Applied Mathematics and Computation 247 (2014) 573–588

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Optimal control in a multistage physiologically structuredinsect population model

http://dx.doi.org/10.1016/j.amc.2014.09.0140096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: delphinepicart@hotmail.com (D. Picart), milner@asu.edu (F. Milner).

Delphine Picart a,⇑, Fabio Milner b

a French National Research Institute of Agronomy, UR1391 ISPA, F-33882 Villenave-dOrnon, Franceb Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85287-1804, United States

a r t i c l e i n f o a b s t r a c t

Keywords:Population dynamicsAge-structured populationPartial differential equationsGrape cultivarPest controlOptimal control theory

We present an age- and stage-structured population model to study some methods ofcontrol of one of the most important grapevine pests, the European grapevine moth. Weconsider control by insecticides that reduce either the proportion of surviving eggs, larvaeor both, as well as chemicals that cause mating disruption, thereby reducing the number ofeggs laid. We formulate optimal control problems with cost functionals related to real-lifecosts in the wine industry, and we prove that these problems admit a unique solution. Wealso provide some numerical examples from simulation.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Crop industries are frequently confronted with pest populations that cause great damage and losses. Many such pests areinsects as, for example, the boll weevil (Anthonomus grandis) in the cotton crop [13], the European grapevine moth (Lobesiabotrana) in the wine industry [8,18], or the apple leaf midge (Dasineura mali) in apple crops [6]. The methods used to controlthese pest populations are mainly application of insecticides (e.g. Bt, growth regulator), mating disruption to a lesser degree[19], and other modern methods currently in the testing process, such as biological control [20] or development of transgenicplant mutants resistant to the particular insect [13].

For many decades Lobesia botrana, the European grapevine moth (EGVM), has been a major concern in vineyards in Eur-ope, North Africa, Asia and —very recently— also California [5]. In this paper, we model the first two control methods of thispest population with the goal of finding strategies that optimize the reduction of pest population size and amount of chem-icals used.

Our model is built from the multistage physiologically structured population model presented in [2,16], for any insectpest whose biological cycle consists of four main stages–egg, larva, female and male. The males are not considered herebecause the additional equation describing their population dynamics can be neglected in the mathematical analysis givenin that paper without any loss of generality. Let ue; ul, and uf be, respectively, the age densities at time t of the egg, larva andadult female moth populations. The dynamics of these populations in their free environment are modeled as in [2,16] by thefollowing so-called size-structured model in which the structure variable a may represent size or another physiologicallyrelevant variable:

574 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

@ue

@t ðt; aÞ þ @@a geðEðtÞ; aÞueðt; aÞ½ � ¼ �beðEðtÞ; aÞueðt; aÞ �meðEðtÞ; aÞueðt; aÞ; ðt; aÞ 2 Xe;

@ul

@t ðt; aÞ þ @@a glðEðtÞ; aÞulðt; aÞ� �

¼ �blðEðtÞ; aÞulðt; aÞ �mlðPl; EðtÞ; aÞulðt; aÞ; ðt; aÞ 2 Xl;

@uf

@t ðt; aÞ þ @@a gf ðEðtÞ; aÞuf ðt; aÞ� �

¼ �mf ðEðtÞ; aÞuf ðt; aÞ; ðt; aÞ 2 Xf ;

8>><>>: ð1:1Þ

where Xk ¼ ½0; T� � ½0; Lk�, for k ¼ e; l; f , the boundary conditions, for t 2 ½0; T�, are given by

geðEðtÞ;0Þueðt;0Þ ¼R Lf

0 bf ðPf ; EðtÞ; aÞuf ðt; aÞda;

glðEðtÞ;0Þulðt;0Þ ¼R Le

0 beðEðtÞ; aÞueðt; aÞda;

gf ðEðtÞ;0Þuf ðt;0Þ ¼R Ll

0 blðEðtÞ; aÞulðt; aÞda

8>>><>>>:

ð1:2Þ

and the initial conditions are

ukð0; aÞ ¼ uk0ðaÞ; a 2 ½0; Lk�; k ¼ e; l; f : ð1:3Þ

The total population for the k-stage is defined by

PkðtÞ ¼Z Lk

0ukðt; aÞda;

where Lk is the maximum age for the stage, and k takes the value e for egg, l for larva and f for female. The variable E cor-responds to the time-dependent vector ðT;H;RÞ modeling the changing climatic (Temperature and Humidity) and environ-mental (food Resource) conditions. The motivations leading to this mathematical framework of this population dynamicsmodel and the explanation about the dependency of the variables ml and bf on the total population Pl; Pf are given inthe paper [2]. We just recall here to help the understanding of the next section that equations of system (1.2) describe allthe key steps of the insect’s biological cycle. In particular, the first equation models the birth dynamics, the second modelsthe hatching dynamics and the third equation is a proxy for the adult flight dynamics.

For k ¼ e; l; f , the initial functions uk0 are assumed non-negative and integrable, the growth functions gk —bounded in the

age variable (with additional conditions specified in the Appendix), and the k-stage mortality functions mk —non-negativeand locally bounded, satisfying the conditions

lima!Lk

Z t

0mkðEðtÞ;Xkðs; t; aÞÞds ¼ 1; a > XkðtÞ;

lima!Lk

Z t

Zkð0;t;aÞmkðEðtÞ;Xkðs; t; aÞÞds ¼ 1; a 6 XkðtÞ;

for t 2 ½0; T� and k ¼ e; f , and the conditions

lima!Ll

Z t

0mlðPl; EðtÞ;Xlðs; t; aÞÞds ¼ 1; a > XlðtÞ;

lima!Ll

Z t

Zlð0;t;aÞmlðPl; EðtÞ;Xlðs; t; aÞÞds ¼ 1; a 6 XlðtÞ;

in order to ensure that all individuals from the first two life stages transition into the next stage by the maximal age for theirstage and that all female adult moths die by the maximal age for the species.

The functions Xk and Zk (for k ¼ e; l; f ) parameterize, respectively with time t and age a as parameters, the characteristiccurves of (1.1) defined by

ðXkÞ0ðsÞ ¼ gkðEðsÞ;XkðsÞÞ;

XkðtÞ ¼ a;

(ðZkÞ

0ðsÞ ¼ gkðEðZkðsÞÞ; sÞ;

ZkðaÞ ¼ t

(

and they are assumed to be Lipschitz continuous in the first variable, with Lipschitz constant mK . The function be models thetransition between the egg and larval stages, whereas the function bl models the transition between the larval and the adultfemale moth stages. The function bf represents the age-specific fertility. These last three functions are non-negative andbounded, and they satisfy

0 6 bk6 bkðEðtÞ; aÞ; bf ðPf ; EðtÞ; aÞ 6 �bk; k ¼ e; l:

Moreover, the fertility function is assumed Lipschitz continuous in the first variable, Pf , with Lipschitz constant bK .In the next section we motivate and describe the control problems we shall analyze in this paper. In Section 3, the

existence of a solution for the resulting optimal control problem is proved whereas in Section 4 the uniqueness of theoptimal control pair is obtained through the use of Ekeland’s principle [3,7]. Section 5 is devoted to the numericalcharacterization of the optimal control through simulations. Finally, in Section 6, we summarize our results and drawsome conclusions.

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 575

2. New models for control

We begin by describing how the two types of control we consider in this paper can be built into the general model (1.1)–(1.3). Then, we describe the cost function we consider and we give a rationale for the form of its functional dependence onthe control(s) and densities.

Insecticides cause a direct decline in egg and larval populations by targeting either population or both. For example, eggpesticides (ovicides) are sprayed just before the onset of the egg laying dynamics in order to kill a maximal number of eggs,thus guiding our model to include a modification of (1.2) as follows. We let v1 be an indicator of the egg pesticide applicationat time t and age a, and we let a1 2 ½0;1� represent its efficiency. More specifically, a1 is the percentage of eggs killed by theovicide per unit time, assumed to be the same for eggs of all ages. As for the control, v1, we shall restrict it to take values in[0,1], with 0 indicating it should not be applied at the particular time and values in (0,1] should be interpreted as the dilutionto use for optimal cost, which we shall assume results in a linear decrease of its efficiency. In the linear case—when there isno density dependence in the fertility function— our simulations seem to indicate that the optimal control actually takesonly the values 0 and 1, thus working just as a switch that turns on and off the application of pesticide.

