J Optim Theory Appl (2013) 157:918–933 DOI 10.1007/s10957-012-0194-3
Exact Null Controllability of a Stage and Age-Structured Population Dynamics System Yuan He · Bedr’Eddine Ainseba
Received: 30 March 2012 / Accepted: 22 September 2012 / Published online: 3 October 2012 © Springer Science+Business Media New York 2012
Abstract This paper is concerned with the exact null controllability of an agedependent life cycle dynamics with nonlocal transition processes arising as boundary conditions. We investigate the controllability for the pest by acting on eggs in a small age interval. The main method is based on the derivation of estimations for the adjoint variables related to an optimal control problem. A fixed point theorem is then used to draw conclusions. Keywords Population dynamics · Integral partial differential equation · Exact null controllability · Characteristics method · Fixed point theorem
1 Introduction European grapevine moth (EGVM) has been the most serious grape pest insect causing important economic damages [1]. The life cycle of the EGVM could be divided into four development stages that are egg, larva, pupa and moth. The first three stages correspond to the insect growth and the last adult stage is devoted to the reproduction. This life cycle is repeated two to five times per year according to the latitude of the vineyard. As a function of temperature and food availability, it lasts about two
Communicated by Viorel Barbu. Y. He () · B.E. Ainseba Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 3 ter Place de la Victoire, 33076 Bordeaux Cedex, France e-mail:
[email protected] B.E. Ainseba e-mail:
[email protected] Y. He School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
J Optim Theory Appl (2013) 157:918–933
919
months during spring and less in summer. In winter the generation diapauses as a pupa lasts about six months. We are interested, in this paper, in the spring or summer generations where the pupa stage lasts one week and we suppose that the pupa stage is included in the larva stage to form an unique stage, the larva stage. This kind of moth reduces not only the amount of berries especially when berries are young in spring, but also their quality by favoring indirect damages as related to different pathogens developing on berries like the gray mold and in several warm vineyards to the black rots on berries [1]. These problems are suspected to increase, and could become more prevalent due to the climatic changes in the future. Thus many biological interventions have been developed to control this pest. Currently, the control procedures for this pest rely mainly on chemical insecticides and slightly on mating disruption (no more than 2 % of the French vine areas in 2007). But pesticides like growth regulators are used to reduce the population size, so that serious pollution damages environment. Researchers are developing some tools to control these insect populations and also to reduce the application of chemical plant health products. One problem accompanied by these control techniques is that their efficiency depends upon the timing of the treatment and its synchrony with few specific steps of the pest life cycle e.g. adult flight, oviposition periods. Then their goal is to predict the periods of appearance of the insect in the vineyard, and the mathematical models with age structure maybe very helpful for this objective. Therefore, considering the economical loss caused by the pest insect, it is meaningful to study the control problem of this Lobesia botrana model (LBM). It is well known that the optimal and exact control problems are widely investigated for agestructured population dynamics by many researchers. In the literature, most of the work is focused on optimal control problems, both almost perfect theory on single species [2, 3] and some results for interacting multi-species. One can see [4–6] and references therein. The null controllability of the linear heat equation on a subset of the domain was established by G. Lebeau and L. Robbiano [7]. Then B. Aïnseba et al. have investigated the local exact controllability for age-dependent linear and nonlinear single-species population model with diffusion, where the birth process is nonlocal. The main proof is based on Carleman’s inequality for the adjoint equation [3, 8, 9]. Viorel Barbu et al. also considered the exact controllability of the linear Lotka–McKendrick model without spatial structure by establishing an observability inequality for the backward adjoint system [10]. However, to our knowledge, there are no results dealing with the exact control problem for an age- and stage-dependent system. We cannot extend the method developed in [10] to