Optimal control problem of a generalized Ginzburg-Landau model equation in population problems

Research Article

Received 1 April 2012 Published online 31 May 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2806MOS subject classification: 49A22; 49K20

Optimal control problem of a generalizedGinzburg–Landau model equation inpopulation problems

Xiaopeng Zhaoa*†, Ning Duanb and Bo Liub

Communicated by G. Ding

In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equationin population. The optimal control under boundary condition is given, the existence of optimal solution to the equation isproved, and the optimality system is established. Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: optimal control; Ginzburg–Landau model equation; optimal solution; optimality condition

1. Introduction

In this paper, we investigate with the generalized Ginzburg–Landu model equation in population

ut C a1D4u� a2D2u� aD2u3C G.u/D 0, .x, t/ 2�� .0, T/, (1)

where DD @@x ,�D .0, 1/, u.x, t/denoting the population density is an unknown function of x 2 Œ0, 1� and t 2 .0, T/. G.u/D jujp�1u .0�

p � 8/ is a given nonlinear function that represents dynamic term or reaction term, a1, a2, and a are the positive physical constants.It is known to all that ‘owing to the disequilibrium of the population distribution in different regions, cities, and provinces within acountry, the population moving policy may still be the feasible measure for adjusting population state; therefore, when forecasting thedeveloping trend of the population in different regions, the population moving function is still an important factor necessary to beconsidered.’ ([1–3]).

For the equation (1), on the basis of physical consideration, the equation is supplemented by the following boundary conditions

u.x, t/D D2u.x, t/D 0, x D 0, 1, (2)

and the initial condition

u.x, 0/D u0.x/, x 2�. (3)

During the past years, many authors have paid much attention on the equation (1). It was Cohen and Murray [4] who first gave thefamous generalized Ginzburg–Landau model equation (1) when they studied the growth and dispersal in populations. In [5], on thebasis of the fixed point principle, Liu and Pao proved the existence of classical solutions for periodic boundary problem. Chen and Lü[6] proved the existence, asymptotic behavior, and blow-up of classical solutions for initial boundary value problem. C. Liu [7] studiedthe instability of the traveling waves of the equation (1). He proved that some traveling wave solutions are nonlinear unstable under H2

perturbations. In [2,3], Wang et al. investigated the time-periodic problem of the equation (1) for 1-dimensional case and 2-dimensionalcase. They also proved the existence and uniqueness of time-periodic generalized solutions and classical solutions. We also noticed thatsome investigations of the generalized Ginzburg–Landau model equation in population problem were studied, such as in [8, 9].

The optimal control plays an important role in modern control theories and has a wider application in modern engineering. Twomethods are used for studying control problems in PDE: one is using a low model method and then changing to an ODE model [10];

aSchool of Science, Jiangnan University, Wuxi 214122, ChinabCollege of Mathematics, Jilin University, Changchun, China*Correspondence to: Xiaopeng Zhao, School of Science, Jiangnan University, Wuxi 214122, China.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 435–446

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X. ZHAO, N. DUAN AND B. LIU

the other is using a quasi-optimal control method [11]. No matter which one is chosen, it is necessary to prove the existence of optimalsolution and establish the optimality system.

Many papers have already been published to study the control problems of nonlinear parabolic equations. In 1991, Yong and Zheng[12] studied the feedback stabilization and optimal control of the Cahn–Hilliard equation in a bounded domain with smooth boundary.In papers wrote by Sang-Uk Ryu and Atsushi Yagi [13, 14], the optimal control problems of Keller–Segel equations and adsorbate-induced phase transition model were considered. Their techniques are based on the energy estimates and the compact method. Theyestablished various a priori estimates for the solutions of equations to show that the classical compact method described systemat-ically by Lions [15] is available. Tian et al. [16–18] also studied the optimal control problems for parabolic equations, such as viscousCamassa–Holm equation, viscous Degasperis–Procesi equation, viscous Dullin–Gottualld–Holm equation, and so on. Recently, Zhaoand Liu [19] considered the optimal control problem for 1D viscous Cahn–Hilliard equation. In their paper, the optimal control underboundary condition was given, and the existence of optimal solution to the equation was proved. There is much literature concernedwith the optimal control problem for parabolic equations, for more recent result, we refer the reader to [20–22] and the referencestherein.

In this paper, we are concerned with distributed optimal control problem

min J.u, w/D1

2kCu� zdk

2S C

ı

2kwk2

L2.Q0/, (4)

subject to 8<:

ut C a1D4u� a2D2u� aD2u3C G.u/D Bw, .x, t/ 2�� .0, T/,u.0, t/D u.1, t/D D2u.0, t/D D2u.1, t/D 0,u.0/D u0.

(5)

The control target is to match the given desired state zd in L2-sense by adjusting the body force w in a control volume Q0 � Q D.0, T/�� in the L2-sense.