The pest population dynamics under the effect of egg pesticide control is modeled by (1.1)–(1.3), except that the bound-ary condition for the egg density is now given by

geðEðtÞ;0Þueðt;0Þ ¼Z Lf

01� a1v1ðt; aÞ½ �bf ðPf ; EðtÞ; aÞuf ðt; aÞda: ð2:1Þ

The number of eggs surviving the control is now computed from the number of eggs laid without the use of control bysubtracting the number of eggs killed by the egg-pesticide.

Larval pesticides are applied during the ‘‘black head’’ stage of egg development. We let v2 be the indicator of larval pes-ticide application at time t and age a, and we let a2 2 ½0;1� represent its efficiency. The pest population dynamics under theeffect of larval pesticide control is modeled by (1.1)–(1.3) with the equation describing the egg population dynamics mod-ified to

@ue

@tðt; aÞ þ @

@ageðEðtÞ; aÞueðt; aÞ½ � ¼ �beðEðtÞ; aÞueðt; aÞ � meðEðtÞ; aÞ þ a2v2ðt; aÞ½ �ueðt; aÞ; ðt; aÞ 2 Xe: ð2:2Þ

The mating disruption consists of disturbing mating by diffusing the female sex pheromone from dispensers therebyreducing the mating rate and, consequently, the number of eggs laid; see [19] for a more complete description of the mech-anism. The pheromone dispensers must be applied before the beginning of mating, typically during the egg laying dynamics.Therefore, this method of control can be modeled in exactly the same way as control by ovicide, and that is the approach weshall take in this paper.

In summary, the population dynamics under these pest population control methods is described by

@ue

@t ðt; aÞ þ @@a geðEðtÞ; aÞueðt; aÞ½ � ¼ � beðEðtÞ; aÞ þmeðEðtÞ; aÞ½ �ueðt; aÞ � a2v2ðt; aÞueðt; aÞ; ðt; aÞ 2 Xe;

@ul

@t ðt; aÞ þ @@a glðEðtÞ; aÞulðt; aÞ� �

¼ �blðEðtÞ; aÞulðt; aÞ �mlðPl; EðtÞ; aÞulðt; aÞ; ðt; aÞ 2 Xl;

@uf

@t ðt; aÞ þ @@a gf ðEðtÞ; aÞuf ðt; aÞ� �

¼ �mf ðEðtÞ; aÞuf ðt; aÞ; ðt; aÞ 2 Xf ;

geðEðtÞ;0Þueðt;0Þ ¼R Lf

0 1� a1v1ðt; aÞ½ �bf ðPf ; EðtÞ; aÞuf ðt; aÞda; t 2 ½0; T�;

glðEðtÞ;0Þulðt;0Þ ¼R Le

0 beðEðtÞ; aÞueðt; aÞda; t 2 ½0; T�;

gf ðEðtÞ;0Þuf ðt;0Þ ¼R Ll

0 blðEðtÞ; aÞulðt; aÞda; t 2 ½0; T�;ukð0; aÞ ¼ uk

0ðaÞ; a 2 ½0; Lk�; k ¼ e; l; f :

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð2:3Þ

We remark that for both pesticides we have assumed here that the control product targets a specific population that isfunction of the real time of the product application: ovicide only affects the newly laid eggs and not the aged eggs, larvicidekills at the egg stage and not at the larval. If we assume rather that the applied product kills any eggs or any larvae, whatevertheir developmental stage, then system (2.3) should be modified by adding the following 3 terms:

(i) �a1v1ðt; aÞueðt; aÞ in the equation related to the egg population dynamics to model the eggs killed by the ovicide,(ii) �a2v2ðt; aÞulðt; aÞ in the equation related to the larval population dynamics to model the larvae killed by the larvicide,

(iii) and the boundary equation of the larval stage is written by

glðEðtÞ;0Þulðt;0Þ ¼Z Le

01� a2v2ðt; aÞ½ �beðEðtÞ; aÞueðt; aÞda;

to model the number of new larvae (i.e. newly hatched eggs) killed as a consequence of the larvicide application.The vector function v ¼ ðv1;v2Þ represents the controls, which take values in the set [0,1] thus maintaining the positivity

of the density functions in (2.3). The mathematical analysis of this system is quite similar to this one developed in [2],

576 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

leading to the existence and uniqueness of solutions uk, for k ¼ e; l; f , of (2.3) and to the following a priori estimates that weshall state without proof: for k ¼ e; l; f and t 2 ½0; T�,

kukðt; �ÞkL1ð½0;Lk �Þ 6�bf TPð0Þe

�bf T ¼ Cð �bf ; TÞPð0Þ;

where Pð0Þ is the sum of all initial populations.Next we establish another technical result, namely the Lipschitz-dependence of the densities on the controls. More

precisely, we have the following result.

Theorem 1. Let v ¼ ðv1;v2Þ and ~v ¼ ð~v1; ~v2Þ represent two controls. Then, there exist positive constants Dk; k ¼ e; l; f , such that

kukvðt; �Þ � uk

~vðt; �ÞkL1ðXkÞ 6 Dkkv � ~vkL1 ;

for any t 2 ½0; T�, where ukv and uk

~v are, respectively, the solutions of (2.3) with the controls v and ~v .

Proof. See the Appendix. h

Let now l represent the financial cost of larval damage per larva per unit time, and let g1=Lf and g2=Le represent, respec-tively, the cost per unit time of carrying out an application of ovicide (and/or mating disruption pheromones) and of larvi-cide. We are then naturally led to study the following optimal control problem:

½P� : Find minðv1 ;v2Þ2K

J ðv1;v2Þ;

where Z Z Z� �

J ðv1; v2Þ ¼ l

Xlulðt; aÞda dt þ g1

LfXfðv1Þ2ðt; aÞda dt þ g2

LeXeðv2Þ2ðt; aÞda dt

and ul is the solution of (2.3) and, K is the convex compact set given by

K ¼ v 2 L1ðXf �XeÞ : v1 : Xf ! ½0;1�; v2 : Xe ! ½0;1�n o

: ð2:4Þ

Note that the integrals forming the cost functional represent, respectively, the total financial cost of grape losses due tothe infestation, and the total cost of the applications of ovicides (and/or mating disruption hormones), and of larvicides. Sinceone does not know the relation between the cost of an intervention and the amount of product used the quadratic form onthe control functions may be appropriate.

In general the control application is equidistributed among eggs/larvae/insects of all ages so that v1 and v2 are age-inde-pendent. In such case we can simplify the form of the functional to

J ðv1; v2Þ ¼ lZ

Xlulðt; aÞdadt þ g1

Z T

0ðv1Þ2ðtÞdt þ g2

Z T

0ðv2Þ2ðtÞdt

� �:

We can find in the literature several results about optimal control for problems governed by parabolic equations [1,12,21],by integro-difference equations [11] and, just like here, by hyperbolic equations [3,4,9,10]. In all these articles, however, thecost functional is expressed as a nonlinear (quadratic) function of the control(s) and the density, whereas this dependency islinear in our problem ½P�. Barbu and Iannelli [3] determined the optimal control to reduce the growth of a population. Theymodified the Gurtin–Mac Camy model to include control functions as factors of the vital rates. Fister and Lenhart [9,10] andBusoni and Matuccia [4] studied the optimal proportion of harvested individuals in two populations. The controls were definedin this case for the age-structured population equations as a functions that could be age- and time-dependent.

3. Existence of a solution

In this section we establish the existence of a solution for problem [P]. This is quite straightforward in contrast with theproof of uniqueness that we leave for the next section as it requires considerably more work.

Theorem 2. The optimal control problem [P] admits a solution ðv1;v2Þ.