the system case to get the key observability inequality. In spite of that, considering the fact that the system is an age- and stage-dependent life cycle dynamics, we are inspired to apply the fixed point theorem in [8, 11] to study the exact null controllability in finite time of the Lobesia botrana model (LBM) with four development stages, by reducing the egg population. The main purpose in our paper is to study the exact null controllability problem by getting the existence of a control with respect to the egg individuals such that the egg population except the small enough age groups can be controlled to be zero in a finite time. This paper is organized as follows: We present the system in Sect. 2. The assumptions and the main result are stated in Sect. 3. Then we give some derivations
920
J Optim Theory Appl (2013) 157:918–933
and the main proof of local exact null controllability in Sect. 4. Concluding remarks are stated in Sect. 5. 2 Presentation of the Model Our system is stated as follows: ⎧ ∂ue (t,a) ∂ue (t,a) + ∂a = −(μe (a) + β e (a))ue (t, a) + χ(a)w(t, a), ⎪ ⎪ ∂t ⎪ ⎪ l ⎨ ∂u (t,a) + ∂ul (t,a) = −(μl (a) + β l (a))ul (t, a), ∂t ∂a ⎪ ∂uf (t,a) + ∂uf (t,a) = −μf (a)uf (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎩ ∂um (t,a) ∂um (t,a) + = −μm (a)um (t, a), ∂t ∂a
(1)
where (t, a) ∈ ]0, T [×]0, A[, A = max{Le , Ll , Lf , Lm }. Here Lk means life expectancy of an individual for k = e, l, f, m, and uk (a, t) represents the age-specific density of the egg, larva, female moth and male moth, respectively. For every k, if A > Lk , we denote uk = 0, β k = 0, μk = 0. The term χ(a)w(t, a) is a control process for egg: χ(a) is the characteristic function of ]0, a ∗ [ (0 < a ∗ < Le ≤ A), which means that our intervention can be restricted to the younger age groups. The boundary conditions are defined by ⎧ Lf ⎪ ue (t, 0) = 0 β f (s)uf (t, s) ds, ⎪ ⎪ ⎪ ⎪ ⎨ ul (t, 0) = Le β e (s)ue (t, s) ds, 0 (2) ⎪ uf (t, 0) = Ll σβ l (s)ul (t, s) ds, ⎪ ⎪ 0 ⎪ ⎪ Ll ⎩ um (t, 0) = 0 (1 − σ )β l (s)ul (t, s) ds, where σ denotes the sex ratio, t > 0. The system is complete with the initial conditions as follows: uk (0, a) = uk0 (a), (3) for k = e, l, f, m. In addition, we state the following conception for this system. The functions μk are the k-stage age-specific per capita mortality functions. The functions β k denote the k-stage age-specific transition functions. In particular, β e models the physiological change between the egg and larva stage, which is called the hatching function. The function β l is the flying function describing the transition between the larva and the moth stage. The function β f models the transition between the moth and the egg stage whose name is the birth function. Note that for each (t, a) the directional derivatives of uk exist, and we can see uk (t + h, a + h) − uk (t, a) , h→0 h
Duk (t, a) = lim
with k = e, l, f, m. It is obvious that for uk smooth enough Duk =
∂uk ∂uk + . ∂t ∂a
(4)
J Optim Theory Appl (2013) 157:918–933
921
3 Setting of the Problem and the Main Result Let L2 = L2 (]0, A[) be the Banach space of equivalence classes of Lebesgue integrable functions, from ]0, A[ in R with the norm: ϕL2 (]0,A[) =
ϕ(a) 2 da
A
1 2
,
0
where A = max{Le , Ll , Lf , Lm }. Let T > 0. For all t ∈ [0, T ], we define the space LT =: C([0, T ], L2 (]0, A[)) as the Banach space of continuous functions from [0, T ], with values in L2 (]0, A[), which is equipped with the norm ϕLT = sup ϕ(t, ·) L2 . 0≤t≤T
Definition 3.1 For all T > 0 and all (a, t) ∈ ]0, Lk [×]0, T [, (ue , ul , uf , um ) is called a solution of (1)–(3) iff it belongs to (LT )4 and it satisfies system (1)–(3), where k = e, l, f, m. Integrating along the characteristic lines (see [12]), we obtain the solution of (1)– (3) for k = l, f, m and T > 0 Π k (a) uk (a − t) Π ku(a−t) , a ≥ t, k u (5) u (t, a) = 0 uk (t − a, 0)Πuk (a), a < t, and ⎧ t e (a) ue (a) ue0 (a − t) ΠΠeu(a−t) + 0 ΠΠ χ(s + a − t)w(s, s + a − t) ds, ⎪ ⎪ u ue (a−t+s) ⎪ ⎪ ⎨ a ≥ t, ue (t, a) = t Πue (a) ⎪ χ(s + a − t)w(s, s + a − t) ds, ⎪ue (t − a, 0)Πue (a) + t−a Πue (a−t+s) ⎪ ⎪ ⎩ a < t, (6) where
Πue,l (a) = e− Πuf,m (a) = e
−
a 0
a 0
(μe,l (τ )+β e,l (τ ))dτ μf,m (τ )dτ
,
(7)
.
Our motivation is to steer the population of egg to an age distribution, using an age- and time-dependent control of eggs. Especially we are able to find a control w corresponding to a removal (eradication) of egg on ]0, a0 [ such that ue (a, T ) = 0 for a fixed T and a ∈ ] , A[. Throughout this paper we impose the following assumptions: (H1) The hatching function β e , the flying function β l and the birth function β f are bounded, non-negative functions. There exist a0 , a1 ∈ ]0, Le [ such that β e = 0 a.e. a ∈ ]0, a0 [ ∪ ]a1 , Le [.
