In this article, we study the null-controllability of an infinite dimensional linear system describing the dynamics of an age-structured single species population. To be specific, we study null-controllability of the classical Lotka-McKendrick system. Let, p(t, a) be the distribution of individuals at age a 0 and t 0. Let a † denotes the highest age attained by the individuals in the population and T be a positive constant. Let β(a) 0 be the natural fertility rate and µ(a) 0 denotes the natural death-rate of the individuals at age a 0. The system in consideration, already studied in [4] and [5], is described by the system where u is the control function, m = χ (a 1 ,a 2 ) is the characteristic function of the interval (a 1 , a 2 ) (0 a 1 < a 2 a † ) and p 0 is the initial population density. We assume that the fertility rate β and the mortality rate µ satisfy the following conditions: (H1) β ∈ L ∞ (0, a † ), β 0 for almost every a ∈ (0, a † ). (H2) µ ∈ L 1 [0, a * ] for every a * ∈ (0, a † ), µ 0 for almost every a ∈ (0, a † ).
These conditions have already been used in [4,5]. For the biological significance of the hypotheses, we refer to [11]. Our main result regarding the null controllability of the system (1.1) is the following Theorem 1.1. Assume that β and µ satisfy the the conditions (H1) − (H3). Furthermore, suppose that the fertility rate β is such that for some a b ∈ (0, a † ) and that a b > a 1 . Let us recall that m is a characteristic function of the interval (a 1 , a 2 ) with 0 a 1 < a 2 a † . Then for every T > a 1 + max{a 1 , a † − a 2 } and for every p 0 ∈ L 2 (0, a † ) there exists a control u ∈ L 2 ((0, T ) × (0, a † )) such that the solution p of the system (1.1) satisfies p(T, a) = 0 for all a ∈ (0, a † Let us now mention some related works from the literature. The null controllability results of the system (1.1) were first obtained by Barbu, Iannelli and Martcheva [4]. They considered the case when the control is supported in the interval (0, a 2 ), i.e when a 1 = 0. The main result of [4] asserts that the system can be steered to any steady state, in large time except for a small interval of ages near zero. Recently, Hegoburu, Magal and Tucsnak [5] considered the system (1.1) with distributed control supported in (0, a 2 ). They proved that the restriction of [4] is unnecessary, i.e., the whole population can be steered into a steady state, provided the individuals do not reproduce at the age near zero. Moreover, they also showed that if initial and final states are positive then the control can be chosen such that the positivity of the state trajectory is preserved.
The main novelty we bring in this work, in contrast to the results of [4,5], is that we do not need to apply control for arbitrary low ages. More precisely, in our case the control is active for ages a ∈ (a 1 , a 2 ), with arbitrary a 1 ∈ [0, a † ) and a 2 ∈ (a 1 , a † ], provided that supp β ∩ [0, a 1 ] = ∅. In other words, we control before the individuals start to reproduce. Moreover, we show our controllability result applies to individuals of all ages, without needing to exclude ages in a neighbourhood of zero. The methodology we employ in proving this result is quite classical. We exploit the fact that null controllability of a linear system is equivalent to the final-state observability inequality of the adjoint system. Before ending this introduction let us mention some controllability results regarding population dynamics model (Lotka-McKendrik type) with spatial diffusion. Ainseba and Aniţa [2] considered the case when the control acts in a spatial subdomain ω and for all ages and initial data close to the target trajectory. In [3], they proved a similar result when the control acts in a spatial subdomain and only for small ages. In [1], Ainseba proved null controllability except for a small interval of ages near zero, with control acting everywhere in the ages but localized in a spatial subdomain. Hegoburu and Tucsnak [6] proved that this restriction is not necessary, i.e, the whole population can be driven to zero. In their case, the control is localized in space variable but active for all ages. Recently, Maity, Tucsnak and Zuazua showed [8] that the same result can be achieved by means of control localized in space variable as well as with respect to age, not necessarily containing a neighbourhood of zero.
The outline of the paper is as follows. In Section 2 we first recall some basic facts about Lotka-McKendrick semigroup. Next, we define the adjoint of the associated semigroup. In Section 3, we prove the final-state observability for the adjoint system. As a consequence we get the proof of the main result.

Lotka-McKendrick Semigroup and its Adjoint.
In this section, with no claim of originality, we recall some existing results on the Lotka-McKendrick Semigroup and its adjoint. We define the operator A : We introduce the control operator B ∈ L(L 2 (0, a † )) defined by Bu(t, ·) = mu(t, ·).