Proof. Let d ¼ minðv1 ;v2Þ2KJ ðv1; v2Þ and let fvn1;vn

2gn2N be a minimizing sequence such that

d < J ðvn1;v

n2Þ 6 dþ 1

n: ð3:1Þ

Since the sequence fvn1;vn

2gn2N belongs to the space K, there exists a convergent subsequence fvnk1 ;v

nk2 gk2N in

L2ðXf Þ � L2ðXeÞ; let ðv�1;v�2Þ be its limit. Consider now the sequence of density functions f½ue�nkgk2N defined by (2.3), givenalong the characteristic curves by

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 577

½ue�nk ðt; aÞ ¼½ue

0�nk ðXeð0; t; aÞÞe�

R t

0½he �nk ðs;Xeðs;t;aÞÞds

; a > XeðtÞ;½ue �nk ðZeð0;t;aÞ;0ÞgeðEðZeð0;t;aÞÞ;0Þ e

�R t

Ze ð0;t;aÞ½he �nk ðs;Xeðs;t;aÞÞds

; a 6 XeðtÞ;

8><>:

where

½he�nk ðt; aÞ ¼ beðEðtÞ; aÞ þmeðEðtÞ; aÞ þ a2vnk2 ðt; aÞ þ @a½ge�nk ðEðtÞ; aÞ;

½ge�nk ðEðtÞ;0Þ½ue�nk ðt;0Þ ¼R Lf

0 ½1� a1vnk1 ðt; aÞ�b

f ð½Pf �nk; EðtÞ; aÞ½uf �nk ðt; aÞda:

(

These relations show explicitly the dependence of the densities on the sequence fvnk1 ;v

nk2 gk2N. Similarly, we can see the

dependence of the density functions f½ul�nkgk2N and f½uf �nkgk2N defined by (2.3) on fvnk1 ;v

nk2 gk2N. From the hypothesis made in

Section 2 we can easily prove (as described in [14]) that the sequence of functions ½Pl�nk ðtÞ ¼

R Ll

0 ½ul�nk ðt; aÞda; k 2 N,

converges strongly to ½Pl��ðtÞ ¼

R Ll

0 ½ul��ðt; aÞda, with ½ul�� solution of (2.3) with ðv�1;v�2Þ, since the sequence is bounded in

H!. Finally, we see that limn!1J ðvn1;vn

2Þ ¼ J ðv�1;v�2Þ and, from the first inequality of (3.1), it follows that ðv�1;v�2Þ is a solutionof ½P�. h

4. Uniqueness of the solution

The method we shall use to establish uniqueness is based on that of Barbu and Iannelli’s paper [3]. We carry out the sameprocedure they used and utilize their notation.

We shall need the following dual problem associated to (2.3),

� @@t p

eðt;aÞ�geðEðtÞ;aÞ @@apeðt;aÞþ beðEðtÞ;aÞþmeðEðtÞ;aÞ½ �peðt;aÞþa2v2ðt;aÞpeðt;aÞ�plðt;0ÞbeðEðtÞ;aÞ¼0; ðt;aÞ2Xe;

� @@t p

lðt;aÞ�glðEðtÞ;aÞ @@aplðt;aÞþplðt;aÞ blðEðtÞ;aÞþmlðPl;EðtÞ;aÞþ@xmlðPl;EðtÞ;aÞulðt;aÞ

h i�pf ðt;0ÞblðEðtÞ;aÞþl¼0; ðt;aÞ2Xl;

� @@t p

f ðt;aÞ�gf ðEðtÞ;aÞ @@apf ðt;aÞþmf ðEðtÞ;aÞpf ðt;aÞ¼ 1�a1v1ðt;aÞ½ �peðt;0Þ� bf ðPf ;EðtÞ;aÞþ@xb

f ðPf ;EðtÞ;aÞuf ðt;aÞh i

; ðt;aÞ2Xf ;

pkðT;aÞ¼0; a2 ½0;Lk�; k¼e; l;f ;

pkðt;LkÞ¼0; t2 ½0;T�; k¼e;l;f :

8>>>>>>>>><>>>>>>>>>:

ð4:1Þ

Using the method of characteristics, we can produce an explicit solution for this system that satisfies the Lipschitz con-ditions given in Theorem 3 below that will be useful later.

Theorem 3. The system (4.1) admits a unique solution and, for t 2 ½0; T�, it satisfies

jpevð:;0Þ � pe

~vð:;0Þj1 6 Gkv � ~vkL1 ; ð4:2Þ

kpev � pe

~vkL2ðXeÞ 6 Hkv � ~vkL1 ; ð4:3Þ

where G and H are positive constants, pev and pe

~v are solutions of (4.1) corresponding, respectively, to the controls v and ~v , andj � j1 denotes the L1 ½0; T�ð Þ-norm.

Proof. See the Appendix. h

The dual problem associated with ½P� is given by (4.1) and its optimality conditions are given in the following lemma.

Lemma 1. The optimality conditions for problem ½P� are

2g1v1ð�; aÞ þ a1peð�;0Þbf ðPf ; �; aÞuf ð�; aÞ ¼ 0; a 2 ½0; Lf �;2g2v2ð�; aÞ þ a2ueð�; aÞpeð�; aÞ ¼ 0; a 2 ½0; Le�;

identically in ½0; T� where v ¼ ðv1;v2Þ is the optimal control, ðuf ;ueÞ satisfy (2.3) for the optimal v, and pe is the solution of (4.1)satisfying the condition

� 2g1

bf Pfmin

6 peðt;0Þ 6 0;

where Pfmin > 0 is the minimum of the adult female population size for t 2 ½0; T�.

Proof. Follows directly from computing the derivatives of the cost functional with respect to the controls. h

578 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

We then prove the uniqueness of solutions to problem ½P�. Following Barbu and Iannelli [3] we define the function

UðvÞ ¼J 0ðvÞ; v 2 K;

þ1; otherwise;

�

so that, for �P 0, there exists v� ¼ ðv1� ;v2

�Þ 2 K satisfying

Uðv�Þ 6 infv2K

UðvÞ þ �; ð4:4Þ

Uðv�Þ 6 infv2K

UðvÞ þffiffiffi�p X2

i¼1

kv i � v i�kL1

!: ð4:5Þ

Computing the derivatives with respect to the vector v of the Lagrangian of the cost functional defined by the expressionin parentheses in (4.5) we get the following relations,

R

Xf 2g1v1�ðt; aÞ þ a1peðt;0Þbf ðPf ; EðtÞ; aÞuf ðt; aÞ

h ih1ðt; aÞdadt þ

ffiffiffi�p R

Xf h1ðt; aÞdadt P 0;RXe 2g2v2

�ðt; aÞ þ a2ueðt; aÞpeðt; aÞ� �

h2ðt; aÞdadt þffiffiffi�p R

Xe h2ðt; aÞdadt P 0;

8<:

for all h ¼ ðh1; h2Þ 2 Tv� ðKÞ, where the set Tv� ðKÞ represents the tangent cone to K at v�. The two expressions in the squarebrackets are the components of a vector in L1ðXf �XeÞ and then, as a consequence of Proposition 5.3 of Barbu and Iannelli[3], we have the existence of a function h� ¼ ðh1

� ; h2�Þ 2 L1ðXf �XeÞ; jhi

�j < 1 for i ¼ 1; 2, such that the functions

2g1v1�ðt; aÞ þ a1peðt;0Þbf ðPf ; EðtÞ; aÞuf ðt; aÞ þ

ffiffiffi�p

h1� ; ðt; aÞ 2 Xf ;

2g2v2�ðt; aÞ þ a2ueðt; aÞpeðt; aÞ þ

ffiffiffi�p

h2� ; ðt; aÞ 2 Xe;

(

are the components of a vector in Nv� ðKÞ, the normal cone of K at v�. We can deduce from the above that the solution v� isgiven by

v1�ðt; aÞ ¼ L �

a12g1

peðt;0Þbf ðPf ; EðtÞ; aÞuf ðt; aÞ �ffiffi�p

2g1h1�ðt; aÞ

� ; ðt; aÞ 2 Xf ;

v2�ðt; aÞ ¼ L �

a22g2

ueðt; aÞpeðt; aÞds�ffiffi�p

2g2h2�ðt; aÞ

� ; ðt; aÞ 2 Xe;

8><>:

where the mapping L is given by

ðLf Þðt; aÞ ¼f ðt; aÞ; g 6 f ðt; aÞ 6 �g;

g; f ðt; aÞ 6 g;�g; f ðt; aÞP �g:

8><>:

We can now prove the following uniqueness result.