922
J Optim Theory Appl (2013) 157:918–933
(H2) The mortality functions μe (a), μl (a), μf (a) and μm (a) are non-negative, locally bounded and satisfy the following conditions:
Lk
μk (a) ds = ∞
0
with k = e, f, l, m. f (H3) The initial distribution u0 = (ue0 , ul0 , u0 , um 0 ) is non-negative, a.e. for a ∈ ]0, A[. These assumptions are biologically meaningful [3, 13–15], so that the existence and uniqueness of a solution of the system (1) is guaranteed. Here we omit the proof. One can refer to [13] and [16]. Roughly speaking, the main result, Theorem 3.1 below, amounts to saying that for T > A − a ∗ with A > a ∗ > 0, the population ue can be controlled to zero in a finite time T . f
Theorem 3.1 We denote u0 ∞ = ue0 ∞ + ul0 ∞ + u0 ∞ + um 0 ∞ , and u0 L∞ (]a ∗ ,A−T [) > 0. Let T > A − a ∗ be arbitrary but fixed, and small enough with 0 < ≤ a0 . Then there exists w ∈ L2 (]0, T [×]0, A[), such that the solution ue of (1)–(3) satisfies ue (T , a) = 0
a.e. a ∈ ] , A[.
(8)
If T < A − a ∗ , then there is no control w such that ue satisfies (1)–(3). 4 Proof of the Main Result First, we choose a number T0 ∈ ]0, min{a0 , a ∗ , A − a ∗ , T − A + a ∗ , A − a1 }[. Define
K = L∞ ]0, A − a ∗ + T0 [ . Let bl ∈ K arbitrary but fixed and for ∀ε > 0, we consider the following optimal control problem: 2 2 e 1 1 w(t, a) dt da + u (t, a) dl , Jε = min (9) 2 Ge ε Γ0 where Ge = ]0, T0 [ × ]0, a ∗ [ ∪ ]0, A − a ∗ + T0 [ × ]0, T0 [, Gl = ]0, A − a ∗ + T0 [ × ]0, Ll [, Gf = ]0, A − a ∗ + T0 [ × ]0, Lf [, Gm = ]0, A − a ∗ + T0 [ × ]0, Lm [, and Γ0 = ]T0 , A − a ∗ + T0 [ × {T0 } ∪ {T0 }× ]T0 , a ∗ [ (see Fig. 1).
J Optim Theory Appl (2013) 157:918–933
923
Fig. 1 An example of domains G and 0 for T > A − a ∗
In addition, w ∈ L2 (Ge ) and ue is the solution subject to the following system: ⎧ ∂ue (t,a) ∂ue (t,a) + ∂a = −(μe (a) + β e (a))ue (t, a) + χ(a)w(t, a), ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂ul (t,a) + ∂ul (t,a) = −(μl (a) + β l (a))ul (t, a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂t ∂uf (t,a) ∂t ∂um (t,a) ∂t
+ +
∂a ∂uf (t,a) ∂a ∂um (t,a) ∂a
(t, a) ∈ Ge , (t, a) ∈ Gl ,
= −μf (a)uf (t, a),
(t, a) ∈ Gf ,
= −μm (a)um (t, a),
(t, a) ∈ Gm , (10)
for (t, a) ∈ Gk and with the following boundary condition: ⎧ Lf ⎪ ue (t, 0) = 0 β f (s)uf (t, s) ds, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ul (t, 0) = bl (t), ⎪ ⎪ ⎨ Ll uf (t, 0) = σ 0 β l (s)ul (t, s) ds, ⎪ ⎪ Lm ⎪ m ⎪ u (t, 0) = (1 − σ ) 0 β l (s)ul (t, s) ds, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uk (0, a) = uk0 (a),
(11)
for t ∈ ]0, A − a ∗ + T0 [. Denote the value of the cost function by Jε (w). Since Jε (w) : L2 (Ge ) → R+ is convex, continuous and lim
wL2 (Ge ) →∞
Jε (w) = ∞,
it means that there is at least one minimum point for Jε (w). As a result, an optimal pair (wε , ueε ) exists for Jε in (9). We define the Lagrange function as follows: L(S) = Jε + I1 + I2 + I3 + I4 ,
924
J Optim Theory Appl (2013) 157:918–933
where ⎧ e e (t,a) ⎪ I1 = Ge q e (t, a) ∂u ∂t(t,a) + ∂u ∂a + (μe (a) + β e (a))ue (t, a) − χ(a)w(t, a) dt da, ⎪ ⎪ ⎪ l ⎪ l ⎪ ⎨I = q l (t, a) ∂u (t,a) + ∂u (t,a) + (μl (a) + β l (a))ul (t, a) dt da, 2
Gl
∂t ∂a ∂uf (t,a) ∂uf (t,a) f ⎪ + ∂a + μf (a)uf (t, a) dt da, I3 = Gf q (t, a) ⎪ ∂t ⎪ ⎪ m ⎪ m ⎪ ⎩ I4 = m q m (t, a) ∂u (t,a) + ∂u (t,a) + μm (a)um (t, a) dt da. G ∂t ∂a