With the above notation, the system (1.1) can be rewritten as Let us first recall some well posedness result of the above system. As already mentioned in the introduction, we are going to use the duality between null controllability of a linear system and final-time observability of the associated adjoint system. Thus it is important to determine the adjoint of the linear operator A. To this aim, we first consider the unbounded operator (A 0 , D(A 0 )) defined by We prove the following lemma Proof. We first verify A 0 is dissipative. For ϕ ∈ D(A 0 ), we have Next, for any f ∈ L 2 (0, a † ) we consider the following problem Then ϕ solves the above equation if and only if We can easily verify that ϕ ∈ D(A 0 ) and hence A 0 generates a C 0 -semigroup on L 2 (0, a † ).
In this section we prove Theorem 1.1. With the notation introduced in Section 2, Theorem 1.1 can be stated as : the pair (A, B) is null controllable in time T > a 1 + max{a 1 , a † − a 2 }. By classical duality argument, the null controllability of the pair (A, B) in time T is equivalent to the final-state observability of the pair (A * , B * ) in time T (see for instance [10, Theorem 11.2.1]). In the remaining part of this section, we show that the pair (A * , B * ) is final-state observable in time T > a 1 + max{a 1 , a † − a 2 }.
For t 0 and q 0 ∈ L 2 (0, a † ), we set where A is defined in (2.1). According to Proposition 2.3, q satisfies the following system: q(0, a) = q 0 (a), a ∈ (0, a † ). Before we start the proof of the above theorem, let us briefly describe the essential steps. We rewrite the system (3.2) as follows: where A 0 is defined in (2.3) and With Duhamel's formula, it is easy to see that with some constant C depending only on T. Thus to prove (3.3), we derive appropriate upper bounds for each of the terms in the right hand side of the above estimate.
(3.7) Then z solves the following system: z(0, a) = q 0 (a), a ∈ (0, a † ). (3.8) We prove the following proposition: Proof. We set a), (t, a) ∈ (0, T ) × (0, a † ). Then z satisfies the adjoint of a transport equation: with z 0 (a) = e − a 0 µ(τ ) dτ q 0 . Therefore z is given by (3.11) With the above explicit expression, for T > max {a 1 , a † − a 2 }, we have which is the standard observability estimate for the one dimensional transport equation. Let us briefly explain how one can obtain the estimate (3.12).
• First of all, since T > max {a 1 , a † − a 2 } , we can always find a * < a 2 such that z(T, a) = 0 for all a ∈ (a * , a 2 ). Thus we need to estimate a 0 z 2 (T, a) da.
• Next, for a ∈ (0, a * ) we note that the trajectory γ(s) = (T − s, a + s), s ∈ [0, T ] (or equivalently the backward characteristics starting from (T, a)) always enters the observation region (a 1 , a 2 ) × (0, T ) (see Fig. 1). Since z is constant along the trajectory γ, one can easily estimate a 0 z 2 (T, a) da by the right hand side of (3.12).
Therefore, using (3.12) we obtain We are going to estimate q(t, 0). We exploit the fact that for (t, a) ∈ (0, T ) × (0, a b ), q remains constant along the characteristics γ(s) = (t − s, s), s t T (or equivalently the backward characteristic starting from (t, 0)). If T > a 1 , the trajectory γ(s) always reaches the observation region (0, T ) × (a 1 , a 2 ) (see the green region in Fig. 2). Without loss of generality, let us assume that T > a b , η < a b and a 2 a b . Figure 2. An illustration of estimate of q(t, 0) with a 2 = a b .
Case 1: Let us fix t ∈ [a b , T ]. We define w(s) = q(s, t − s), s ∈ (t − a b , t). (3.17) Then ∂w ∂s = 0 for all s ∈ (t − a b , t). In particular, w(s) = constant for all s ∈ (t − a b , t). Since a 1 < a b we have Therefore, Integrating with respect to t over [a b , T ] we obtain Case 2: Let us fix t ∈ (η, a b ). We define w 1 (s) = q(s, t − s), s ∈ (0, t).
Then ∂w 1 ∂s = 0 for all s ∈ (0, t). In particular, w(s) = constant for all s ∈ (0, t). Since t > a 1 we obtain Therefore, Integrating with respect to t over [η, a b ] we get  Note that, q(t, 0) = q(t, 0). Thus from the above estimate we clearly get (3.14). Now, we are in a position to prove Theorem 3.1 and consequently our main result in Theorem 1.1.