Theorem 4. Let us define the following positive conditions,

c1 ¼ �bf kuf kL1Gþ �bf jpeð�; 0Þj1Df þ bK jpeð�;0Þj1kuf kL1Df ;

c2 ¼ kpekL1De þ ðLTÞ1=2kuekL1H;

(

where Df and De are positive constants given in Theorem 1 whereas G and H are those given in Theorem 3. If, 0 6 c12g1; c2

2g2< 1, then

problem ½P� has a unique solution v ¼ ðv1;v2Þ 2 K.The above conditions ðc1; c2Þ guaranteeing the uniqueness are dependent on the density functions ue and uf that are solu-

tions of the system (2.3) and linked with the dual variables pe and, independent on the control functions ðv1; v2Þ. Since allterms of these conditions are positive then we can simplify them as follow

�bf kuf kL1G < 2g1; and ðLTÞ1=2kuekL1G < 2g2;

because H ¼ G (see Annex) and deduce this condition on the maximal value on the female density functions,

kuf kL1 6g1

g2

ðLTÞ1=2

gePf

max;

that requires to be lesser than the maximal number of female Pfmax times a positive constant.

Proof. Let E be the Banach space ðL1ðXf �XeÞÞ, and consider the mapping ðFvÞðtÞ : E ! E given by

ðF 1vÞðt; aÞ ¼ L � a12g1

pevðt;0Þb

f ðPfv ; EðtÞ; aÞu

fvðt; aÞ

� ;

ðF 2vÞðt; aÞ ¼ L � a22g2

uevðt; aÞpe

vðt; aÞ�

;

8><>: ð4:6Þ

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 579

where ðpe~v ;u

e~v ;u

f~vÞ and ðpe

v ;uev ;u

fv Þ are solutions of (4.1) and (2.3) with the controls ~v and v, respectively. First, we prove that

the mapping Fv is a contraction. Let v ¼ ðv1;v2Þ and ~v ¼ ð~v1; ~v2Þ be two controls; then, from (4.6) we deduce that

jF 1vðt; aÞ � F 1 ~vðt; aÞj 6 12g1

pe~vðt;0Þ � pe

vðt;0Þ jbf ðPf

v ; EðtÞ; aÞuf~vðt; aÞj

þ 12g1jpe

vðt;0Þj bf ðPf

v ; EðtÞ; aÞ � bf ðPf~v ; EðtÞ; aÞ

juf~vðt; aÞj þ

�bf

2g1jpe

vðt;0Þj uf~vðt; aÞ � uf

vðt; aÞ ;

jF 2vðt; aÞ � F 2 ~vðt; aÞj 6 12g2

ue~vðt; sÞ � ue

vðt; sÞ jpe

vðt; sÞj þ1

2g2ue

~vðt; sÞ pe~vðt; sÞ � pe

vðt; sÞ :

Using Theorems 1 and 3, the above inequalities become

jF 1vðt; aÞ � F 1 ~vðt; aÞj 6 12g1

�bf kuf~vkL1Gþ �bf jpe

vð:;0Þj1Df þ bK jpevð:;0Þj1ku

f~vkL1Df

h ijj~v � vjjL1 ;

jF 2vðt; aÞ � F 2 ~vðt; aÞj 6 12g2

kpevkL1De þ ðLTÞ1=2kue

~vkL1Hh i

k~v � vkL1 ;

where the estimate of the L2-norm of the dual variable pe is given in the Appendix. Let us denote by c1 and c2, respectively,the terms inside the square brackets of the first and second inequalities above, so that those inequalities are now rewritten as

jF 1vðt; aÞ � F 1 ~vðt; aÞj 6 c12g1k~v � vkL1 ;

jF 2vðt; aÞ � F 2 ~vðt; aÞj 6 c22g2k~v � vkL1 :

(ð4:7Þ

Since we have

0 6c1

2g1;

c2

2g2< 1; ð4:8Þ

the mapping F is a contraction admitting a unique fixed point, say v�. Next, we prove that the solution of ½P0� is unique, and itis v�. Indeed, for i ¼ 1; 2, we have

kv�i � v i�kL1 6 F iðv�i Þ � F 1ðv i

�Þ þ ffiffiffi

�p

2gijhi�ðt; aÞj;

which implies, using (4.7), that for i ¼ 1; 2,

kv�i � v i�kL1 6 1� ci

2gi

� ��1 ffiffiffi�p

2gi:

Finally, we conclude that lim�!0

v� ¼ v� and, by inequality (4.4), v� is the unique minimizer of ½P0� and, therefore, also theunique minimizer of ½P�. h

5. Numerical characterization of the controls

We present next the results from a numerical simulation and the corresponding optimal controls for the resulting prob-lem ½P�.

We combine a Quasi-Newton (QN) method to compute the optimal solutions of problem ½P� with a Finite Difference pro-cedure to approximate the egg, larval and adult density functions of system (2.3). One can prove the convergence of thismethod in the same way as in [15]. We consider the analogous discrete adjoint problem, that is (using the same notationgiven in [15])

ke;n0�1i0

¼ DtDa

ke;n0i0þ1

ve;n0i0

1þDtðbeþmeÞn0þ1i0þ1

þDta2vn0þ12;i0þ1

þ ke;n0i0

1�DtDav

e;n0i0

� 1þDtðbeþmeÞn0þ1

i0þDta2vn0þ1

2;i0

þ Dtk

l;n01 b

e;n0i0

1þDtðblþmlÞn0þ11

;

kl;n0�1i0

¼ DtDa

kl;n0i0þ1

v l;n0i0

1þDtðblþmlÞn0þ1i0þ1

þ kl;n0i0

1�DtDav

l;n0i0

� 1þDtðblþmlÞn0þ1

i0

þ Dtk

f ;n01

bl;n0i0

1þDtmf ;n0þ11

;

kf ;n0�1i0

¼ DtDa

kf ;n0i0þ1

v f ;n0i0

1þDtmf ;n0þ1i0þ1

þ kf ;n0i0

1�DtDav

f ;n0i0

� 1þDtm

f ;n0þ1i0

þ Dtk

e;n01 ð1�a1ðv1Þ

n0i0Þbf ;n0

i0

1þDtðbeþmeÞn0þ11 þDta2ðv2Þ

n0þ11

;

8>>>>>>>>>>>><>>>>>>>>>>>>:

for 1 6 i0 6 Na and 1 6 n0 6 Nt .The parameters of the model are chosen so that the growth of the population is chronological and constant in time as

observed under laboratory conditions [16,18] (120 eggs per female on average, no mortality and no competition for food)that is

Fig. 1.initializ

580 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

gkðEðtÞ; aÞ � 1; mkðEðtÞ; aÞ � 0; for k ¼ e; l; f ;

making the model age-structured; we use truncated normal distributions for the fertility and stage-transition functionsgiven by

beðt; aÞ ¼ d1e�a�7:50:35ð Þ2 ¼ blðt; aÞ; bf ðPf ; EðtÞ; aÞ ¼ d2e�

a�3:50:35ð Þ2 ;

with d1 ¼ 16:12; d2 ¼ 19:34, and Le ¼ Lf ¼ Ll ¼ 10. In order to be able to present in a single graph the population dynamics ofall classes we assume the fertility rate to be one tenth of its actual value, i.e. 12 eggs by female. The only growth variationsconsidered in these examples are inside cohorts as measured and showed in [16,18].

We computed numerically the optimal control for a population initialized by a cohort of one hundred females with trun-cated-normal age distribution, that is

uf ð0; aÞ ¼ d3e�a�10:25ð Þ

2

; d3 ¼ 225:67:

The temporal dynamics of these females is represented in Fig. 1 by the dotted line, that of their eggs by the solid line, andthat of the eggs that become larvae after few days by the dash-dotted line. The total number of larvae during the wholeexperiment is 1200 individuals.

The financial cost of larval damages per larva per unit of time is assumed to equal one unit, i.e. l ¼ 1, and the multiplierfor the application of ovicide and larvicide is also set equal to unity, i.e. g1 ¼ g2 ¼ 1. For simplicity we take the cost of anovicide application to be the same as the cost of a mating disruption use, though this is not true in reality. The coefficientsof pesticide efficiency, a1 and a2, are assumed identical and set to the maximal value 1. The issue here is not to discuss theactual cost or performance of the pest control methods but rather to show to numerically realize the solution of the optimalcontrol problem ½P�.

We focus on just the first generation of the simulated population dynamics and determine through the optimal controlproblem [P] presented in Section 2 the best way to use the whole array of chemical products described to minimize the eco-nomic damage caused by this insect population. The age- and time-discretization steps are chosen equal to 0.1 and the QNalgorithm is initialized with the step-functions represented by the dashed-dotted curves of Fig. 2.