(12) Here q e , q l , q f , q m are the adjoint variables with respect to ue , ul , uf , um , representing the fluctuations of the population of Lobesia botrana model (LBM). The vector S ∗ = (w, q) is an optimum of L if and only if the gradient of the Lagrange function is zero at the optimum, where q = (q e , q l , q f , q m ). The derivative of the Lagrange function with respect to the variables q at the optimum S ∗ gives the evolution problem (10). From integrations by parts for I{k=1,2,3,4} and differentiating the Lagrangian at point S ∗ with respect to the densities ue , ul , uf , um (see [17]), we get the dual (backward) system of (10) and (11) as follows: ⎧ ∂q e (t,a) ∂q e (t,a) + ∂a = (μe (a) + β e (a))q e (t, a), ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎨ ∂q l (t,a) + ∂q l (t,a) = (μl (a) + β l (a))q l (t, a) − σβ l (a)q f (t, 0), ∂t
∂a
f ∂q f (t,a) ⎪ ⎪ + ∂q ∂a(t,a) = μf (a)q f (t, a) − β f (a)q e (t, 0), ⎪ ∂t ⎪ ⎪ ⎩ ∂q m (t,a) ∂q m (t,a) + ∂a = μm (a)q m (t, a), ∂t
(13)
with (t, a) ∈ Gk , k = e, l, f, m, corresponding to the boundary conditions ⎧ l q (t, a) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ q f (t, a) = 0, q m (t, a) = 0, ⎪ ⎪ q e (t, a) = 0, ⎪ ⎪ e ⎩ e q (t, a) = − u (t,a) ε ,
t = A − a ∗ + T0 , or a = Ll , t = A − a ∗ + T0 , or a = Lf , t = A − a ∗ + T0 , or a = Lm , (t, a) ∈ Γ \ Γ0 , (t, a) ∈ Γ0 ,
(14)
with Γ = {A − a ∗ + T0 }×]0, T0 [ ∪ ]0, T0 [×{a ∗ } ∪ Γ0 . Using Ekeland variational principle one can obtain the above optimality system as in the work of Barbu and Iannelli for the scalar population dynamics case [18]. The existence and uniqueness of the dual system are easy to check by the method of characteristics. Here we omit the details, and denote by ukε , qεk the solution of (10) with (11), (13) with (14), respectively, for k = e, l, f, m. One can easily see that q m is identically zero. At the same time, it is known that qεe satisfies wε (t, a) = χ(a)qεe (t, a),
a.e.
(t, a) ∈ ]0, A − a ∗ + T0 [ × ]0, A[.
(15)
Multiplying the first equation of (13) by ueε , and then integrating on Ge :
Ge
ueε
e ∂qεe (t, a) ∂qεe (t, a) e e + − μ (a) + β (a) qε (t, a) dt da = 0. ∂t ∂a
(16)
J Optim Theory Appl (2013) 157:918–933
925
Using integrations by parts, then using (10), (11) and (14), we obtain a ∗ 0
A−a ∗ +T0
0
wε (t, a) 2 dt da + 1 ε
A−a ∗ +T0
=− 0
e u (t, a) 2 dl ε
Γ0
qεe (t, 0)ueε (t, 0) dt −
a∗
0
qεe (0, a)ueε (0, a) da.
(17)
f
Similarly multiplying the remaining equations of (13) by ulε , uε , and then integrating on Gk , k = l, f , respectively, we have
A−a ∗ +T0
0
=
qεf (t, 0)ufε (t, 0) dt
A−a ∗ +T0
0 A−a ∗ +T0
0
=
Ll
+
qεf (t, 0)ufε (t, 0) dt +
A−a ∗ +T0
0
qεl (t, 0)ulε (t, 0) dt
0
0 Lf
qεl (0, a)ulε (0, a) da,
(18)
qεf (0, a)ufε (0, a) da
qεe (t, 0)ueε (t, 0) dt.
(19)
Combining the above three equations (17), (18) and (19), we obtain a ∗ 0
A−a ∗ +T0
0
A−a ∗ +T0
=− 0
a∗
− =−
wε (t, a) 2 dt da + 1 ε
qεl (t, 0)ulε (t, 0) dt −
e u (t, a) 2 dl + ε
Γ0
Ll
0
0
Lf
qεf (0, a)ufε (0, a) da
qεl (0, a)ulε (0, a) da
qεe (0, a)ueε (0, a) da
0 A−a ∗ +T0
0
qεf (t, 0)ufε (t, 0) dt
−
a∗
0
qεe (0, a)ueε (0, a) da.
Let S e and S f be arbitrary characteristic lines of the first and third equation, respectively, in (13), S e = (γ + t, θ + t); t ∈ ]0, T0 [, (γ , θ ) ∈ {0}× ]0, a ∗ − T0 [ ∪ ]0, A − a ∗ [ ×{0} , S f = (t, π + t); t ∈ ]0, A − a ∗ + T0 [, π ∈ ]0, a ∗ − T0 [ . Define w˜ ε (t) = wε (γ + t, θ + t),
t ∈ ]0, T0 [,
u˜ eε (t) = ueε (γ + t, θ + t),
t ∈ ]0, T0 [,
q˜εe (t) = qεe (γ + t, θ + t),
t ∈ ]0, T0 [,
926
J Optim Theory Appl (2013) 157:918–933
μ˜ e (t) = μe (θ + t),
t ∈ ]0, T0 [,
β˜ e (t) = β e (θ + t),
t ∈ ]0, T0 [,
χ˜ (t) = χ(θ + t),
t ∈ ]0, T0 [,
u˜ fε (t) = ufε (t, π + t),
t ∈ ]0, A − a ∗ + T0 [,
q˜εf (t) = qεf (t, π + t),
t ∈ ]0, A − a ∗ + T0 [,
μ˜ f (t) = μf (π + t),
t ∈ ]0, A − a ∗ + T0 [,
β˜ f (t) = β f (π + t),
t ∈ ]0, A − a ∗ + T0 [.
Note that (u˜ eε , w˜ ε ) satisfies
d u˜ eε (t) = − μ˜ e (t) + β˜ e (t) u˜ eε (t) + χ(t) ˜ w˜ ε (t), dt ⎧ ⎨ Lf β f (a)uf (γ , a) da, θ = 0, e u˜ ε (0) = 0 ⎩ue (θ ), γ = 0; 0 q˜εe satisfies
⎧ e ⎨ d q˜ε (t) = (μ˜ e (t) + β˜ e (t))q˜ e (t), ε dt ⎩ q˜ e (T ) = ε 0
−u˜ eε (T0 ) ε
t ∈ ]0, T0 [,
t ∈ ]0, T0 [,
(20)
(21)
(22)
,
and w˜ ε (t) = χ(t) ˜ q˜εe (t), f
a.e.
t ∈ ]0, T0 [.
(23)
f
Similarly, u˜ ε and q˜ε satisfy the following equations, respectively: f
d u˜ ε (t) = −μ˜ f (t)u˜ fε (t), t ∈ ]0, T0 [, dt ⎧ l ⎨ L σβ l (a)ul (γ , a) da, θ = 0, f u˜ ε (0) = 0 ⎩uf (θ ), γ = 0;
(24)
(25)
0
⎧ f ⎨ d q˜ε (t) = μ˜ f (t)q˜ f (t) − β˜ f (t)q e (γ , 0), ε dt ⎩ f q˜ε (A − a ∗ + T0 ) = 0.
t ∈ ]0, T0 [,
(26)
Multiplying the first equation in (22) by u˜ eε (t), and the first equation in (26) by respectively, integrating on ]0, A − a ∗ + T0 [, then applying (20), (21), (24), and (25), we have f u˜ ε (t)
0
T0
w˜ ε (t) 2 dt + 1 u˜ e (T0 ) 2 ≤ −q˜ f (0)u˜ f (0) − q e (0, θ )ue (θ ). ε ε ε 0 ε ε
J Optim Theory Appl (2013) 157:918–933
927
By Young’s inequality, we obtain
w˜ ε (t) 2 dt + 1 u˜ e (T0 ) 2 ε ε 0 2 2 δ 2 2 δ 1 1 ≤ q˜εf (0) + u˜ fε (0) + qεe (0, θ ) + ue0 (θ ) , 2 2δ 2 2δ T0
(27)
with δ being any positive number. Using the constant variation formula to (22), we have A−a ∗ +T0 f
μ˜ (s) ds f q˜ε (0) q˜εf A − a ∗ + T0 − e 0 A−a ∗ +T0
t f =− e 0 μ˜ (s) ds β˜ f A − a ∗ + T0 − t qεe (γ , 0) dt, 0
which means e
A−a ∗ +T0 0
μ˜ f (s) ds f q˜ε (0) = −
Then
A−a ∗ +T0 t
e
0
μ˜ f (s) ds ˜ f
β
0
A − a ∗ + T0 − t qεe (γ , 0) dt.
f 2 q˜ (0) ≤ C q e (γ , 0) 2 . ε
(28)
ε
Next multiplying the equation in (22) by q˜εe , we obtain 2 1 d(q˜εe )2 e = μ˜ (t) + β˜ e (t) q˜εe (t) ≥ 0, 2 dt
a.e. t ∈ ]0, T0 [,
(29)
w˜ ε (τ ) 2 dτ.