The optimal solution computed for problem ½P� is given in Fig. 2. According to it, the optimal strategy to control a pop-ulation with dynamics as in Fig. 1 is to continuously spray ovicides during the egg laying dynamics (or to combine an eggpesticide application with a use of pheromone dispensers provided that the efficiency of the combination of these chemicalproducts is maximal). In addition to that a larvicide application seems to be required to totally eliminate the insects but,actually, when looking at the integral of this solution v2 (cf. Table 1) the amount of product is almost null.

Population dynamics of females (dotted line), eggs (solid line) and larvae (dashed-dotted line) without any control applied. The population ised with 100 females.

0 102 4 12 14 16 181 3 6 85 7 9 11 13 15 170

1

0.2

0.4

0.6

0.8

0.1

0.3

0.5

0.7

0.9

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

Time (Days)

v1

0 102 4 6 8 12 14 16 181 3 5 7 9 11 13 15 170

0.2

0.4

0.1

0.3

0.5

0.02

0.04

0.06

0.08

0.12

0.14

0.16

0.18

0.22

0.24

0.26

0.28

0.32

0.34

0.36

0.38

0.42

0.44

0.46

0.48

Time (Days)

v2

Fig. 2. Optimal control (v1;v2 in solid curves) for the population of Fig. 1, determined from problem ½P�. The horizontal axis represents the time. The dash-dotted curves are the QN initial functions.

Table 1Values of the three integrals that form the cost functional J at the optimum ðv1;v2Þ of the control problem ½P� presented in Section 2, and the minimum valueof the functional J .

Component lR T

0 Plðv1 ;v2ÞðtÞdt g1

Lf

R T0 ðv1ðtÞÞ2dt g2

Le

R T0 ðv2ðtÞÞ2dt J

Value 0.139 0.2534 0.00271 0.39511

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 581

Table 1 below shows the contributions to the three components of the cost function for the optimal solution ðv1;v2Þ ofproblem ½P�.

6. Concluding remarks

In this paper we described and analyzed a mathematical model that may realistically guide the ‘‘best’’ choice of controlstrategies for one of the most ravaging pests that damages the grapes and decreases production in European and Americanvineyards, specifically that caused by the European grapevine moth (Lobesia botrana).

Our model for control of the pest is based on the Lobesia botrana Model published in [2]. We considered a reduced versionof it with modifications to both one of the differential equations and to the boundary condition for the egg density in order toaccount, respectively, for the application of larval pesticides and that of ovicides and/or pheromones that disrupt and reducemating and thus reduce egg production too. These three techniques are, in fact, the most widely used in many vineyards inEurope, see [8,17,19] for example. We constructed a cost function that is amenable to matching with real-life financial costsby allowing linear dependence on the density of the pest and nonlinear dependence on the control. This is in sharp contrastwith the majority of models for optimal control where the dependence of the cost on the relevant density and on the controlis quadratic so that classical techniques based on a Lagrangian and a dual problem can be applied to show existence of anoptimal control. This is one of the novel features in our results, and one that is very desirable when trying to apply a math-ematical model to a real-life situation.

We show existence of an optimal control that describes when to apply the chosen chemical to minimize financial lossesand, under some additional technical conditions, we show that such control is unique. We also provided a numerical illus-tration of the optimal control pair that consists of continuous functions for a given population dynamics.

These results provide a solid ground on which vineyard managers can base a treatment protocol for this pest that willminimize financial losses. Even though our model includes the possible use of mating disruption as a control measure, dif-ficulties arise when considering its efficiency. On the other hand, while the use of pesticides has a clear direct financial costgiven by the purchase and application costs for the chemical, it also has another cost, more subtle and harder to model andevaluate, that we could refer to as ‘‘environmental cost’’ due to the poisoning of the soil. We shall address these and otherissues related to the modeling of the coefficients in the control terms and in the cost function in a forthcoming paper.

582 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

Acknowledgement

The work presented here was funded by Arizona State University as postdoc position in 2010 and derives from the doc-toral thesis of Delphine Picart realized from 2006 to 2009 in collaboration with the research unit SAVE (www.bordeaux-aqui-taine.inra.fr/santevegetale) of the French National Research Institute of Agronomy.

Appendix A

For the proofs presented in the following we make some additional hypotheses. The growth functions gk, for k ¼ e; l; f ,are non-negative and bounded away from zero at the initial age 0,

0 < gk6 gkðEðtÞ;0Þ 6 �gk; k ¼ e; l; f :

The birth function bf ðPf ; t; aÞ is differentiable in the first variable Pf and its derivative is Lipschitz continuous with con-stant bK such that

jbf ðPfv ; EðtÞ; aÞ � bf ðPf

~v ; EðtÞ; aÞj þ j@xbf ðPf

v ; EðtÞ; aÞ � @xbf ðPf

~v ; EðtÞ; aÞj 6 bKkufv � uf

~vkL1 ;

where Pfv ðtÞ ¼

R Lf

0 ufv ðt; aÞda and uf

v satisfies (2.3) with the control v.The larval mortality function mlðPl; EðtÞ; aÞ is differentiable in the first variable Pl and its derivative is Lipschitz continuous

with constant mK such that

jmlðPlv ; EðtÞ; aÞ �mlðPl

~v ; EðtÞ; aÞj þ j@xmlðPlv ; EðtÞ; aÞ � @xmlðPl

~v ; EðtÞ; aÞj 6 mKkulv � ul

~vkL1 ;

where Plv ðtÞ ¼

R Ll

0 ulv ðt; aÞda and ul

v satisfies (2.3) with the control v.

A.1. Continuity of the state variables with respect to the controls

Proof of Theorem 1. Let us introduce the following change of variables,

ukvðt; aÞ ¼ e�ktuk

vðt; aÞ; ðt; aÞ 2 Xk; k ¼ e; l; f ;

where ukv satisfies (2.3) and the new variables uk

v satisfy the following system,

@@t ue

vðt; aÞ þ @@a geðEðtÞ; aÞue

vðt; aÞ� �

þ kuevðt; aÞ þ beðEðtÞ; aÞ þmeðEðtÞ; aÞð Þue

vðt; aÞa2v2ðt; aÞuevðt; aÞ ¼ 0;

@@t ul

vðt; aÞ þ @@a glðEðtÞ; aÞul

vðt; aÞ� �

þ kulvðt; aÞ þ blðEðtÞ; aÞul

vðt; aÞ þmlðPl; EðtÞ; aÞulvðt; aÞ ¼ 0;

@@t uf

vðt; aÞ þ @@a gf uf

vðt; aÞh i

þ kufvðt; aÞ þmf ðEðtÞ; aÞuf

vðt; aÞ ¼ 0;

geðEðtÞ;0Þuevðt; 0Þ ¼

R Lf

0 ð1� a1v1ðt; aÞÞbf ðPf ; EðtÞ; aÞufvðt; aÞda;

glðEðtÞ;0Þulvðt;0Þ ¼

R Le

0 beðEðtÞ; aÞuevðt; aÞda;

gf ðEðtÞ;0Þufvðt;0Þ ¼

R Ll

0 blðEðtÞ; aÞulvðt; aÞda;

ukvð0; aÞ ¼ e�ktuk

0ðaÞ; k ¼ e; l; f :

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ðA:1Þ

Now, let v and ~v be two controls and let ukv and uk

~v , respectively, be the corresponding solutions of (A.1), for k ¼ e; l; f . Thecorresponding differences between these solutions, denoted uk, satisfy the system (A.2) below,

@@t u

eðt;aÞþ @@a geðEðtÞ;aÞueðt;aÞ½ �þkueðt;aÞþðbeðEðtÞ;aÞþmeðEðtÞ;aÞÞueþa2 ~v2ðt;aÞueþa2ðv2ðt;aÞ� ~v2ðt;aÞÞue

v ¼0;

@@t u

lðt;aÞþ @@a glðEðtÞ;aÞulðt;aÞ� �

þkulðt;aÞþðblðEðtÞ;aÞþmlv Þulðt;aÞþðml

v�ml~v Þul

~v ¼0;