(30)
which leads to the result e 2 q˜ (0) ≤ C ε
0
T0
e 2 q˜ (τ ) dτ = C ε
T0
0
Substituting (28) and (30) into (27),
T0
δC w˜ ε (t) 2 dt + 1 u˜ e (T0 ) 2 ≤ 1 u˜ f (0) 2 + ue (θ ) 2 , 1− ε ε 0 2 ε 2δ 0 where C is a constant satisfying δC 2 < 1, independent on ε. Then we get T0
w˜ ε (t) 2 dt ≤ 1 u˜ f (0) 2 + ue (θ ) 2 ≤ 1 u˜ f (0) 2 + C . ε ε 0 2δ 2δ 0 Note that here and after we denote several constants independent of all variables by the same C. Recalling that (21) holds, it is obvious that T0
2 w˜ ε (t) 2 dt ≤ 1 uf 2 ∞ w˜ ε L2 (]0,T [) = 0 L (]0,A[) + C ≤ M1 , 0 2δ 0
928
J Optim Theory Appl (2013) 157:918–933
as γ = 0. If π = 0, then we substitute uf given by the formula (5) in (21), and get w˜ ε 2L2 (]0,T
0
Ll
l 1 u (γ , a)β l (a) 2 da + C ε 2δ 0 l 2 γ β ∞ l L (]0,A[) u (γ − a, 0)Π l (a) 2 da ≤ u ε 2δ 0 A l Πul (a) 2 C + u0 (a − γ ) Π l (a − γ ) da + β l 2
≤ [)
γ
≤
L∞ (]0,A[)
u
l 2 β ∞
L (]0,A[)
2δ
l 2 b A − a ∗ + ul 2 ∞ 0 L (]0,A[) A + K
C
β l 2L∞ (]0,A[)
≤ M1 . According to the property of relatively weak compactness in L2 , there exists a subsequence (also denoted by w˜ ε ) such that
weakly in L2 ]0, T0 [ as ε → 0.
w˜ ε → w˜
f
In what follows we multiply the equation in (24) by u˜ ε ≥ 0, thus f
2 1 d(u˜ ε )2 = −μ˜ f (t) u˜ fε (t) ≤ 0, 2 dt
a.e. t ∈ ]0, T0 [,
which implies that f u˜ (T0 ) 2 ≤ u˜ f (t) 2 ≤ u˜ f (0) 2 . ε
Thus
ε
T0
ε
f 2 u˜ (t) dt ≤ C u˜ f (0) 2 . ε
0
ε
From (25), obviously we have
f 2 u˜ 2 ε
L (]0,T0
= [)
T0
0
f 2 u˜ (t) dt ≤ C uf 2 ∞
0 L (]0,A[) ,
ε
as γ = 0. For π = 0, applying (5) in (25), we get f 2 u˜ 2 ε L (]0,T
0
≤ Cσ [)
0
ul (γ , a)β l (a) 2 da
A
ε
2 ≤ C β l L∞ (]0,A[) + γ
γ 0
l u (γ − a, 0)Π l (a) 2 da u ε
2 A ul (a − γ ) Πul (a) da 0 Π l (a − γ ) u
J Optim Theory Appl (2013) 157:918–933
929
Cβ l 2L∞ (]0,A[) l 2 b A − a ∗ + ul 2 ∞ 0 L (]0,A[) A K 2δ ≤ M2 ,
≤
where C1 , C2 , C3 are constants independent of ue , ul , uf , um . We also have f 2 T0 d u˜ ε f μ˜ (t)u˜ f (t) 2 dt ≤ M u˜ f 2 2 = ε ε L (]0,T0 [) , dt 2 0 L (]0,T0 [) f
where M is a constant. Hence there exists a subsequence (also denoted by u˜ ε (t)) such that
u˜ fε (t) → u˜ f (t) weakly in W 1,2 ]0, T0 [ as ε → 0. In addition, we apply (5) and get the following result, which is similar to the f estimation for u˜ ε 2L2 (]0,T [) : 0
e 2 u˜ 2
ε L (]0,T0 [)
≤C γ
A e ue (a − γ ) Πu (a) 0 Π e (a − γ ) u
2 Π (a) χ(s + a − γ )wε (s, s + a − γ ) ds da 0 Πue (a − γ + s) γ γ e Πue (a) u (γ − a, 0)Πue (a) + χ(s + a − γ ) + 0 γ −a Πue (a − γ + s) 2 × wε (s, s + a − γ ) ds da
+
≤C
γ
ue
γ 0
0
A
2 (1 + ζ ) β f (s)ufε (γ − a, s) ds da
2 + (1 + ζ )A ue0 L∞ (]0,A[)
A γ 1 χ(s + a − γ )wε (s, s + a − γ ) 2 ds da + 1+ ζ γ 0
γ γ 2 1 χ(s + a − γ )wε (s, s + a − γ ) ds da + 1+ ζ 0 γ −a 2 2 ≤ C C1 β f L∞ (]0,A[) M2 + C2 ue0 L∞ (]0,A[) + C3 w˜ ε 2L2 (]0,T [) , 0
where C1 , C2 , C3 are constants independent of ue , ul , uf , um , and ζ is a positive number. e 2 T
0 d u˜ ε 2 e 2 1 e e ≤ ˜ 1+ dt μ˜ (t) + β (t) u˜ ε + (1 + δ) χ˜ w˜ ε (t) dt δ
0 2 1 M2 μ˜ e (t) + β˜ e (t) ∞ + (1 + δ)M1 , ≤ 1+ δ
930