@@t u

f ðt;aÞþ @@a gf ðEðtÞ;aÞuf ðt;aÞ� �

þkuf ðt;aÞþmf ðEðtÞ;aÞuf ðt;aÞ¼0;

geðEðtÞ;0Þueðt;0Þ¼R Lf

0 ð1�a1 ~v1ðt;aÞÞbfv uf ðt;aÞda�

R Lf

0 a1ðv1� ~v1Þðt;aÞbfv uf

vðt;aÞdaþR Lf

0 ð1�a1 ~v1ðt;aÞÞðbfv�bf

~v Þuf~vðt;aÞda;

glðEðtÞ;0ÞulÞðt;0Þ¼R Le

0 beðEðtÞ;aÞueðt;aÞda;

gf ðEðtÞ;0Þuf ðt;0Þ¼R Ll

0 blðEðtÞ;aÞulðt;aÞda;

ukð0;aÞ¼0; k¼e; l;f ;

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

ðA:2Þ

where ml

v ¼ mlðPlv ; EðtÞ; aÞ and bf

v ¼ bf ðPfv ; EðtÞ; aÞ. We multiply the first, second and third equations of (A.2), respectively, by

ue; ul and uf , then we integrate them on the corresponding domain Xk to get the following system,

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 583

kR

Xe ðueÞ2ðt; aÞdadt � 12

R T0 geðEðtÞ;0Þ ueðt;0Þð Þ2dt 6

RXe ~v2ðt; aÞ � v2ðt; aÞ½ �ue

vðt; aÞueðt; aÞdadt

kR

Xl ðulÞ2ðt; aÞdadt � 12

R T0 glðEðtÞ; 0Þ ulðt; 0Þ

� �2dt 6

RXl ðml

v �ml~vÞul

~vðt; aÞulðt; aÞdadt

kR

Xf ðuf Þ2ðt; aÞdadt � 12

R T0 gf ðEðtÞ;0Þ uf ðt;0Þ

� �2dt 6 0:

8>><>>:

These can be modified by applying Young’s inequality to the integrals of the first and second equations on the right side toobtain system (A.3).

kR

Xe ðueÞ2ðt; aÞdadt 6 12

R T0 geðEðtÞ;0Þ ueðt;0Þð Þ2dt þ �1kue

v k2L2

2

RXe ðueÞ2ðt; aÞdadt þ 1

2�1

RXe j~v2ðt; aÞ � v2ðt; aÞj2dadt

kR

Xl ðulÞ2ðt; aÞdadt 6 12

R T0 glðEðtÞ;0Þ ulðt;0Þ

� �2dt þmK Llkul

vkL2

RXl ðulÞ2ðt; aÞdadt

kR

Xf ðuf Þ2ðt; aÞdadt 6 12

R T0 gf ðEðtÞ;0Þ uf ðt;0Þ

� �2dt:

8>>>>>><>>>>>>:

ðA:3Þ

Next we use the boundary conditions given in (A.2) for the densities ukðt;0Þ (k ¼ e; l; f ) to evaluate the first integrals onthe right hand sides of the three inequalities above to obtain

12

R T0 geðEðtÞ;0Þ ueðt;0Þð Þ2dt¼ 1

2

R T0

1ge ðEðtÞ;0Þ

R Lf

0 ð1�a1 ~v1ðt;aÞÞbfv uf ðt;aÞdaþa1

R L0 ð~v1�v1Þðt;aÞbf

v ufv ðt;aÞdaþ

R Lf

0 ð1�a1 ~v1ðt;aÞÞðbfv �bf

~v Þuf~v ðt;aÞda

h i2dt;

12

R T0 glðEðtÞ;0Þ ulðt;0Þ

� �2dt¼ 1

2

R T0

1glðt;0Þ

R Le

0 beðt;aÞueðt;aÞdah i2

dt;

12

R T0 gf ðEðtÞ;0Þ uf ðt;0Þ

� �2dt¼ 1

2

R T0

1gf ðt;0Þ

R Ll

0 blðt;aÞulðt;aÞdah i2

dt:

8>>>>>><>>>>>>:

Developing the squares in the above expressions, and applying Young’s and Cauchy–Swartz’s inequalities, we see that

12

R T0 geðEðtÞ;0Þ ueðt;0Þð Þ2dt 6 1

2ge

R T0 kb

f k2L2 þ 2�bf bK Lf kuf

~vðtÞkL1 þ ðbKÞ2Lf kuf

~vðtÞk2L1 þ �2kbf k2

L2

h

þ�3Lf bKkuf~vðtÞkL1

i R Lf

0 ðuf Þ2ðt; aÞdadt þ ð�bf Þ2

2ge kufvðtÞk2

L21�2þ bK�3þ 1

h i RXf ~v1ðt; aÞ � v1ðt; aÞ½ �2dadt;

12

R T0 glðEðtÞ;0Þ ulðt;0Þ

� �2dt 6

kbek2L2

2gl

RXe ðueÞ2ðt; aÞdadt;

12

R T0 gf ðEðtÞ;0Þ uf ðt;0Þ

� �2dt 6

kblk2L2

2gf

RXl ðulÞ2ðt; aÞdadt:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ðA:4Þ

We now substitute the estimates (A.4) into (A.3) to obtain the following inequalities:

k� �1kuevk

2L2

2

�RXe ðueÞ2ðt;aÞdadt6 A

2ge

RXf ðuf Þ2ðt;aÞdadtþ B

2ge

RXf j~v1ðt;aÞ�v1ðt;aÞj2dadtþ 1

2�1

RXf ~v2ðt;aÞ�v2ðt;aÞj j2dadt

k�mK LlkulvkL2

� RXl ðulÞ2ðt;aÞdadt6

kbek2L2

2gl

RXe ðueÞ2ðt;aÞdadt

kR

Xf ðuf Þ2ðt;aÞdadt6kblk2

L2

2gf

RXl ðulÞ2ðt;aÞdadt;

8>>>>>>>>><>>>>>>>>>:

ðA:5Þ

where

A ¼ kbf k2L2 þ 2�bf bK Lf kuf

~vðtÞkL1 þ ðbKÞ2Lf kuf

~vðtÞk2L1 þ �2kbf k2

L2 þ �3Lf bKkuf~vðtÞkL1

h i> 0

and,

B ¼ ð�bf Þ2kufvðtÞk

2L2

1�2þ bK

�3þ 1

� �> 0: ðA:6Þ

If we now set

�1 ¼2ðk� 1Þkue

vk2L2

; �2 ¼ 1; �3 ¼ 1

and assume k > mlK Lf kul

vkL2 and it also satisfies the condition

kðk�mK LlkulvkL2 Þ >

Akblk2L2kbek2

L2

8gegf gl> 0; ðA:7Þ

584 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

then, from (A.5) we derive the following inequality:

ZXeðueÞ2ðt; aÞdadt 6

C�1B2ge

ZXfj~v1ðt; aÞ � v1ðt; aÞj2dadt þ C�1

2�1

ZXfj~v2ðt; aÞ � v2ðt; aÞj2dadt;

(

where

C ¼ 1� A2kge

kblk2L2

2gf

kbek2L2

2gl

1

k�mK LlkulvkL2

� 0@

1A ðA:8Þ

and it then follows that the L2-norm of the density ue can be bounded as follows:

kuekL2ðXeÞ 6 C�1=2 B2ge

!2

þ kuevk

2L2

4ðk� 1Þ

!224

35

1=2

k~v � vkL1 :

Finally, we conclude the proof of Theorem 1 for k ¼ e using the constants

De1 ¼ LT

B2Cge

!1=2

;

De2 ¼

LTkuevkL2

21

Cðk� 1Þ

� �1=2

;

where the constants B and C are defined, respectively, in (A.6) and (A.8), and k satisfies the condition (A.7).Similarly, in order to derive the two other inequalities of Theorem 1 (for k ¼ e; l) we use the inequalities (A.5) with the

above definition of �i; i ¼ 1;2;3, to finally conclude that the corresponding constants Dl and Df are given by

Dl1 ¼

LTkbekL2

2B

gegl

!C�1

k�mK LlkulvkL2

!" #1=2

;

Dl2 ¼

LTkbekL2kuevkL2

21

glðk� 1Þ

!C�1

k�mK LlkulvkL2

!" #1=2

;

Df1 ¼

LTkbekL2kblkL2

2B

2geglgf

!ðkCÞ�1

k�mK LlkulvkL2

!" #1=2

;

Df2 ¼

LTkbekL2kblkL2kuevkL2

21

glgf ðk� 1Þ

!ðkCÞ�1

k�mK LlkulvkL2

!" #1=2

and k satisfies the condition (A.7).