J Optim Theory Appl (2013) 157:918–933
in which δ is a positive number. Therefore, there exists a subsequence (also denoted by u˜ eε (t)) such that
u˜ eε (t) → u˜ e (t) weakly in W 1,2 ]0, T0 [ as ε → 0, and (u˜ e , w) ˜ satisfies (20), with u˜ e (T0 ) = 0, and u˜ f is a solution of (24). One can see [19]. We extend w to w˜ on each characteristic line by 0. Then it is known that w˜ ∈ L2 (]0, T [×]0, A[). Let ue be the solution of (10) and (11), which is located on ]0, A − a ∗ + T0 [×]0, A[. Since ue = 0 on Γ0 and w = 0 outside G, it can be concluded that ue = 0, a.e. (t, a) ∈ S = (t, a); T0 < t < A − a ∗ + T0 , T0 < a < t + a ∗ − T0 ,
ue A − a ∗ + T0 , a = 0, a.e. a ∈ ]T0 , A[. Due to (5), we have e 2 2 2
u ∞ ≤ C ue (0, ·) L∞ (]0,A[) + ue (·, 0) L∞ (]0,A−a ∗ +T [) + w2L2 (Q) L (Q) 0 f 2 2
2 ≤ C ue0 L∞ (]0,A[) + u0 L∞ (]0,A[) + bl L∞ (]0,A−a ∗ +T [) 0 l 2 2 (31) + u0 L∞ (]0,A[) + wL2 (Q) , where Q = ]0, A − a ∗ + T0 [ ×]0, A[, C represents different constants independent of variables. In the following part, we prove the exact null controllability result by a fixed point technique. For any bl ∈ K, we denote by Le
Φ bl := β e (a)ue (t, a) da ⊂ L2 ]0, A − a ∗ + T0 [ 0
such that
ue
satisfies (31) and ue = 0,
a.e. (t, a) ∈ S
with ue (A − a ∗ + T0 , a) = 0, a.e. a ∈ ]T0 , Le [. We consider the following two cases: (1) t > T0 . It is known that β e = 0, a.e. a ∈ ]0, a0 [ ∪ ]a1 , Le [, which implies that β e (a) = 0 for a ∈ ]0, T0 [. Further from the condition (32), we have t+a ∗ −T0 β e (a)ue (t, a) da = 0. T0
Then
Le 0
β e (a)ue (t, a) da =
a1
t+a ∗ −T0
β e (a)ue (t, a) da.
(32)
J Optim Theory Appl (2013) 157:918–933
931
(a) For 0 < t + a ∗ − T0 < a, we have ue (t, a) = ue0 (a − t) ΠΠeu(a−t) . Obviously, u Le e e l 0 β (a)u (t, a) da does not depend on b . e
(2) 0 < t < T0 . Once again we use the fact that β e = 0, a.e. a ∈ ]0, a0 [ ∪ ]a1 , Le [. Thus we obtain Le β e (a)ue (t, a) da 0
=
T0
β e (a)ue (t, a) da +
0
=
Le −T0
β e (a)ue (t, a) da +
Le −T0
T0 Le −T0
Le
β e (a)ue (t, a) da
β e (a)ue (t, a) da,
T0 (a) which does not depend on bl , with ue (t, a) = ue0 (a − t) ΠΠlu(a−t) holding, for 0 < t < u T0 < a < A − T0 . Moreover, Le −T 0 e e e ≤ C β e ∞ u β (a)u (t, a) da . (33) L (]0,A[) 0 L∞ (]0,A[) e
T0
In summary, it is obvious that Φ(bl ) is a contraction and admits a fixed point. Next we choose a fixed point for the multivalued function Φ as follows. It is known that Le β e (a)ue (t, a) da 0
=
T0
β e (a)ue (t, a) da +
0
=
Le −T0
T0 Le −T0
β e (a)ue (t, a) da +
Le
Le −T0
β e (a)ue (t, a) da
β e (a)ue (t, a) da,
T0
a.e. t ∈ ]A − a ∗ , A − a ∗ + T0 [. Furthermore, the condition (32) implies Le −T0 β e (a)ue (t, a) da = 0. T0
Therefore, for any w we can choose 0, t ∈ ]A − a ∗ , A − a ∗ + T0 [, l b (t) = Le e e ∗ 0 β (a)u (t, a) da, t ∈ ]0, A − a [, as a fixed point of the multivalued function Φ. We find that there exists w ∈ L2 (]0, A − a ∗ + T0 [×]0, A[) with w = 0 in ]A − a ∗ , A − a ∗ + T0 [×]a ∗ , A[ such that ue subject to (1) satisfies
ue A − a ∗ + T0 , a = 0, a.e. a ∈ ]T0 , A[.