A.2. Continuity of the dual variables with respect to the controls

Before giving the proof of Theorem 3 we shall estimate the L2-norm of the dual variable pe.Let us introduce the following change of variables,

pkvðt; aÞ ¼ ektpk

vðt; aÞ k ¼ e; l; f ;

where pkv satisfies (4.1) with the control v ¼ ðv1;v2Þ, and pk

v satisfies system (A.9) below.

� @@t p

ev ðt;aÞ�geðEðtÞ;aÞ @

@a pev ðt;aÞþkpe

vðt;aÞþ beðEðtÞ;aÞþmeðEðtÞ;aÞ½ �pev ðt;aÞ ¼þa2v2ðt;aÞpe

vðt;aÞþ plv ðt;0Þb

eðEðtÞ;aÞ;

� @@t p

lv ðt;aÞ�glðEðtÞ;aÞ @

@a plv ðt;aÞþkpl

vðt;aÞþ blðEðtÞ;aÞþmlðPlv ;EðtÞ;aÞþ@xmlðPl

v ;EðtÞ;aÞulv ðt;aÞ

h ipl

v ðt;aÞ ¼ pfvðt;0ÞblðEðtÞ;aÞ�lekt ;

� @@t p

fv ðt;aÞ�gf ðEðtÞ;aÞ @

@a pfv ðt;aÞþkpf

v ðt;aÞþmf ðEðtÞ;aÞpfv ðt;aÞ¼ 1�a1v1ðt;aÞ½ �pe

v ðt;0Þ bf ðPfv ;EðtÞ;aÞþbf ðPf

v ;EðtÞ;aÞ0ul

vðt;aÞh i

;

pkv ðT;aÞ¼0; k¼ e; l; f ;

pkv ðt;L

kÞ¼0; k¼ e; l; f ;

8>>>>>>>>><>>>>>>>>>:

ðA:9Þ

where Pk

v ðtÞ ¼R Lk

0 ukv ðt; aÞda; k ¼ l; f , and uk

v is given by (2.3). We next multiply the first three equations of (A.9), respectivelyby pe

v ; plv and pf

v , and then we integrate the resulting equations on the corresponding sets Xk (k ¼ e; l; f ) to see that

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 585

12

R T0 ðpe

vÞ2ðt;0Þdt þ k

RXe ðpe

vÞ2ðt; aÞdadt 6

RXe pl

vðt;0ÞbeðEðtÞ; aÞpe

vðt; aÞdadt;12

R T0 ðpl

vÞ2ðt;0Þdt þ k

RXl ðpl

vÞ2ðt; aÞdadt 6

RXl pf

vðt;0ÞblðEðtÞ; aÞplvðt; aÞdadt;

12

R T0 ðp

fvÞ

2ðt;0Þdt þ k

RXf ðpf

vÞ2ðt; aÞdadt 6

RXf pe

vðt;0Þ bf ðPf ; EðtÞ; aÞ þ bf ðPf ; EðtÞ; aÞ0ulðt; aÞ

h ipf

vðt; aÞdadt:

8>>><>>>:

ðA:10Þ

We apply now Young’s and Cauchy Schwartz’s inequalities to the right-hand sides of the above inequalities to obtain thebounds

RXe pl

vðt;0ÞbeðEðtÞ; aÞpe

vðt; aÞdadt 6�1kbek2

L2

2

RXe ðpe

vÞ2ðt; aÞdadt þ 1

2�1

R T0 ðpl

vÞ2ðt;0Þdt;R

Xl pfvðt;0ÞblðEðtÞ; aÞpl

vðt; aÞdadt 6�3kblk2

L2

2

RXl ðpl

vÞ2ðt; aÞdadt þ 1

2�3

R T0 ðp

fvÞ

2ðt;0Þdt;R

Xf pevðt;0Þ bf ðPf ; EðtÞ; aÞ þ bf ðPf ; EðtÞ; aÞ

0ul

vðt; aÞh i

pfvðt; aÞdadt

6�5kbf k2

L2

2 þ �6bfPkul

v ðtÞk2L2

2

�RXf ðpf

vÞ2ðt; aÞdadt þ 1

2�5þ bf

P2�6

�R T0 ðpe

vÞ2ðt;0Þdt:

8>>>>>>>>><>>>>>>>>>:

Next, by setting

�1 ¼k� 1

kbek2L2

; �3 ¼2k

kblk2L2

; �5 ¼2ðk� 1Þkbf k2

L2

; �6 ¼2

kulvðtÞk

2L2

;

system (A.10) becomes

12

R T0 ðpe

vÞ2ðt;0Þdt þ

RXe ðpe

vÞ2ðt; aÞdadt 6 1

2�1

R T0 ðpl

vÞ2ðt;0Þdt;

12

R T0 ðpl

vÞ2ðt;0Þdt 6 1

2�3

R T0 ðp

fvÞ

2ðt;0Þdt;

12

R T0 ðp

fvÞ

2ðt;0Þdt 6 1

2�5þ bf

P2�6

�R T0 ðpe

vÞ2ðt; 0Þdt;

8>>>>><>>>>>:

that leads immediately to the single inequality

ZXeðpe

vÞ2ðt; aÞdadt 6

12�1�3

1�5þ bf

P

�6

!Z T

0ðpe

vÞ2ðt;0Þdt:

If k > 1 and k satisfies the condition

8kðk� 1Þ2 > kbek2L2kblk2

L2kbf k2L2 ;

then, using Lemma 1, we see that the L2-norm of dual variable pev is bounded as follows,

kpevðtÞkL2ðXeÞ 6

2g1

bf Pfmin

<1;

where Pfmin ¼mint�0fPf ðtÞg for all time. h

Proof of (4.2) in Theorem 3:Let v and ~v be two controls and, pk

v and pk~v (k ¼ e; l; f ) the corresponding solutions of (A.9). We now denote by pk the dif-

ference between pkv and pk

~v , and these functions satisfy the following system

� @@t p

eðt;aÞ�geðEðtÞ;aÞ @@a peðt;aÞþkpeðt;aÞþðbeðEðtÞ;aÞþmeðEðtÞ;aÞþa2 ~v2ðt;aÞÞpeðt;aÞ¼ plðt;0ÞbeðEðtÞ;aÞ�a2ðv2ðt;aÞ� ~v2ðt;aÞÞpe

v ðt;aÞ;

� @@t p

lðt;aÞ�glðEðtÞ;aÞ @@a plðt;aÞþkplðt;aÞþ blðEðtÞ;aÞþml

~v ðEðtÞ;aÞþ@xml~v ul

~v ðt;aÞ�

plðt;aÞ

þðmlv �ml

~v Þplv ðt;aÞþð@xml

v ulv �@xml

~v ul~v Þpl

v ðt;aÞ¼ pf ðt;0ÞblðaÞ;

� @@t p

f ðt;aÞ�gf ðEðtÞ;aÞ @@a pf ðt;aÞþkpf ðt;aÞþmf ðEðtÞ;aÞpf ðt;aÞ¼ ð1�a1 ~v1ðt;aÞÞpeðt;0Þ� bf

~v þ@xbf~v ul

~v

h i

�a1 v1ðt;aÞ� ~v1ðt;aÞð Þ bfv þ@xb

fv ul

v

h ipe

v ðt;0Þþð1�a1 ~v1ðt;aÞÞpev ðt;0Þ bf

v �bf~v þul

v@xbfv �ul

~v@xbf~v

h i;

pkðT;aÞ¼0; k¼ e; l; f ;

pkðt;LkÞ¼0; k¼ e; l; f ;

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

where PkðtÞ ¼R Lk

0 ukv ðt; aÞda (k ¼ l; f ) and uk

v is given by (2.3). We multiply now the first three equations above, respectively bype; pl and pf and we integrate the resulting equations on the corresponding domain Xk to get

586 D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588

12

R T0 ðpeÞ2ðt;0Þdtþk

RXe ðpeÞ2ðt;aÞdadt6

RXe plðt;0ÞbeðEðtÞ;aÞpeðt;aÞ dadtþ

RXe ~v2ðtÞ�v2ðtÞ½ �pe

v ðt;aÞpeðt;aÞ dadt;

12

R T0 ðplÞ2ðt;0Þdtþk

RXl ðplÞ2ðt;aÞdadt6

RXl pf ðt;0ÞblðEðtÞ;aÞplðt;aÞdadt þRXl ðml

v �ml~vþul

v@xmlv �ul

~v@xml~v Þ

� �pl

v ðt;aÞplðt;aÞ dadt;

12

R T0 ðpf Þ2ðt;0Þdtþk

RXf ðpf Þ2ðt;aÞdadt6

RXf peðt;0Þ bf

~vþuf~v@xb

f~v

h ipf ðt;aÞ

dadtþR

Xf ~v1ðtÞ�v1ðtÞð Þpev ðt;0Þ bf

v þufv@xb

fv

h ipf ðt;aÞ

dadt

þR

Xf pev ðt;0Þ bf

v�bf~v þuf

~v@xbfv x�uf

~v@xbf~v

h ipf ðt;aÞ

dadt:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ðA:11Þ

Applying Young’s and Cauchy Schwartz’s inequalities to the integrals on the right-hand sides of these inequalities we areled to the following relations,

R

Xe plðt;0ÞbeðEðtÞ; aÞpeðt; aÞ dadt 6 �1

2 kbek2

L2

RXe ðpeÞ2ðt; aÞdadt þ 1

2�1

R T0 ðplÞ2ðt;0Þdt;R

Xe ~v2ðt; aÞ � v2ðt; aÞ½ �pevðt; aÞpeðt; aÞ

dadt 6 �22 kpe

vk2L2

RXe ðpeÞ2ðt; aÞdadt þ 1

2�2

RXe ~v2ðt; aÞ � v2ðt; aÞ½ �2dadt;

8<: ðA:12Þ

for the first inequality in (A.11);

RXl pf ðt;0ÞblðEðtÞ; aÞplðt; aÞ dadt 6 �3

2 kblk2

L2

RXl ðplÞ2ðt; aÞdadt þ 1

2�3

R T0 ðpf Þ2ðt;0Þdt;

RXl ðml

v �ml~v þ ul

v@xmlv � ul

~v@xml~vÞ

� �pl

vðt; aÞplðt; aÞ dadt 6 mK ðDlÞ

2

2�4kv � ~vk2

L1 þmK�4

2 kplvk

2L2

RXl ðplÞ2ðt; aÞdadt;

8><>: ðA:13Þ

for the second inequality in (A.11);

RXf peðt;0Þ bf

~v þ uf~v@xb

f~v

h ipf ðt; aÞ

dadt 6 bKkuf~vkL1

Lf �52

RXf ðpf Þ2ðt; aÞdadt þ bKku

f~vkL1

2�5

R T0 ðpeÞ2ðt;0Þdt;

RXf ~v1ðt; aÞ � v1ðt; aÞ½ �pe

vðt; 0Þ bfv þ ul

v@xbfv

h ipf ðt; aÞ

dadt 6 jpevð:; 0Þj1bKkuf

vkL1

�R

Xf1

2�6~v1ðt; aÞ � v1ðt; aÞ½ �2 þ Lf �6

2 ðpf Þ2ðt; aÞ�

dadt;

RXf pe

vðt;0Þ bfv � bf

~v þ uf~v@xb

fvx� uf

~v@xbf~v

h ipf ðt; aÞ

dadt 6 jpevð:;0Þj1bK � ðDf Þ

2

2�7k~v � vk2

L1 þLf �7

2

RXf ðpf Þ2ðt; aÞdadt

�;

8>>>>>>>>>>>><>>>>>>>>>>>>:

ðA:14Þ

for the last one. Next, we choose �1 and �2 in the inequalities (A.12) as

�1 ¼2ðk� 1Þkbek2

L2

; �2 ¼2

kpevk

2L2

and then the first relation of (A.11) becomes

12

Z T

0ðpeÞ2ðt;0Þdt 6

12�1

Z T

0ðplÞ2ðt; 0Þdt þ 1

2�2

ZXe

v2ðt; aÞ � ~v2ðt; aÞ½ �2dadt: ðA:15Þ

Similarly, we choose �3 and �4 in the inequalities (A.13) as

�3 ¼2ðk� 1Þkblk2

L2

; �4 ¼2

mKkplvk

2L2

and the second relation of (A.11) becomes

12

Z T

0ðplÞ2ðt;0Þdt 6

12�3

Z T

0ðpf Þ2ðt; 0Þdt þmK

�4ðDlÞ

2kv � ~vk2

L1 : ðA:16Þ

Finally, we choose �5; �6 and �7 in the inequalities of (A.14) as

�5 ¼2 k� 2ð Þ

bK Lf kuf~vk

2L1

; �6 ¼2

Lf bKkufvk2

L1 jpevð:;0Þj1

; �7 ¼2

Lf bK jpevð:;0Þj1

and the third relation of (A.11) becomes

12

Z T

0ðpf Þ2ðt;0Þdt 6

bK jpevð:;0Þj1ku

fvkL1

2�6

ZXf

~v1ðt; aÞ � v1ðt; aÞ½ �2dadt þ bKkuf~vkL1

2�5

Z T

0ðpeÞ2ðt;0Þdt

þ bK jpevð:;0Þj12�5

ðDf Þ2k~v � vk2

L1 : ðA:17Þ

D. Picart, F. Milner / Applied Mathematics and Computation 247 (2014) 573–588 587

We substitute this last bound (A.17) in the inequality (A.16) to get

12

Z T

0ðplÞ2ðt;0Þdt 6

1�3

bK jpevð:;0Þj1ku

fvkL1

2�6

ZXf

~v1ðt; aÞ � v1ðt; aÞ½ �2dadt

"

þbKkuf

~vkL1

2�5

Z T

0ðpeÞ2ðt;0Þdt þ bK jpe

vð:;0Þj12�5

ðDf Þ2k~v � vk2

L1

#þmK

�4ðDlÞ

2kv � ~vk2

L1

and now substitute this inequality into (A.15) to see that

12�bKkuf

~vkL1

2�1�3�5

!Z T

0ðpeÞ2ðt;0Þdt6

1�1�3

bK jpevð:;0Þj1ku

fvkL1

2�6�Z

Xf~v1ðt;aÞ�v1ðt;aÞ½ �2dadtþbK jpe

vð:;0Þj12�5

ðDf Þ2k~v�vk2

L1

" #

þ mK

�1�4ðDlÞ

2kv� ~vk2

L1 þ1

2�2

ZXe

~v2ðt;aÞ�v2ðt;aÞ½ �2dadt:

If we take k such that

8ðk� 2Þðk� 1Þ2 > kbek2L2kblk2

L2 bK Lf kuf~vkL1 ;

then inequality (A.18) can be rewritten as

Z T

0ðpeÞ2ðt;0Þdt 6

1�1�3

bK jpevð:;0Þj1ku

fvkL1

2�6�Z

Xf~v1ðt; aÞ � v1ðt; aÞ½ �2dadt þ bK jpe

vð:; 0Þj12�5

ðDf Þ2k~v � vk2

L1

" #

þ mK

�1�4ðDlÞ

2kv � ~vk2

L1 þ1

2�2

ZXe

~v2ðt; aÞ � v2ðt; aÞ½ �2dadt: ðA:18Þ

We now let

G1 ¼bK jpe

v ð:;0Þj1kufv kL1

2�1�3�6þ bK jpe

v ð:;0Þj12�1�3�5

ðDf1Þ

2þ mK�1�4ðDl

1Þ2

� �1=2

;

G2 ¼ bK jpev ð:;0Þj1

2�1�3�5ðDf

2Þ2þ mK�1�4ðDl

2Þ2þ 1

2�2

h i1=2ðA:19Þ

and then (A.18) leads to the first inequality of Theorem 3, that is

jpevðt;0Þ � pe

~vðt;0Þj 6 jpeðt; 0Þj 6 G � kv � ~vkL1 ;

for all t, with G ¼ maxfG1;G2g. h

Proof of (4.3) of theorem (3):We consider systems (A.11)–(A.14), and we choose the constants �i, for i ¼ 1; . . . ;7 as in the previous proof except that �1

is given now by

�1 ¼2ðk� 2Þkbek2

L2

:

The same argument just used for the first estimate in Theorem 3 now leads to the second estimate in the theorem, withH ¼ G defined in (A.19). h

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