932
J Optim Theory Appl (2013) 157:918–933
Fig. 2 An example of domain U when T < A − a ∗
Then we denote = T0 small enough, which is right because of the definition of T0 . Letting T = A − a ∗ + T0 , it completes the argument of Theorem 3.1. Now we consider the second condition if T < A − a ∗ , which implies a ∗ < A. Assume that u0 L∞ (]a ∗ ,A−T [) > 0, then there exists w ∈ L2 (]0, T [×]0, A[) such that the solution ue (t, a) of (1) satisfies(8). Since χw = 0 when a ∈ ]a ∗ , A[, it is concluded that ue (t, a) independent of w ∈ U , where U =: (t, a); 0 < t < a − a ∗ , a ∗ < a < A (see Fig. 2). Further, uk (t, a) with k = e, l, f, m also satisfies the following system: ⎧ e ∂u (t,a) ∂ue (t,a) e e e ⎪ ⎪ ⎪ l∂t + l∂a = −(μ (a) + β (a))u (t, a), ⎪ ⎪ (t,a) ⎪ ⎪ ∂u ∂t(t,a) + ∂u ∂a = −(μl (a) + β l (a))ul (t, a), ⎨ f
f
∂u (t,a) + ∂u ∂a(t,a) = −μf (a)uf (t, a), ∂t ⎪ ⎪ m ⎪ ∂u (t,a) ∂um (t,a) m m ⎪ ⎪ ⎪ ∂t + ∂a = −μ (a)u (t, a), ⎪ ⎩ k k u (0, a) = u0 (a) for k = e, l, f, m,
where (t, a) ∈ U . Since u0 L∞ (]a ∗ ,A−T [) > 0, applying the backward uniqueness result, obviously, it leads to the conclusion that e u (T , ·) ∞ > 0, L (]0,A[) which is a contradiction to (8). That completes the proof of Theorem 3.1.
5 Concluding Remarks We study the exact null controllability of an age-dependent life cycle dynamics with nonlocal transition process arising as boundary conditions in the finite time, by manipulating on the eggs in a small age interval. The motivation of this work is to reduce the population of eggs, except the small enough age groups, to zero at a certain moment in the future, using an age- and time-dependent control of eggs. It is described by a control corresponding to a removal (eradication) of eggs. We construct an optimal control problem, and apply the fixed point technique for a multivalued function. Finally, the backward uniqueness result is used to draw conclusions.
J Optim Theory Appl (2013) 157:918–933
933
References 1. Thiéry, D., Esmenjaud, D., Kreiter, S., Martinez, M., Sforza, R., Van Helden, M., Yvon, M.: Les insectes de la vigne: les tordeuses nuisibles à la vigne. In: Kreiter, S. (ed.) Ravageurs de la vigne, pp. 214–246, Féret (2008). Deuxième édn. 2. Ainseba, B.E., Ani¸ta, S., Langlais, M.: Optimal control for a nonlinear age-structured population dynamics model. Electron. J. Differ. Equ. 28, 1–9 (2002) 3. Ani¸ta, S.: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic, Dordrecht (2000) 4. Luo, Z.X., He, Z.R., Li, W.T.: Optimal birth control for predator-prey system of three species with age-structure. Appl. Math. Comput. 155, 665–685 (2004) 5. Luo, Z.X., He, Z.R., Li, W.T.: Optimal birth control for an age-dependent n-dimensional food chain model. J. Math. Anal. Appl. 287, 557–576 (2003) 6. Picart, D., Ainseba, B.E., Milner, F.: Optimal control problem on insect pest populations. Appl. Math. Lett. 24, 1160–1164 (2011) 7. Lebeau, G., Robbiano, L.: Contrôle exact de l’equation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995) 8. Ainseba, B.E., Ani¸ta, S.: Internal exact controllability of the linear population dynamics with diffusion. Electron. J. Differ. Equ. 112, 1–11 (2004) 9. Ainseba, B.E., Iannelli, M.: Exact controllability of a nonlinear population-dynamics problem. Differ. Integral Equ. 16, 1369–1384 (2003) 10. Barbu, V., Iannelli, M., Martcheva, M.: On the controllability of the Lotka–McKendrick model of population dynamics. J. Math. Anal. Appl. 253, 142–165 (2001) 11. Ainseba, B.E., Ani¸ta, S.: Local exact controllability of the age-dependent population dynamics with diffusion. Abstr. Appl. Anal. 6, 357–368 (2001) 12. Calsina, A., Saldaña, J.: A model of physiologically structured population dynamics with a nonlinear individual growth rate. J. Math. Biol. 33, 335–364 (1995) 13. Ainseba, B.E., Picart, D.: Parameter identification in multistage population dynamics model. Nonlinear Anal. RWA 12, 3315–3328 (2011) 14. Ainseba, B.E., Picart, D., Thiéry, D.: An innovative multistage, physiologically structured, population model to understand the European grapevine moth dynamics. J. Math. Anal. Appl. 382, 34–46 (2011) 15. Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985) 16. Fekih, S., Ainseba, B.E., Bouguima, S.M.: Global existence for an age-structured model with vertical transmission. Submitted 17. Picart, D.: Modélisation et estimation des parametrès liés au succès reproducteur d’un ravageur de la vigne. (Lobesia botrana DEN.& SCHIFF.), Ph.D. Thesis 18. Barbu, V., Iannelli, M.: Optimal control of population dynamics. J. Optim. Theory Appl. 102, 1–14 (1999) 19. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)