In the following, we introduce some notations that will be used throughout the paper. For fixed T > 0, let�D .0, 1/, QD�� .0, T/,we also let Q0 � Q be an open set with positive measure, V D H2

per.0, 1/, U D H1per.0, 1/ and H D L2.0, 1/, let V�, U� and H� are dual

spaces of V , U, and H. Then, we obtain

V ,! U ,! HD H� ,! U� ,! V�.

Clearly, each embedding being dense.The extension operator B 2 L

�L2.Q0/, L2.0, T ; H/

�, which is called the controller is introduced as

BqD

�q, q 2 Q0,0, q 2 Q nQ0.

We supply H with the inner product .�, �/ and the norm k � k, and define a space W.0, T ; V/ as

W.0, T ; V/D fy : y 2 L2.0, T ; V/, yt 2 L2.0, T ; V�/g,

which is a Hillbert space endowed with common inner product.This paper is organized as follows. In the next section, we prove the existence and uniqueness of the weak solution to problem

(1)–(3) in a special space. We also discuss the relation among the norms of weak solution, initial value, and control item; in Section 3,we consider the optimal control problem of problem (1)–(3) and prove the existence of optimal solution; in Section 4, the optimalityconditions for problem (1)–(3) is showed, and the optimality system is derived.

In the following, the letters c, ci , (iD 1, 2, � � � ) will always denote positive constants different in various occurrences.

2. Existence and uniqueness of weak solution

In this section, we prove the existence and uniqueness of weak solution for the following equation

ut C a1D4y � a2D2u� aD2u3C G.u/D Bw, .x, t/ 2 Q, (6)

with the periodic boundary value conditions

u.0, t/D u.1, t/D D2u.0, t/D D2u.1, t/D 0, (7)

and the initial value condition

u.x, 0/D u0.x/, (8)

where x 2 .0, 1/, t 2 Œ0, T�, Bw 2 L2.0, T ; H/, and a control w 2 L2.Q0/.

43

6



We shall give the definition of the weak solution in the space W.0, T ; V/ for the problem (6)–(8).

Definition 2.1For all � 2 V and u.0/D u0 2 H1

0.0, 1/, a function u.x, t/ 2W.0, T ; V/ is called a weak solution to problem (6)–(8), if

d

dt.u, �/C a1.D

2u, D2�/C a2.Du, D�/C a.Du3, D�/C .G.u/, �/D .Bw, �/. (9)

Now, we give Theorem 2.1, which ensures the existence of a uniqueness weak solution to problem (6)–(8).

Theorem 2.1Let u0 2 U, Bw 2 L2.0, T ; H/, then the problem (6)–(8) admits a unique weak solution u.x, t/ 2W.0, T ; V/.

Proof 2.1Galerkin method is applied to the proof.

Denote A D��@2

x

�2as a differential operator, let f ig

1iD1 denote the eigenfunctions of the operator A D

��@2

x

�2. For n 2 N, define

the discrete ansatz space by

Vn D spanf 1, 2, � � � , ng � V .

Let un.t/D un.x, t/DPn

iD1 uni .t/ i.x/ require un.0, �/! u0 in H holds true.

By analyzing the limiting behavior of sequences of smooth function fung, we can prove the existence of a weak solution to theproblem (6)–(8).

Performing the Galerkin procedure for the equation (6), we obtain

unt C a1D4un � a2D2un � aD2u3nC G.un/D Bw, (10)

with the boundary value conditions

un.x, t/D D2un.x, t/, x D 0, 1, (11)

and the initial value condition

un.x, 0/D un,0.x/. (12)

It is easy to see that the equation (10) is an ordinary differential equation and according to ODE theory, there exists a unique solutionto the equation (10) in the interval Œ0, tn/. What we should do is to show that the solution is uniformly bounded when tn ! T . We alsoneed to show the times tn that there are not decaying to 0 as n!1.

Therefore, we shall prove the existence of solution in the following steps.Step 1, multiplying the equation (10) by un, integrating with respect to x on .0, 1/, we deduce that

1

2

d

dtkunk

2C a1kD2unk2C a2kDunk

2C a�

Du3n, Dun

�C�junj

p�1un, un

�D .Bw, un/.

Noticing that

a�

Du3n, Dun

�D 3a

�u2

nDun, Dun

�D

Z 1

0u2

n.Dun/2dx � 0,

and

�junj

p�1un, un

�D

Z 1

0junj

pC1dx � 0.

Hence

1

2

d

dtkunk

2C a1kD2unk2C a2kDunk

2 � kBwkkunk �1

2kunk

2C1

2kBwk2.

Because Bw 2 L2.0, T ; H/ is the control item, we can assume that kBwk � M, where M is a positive constant. Thus, a simple calculationshows that

d

dtkunk

2C 2a1kD2unk2C 2a2kDunk

2 � kunk2CM2. (13)

Using Gronwall’s inequality, we obtain

kunk2 � etkun0k

2dxCM2 � eTZ 1

0u2

n0dxCM2 D c21, 8t 2 Œ0, T�. (14)


43

7


Step 2, multiplying the equation (10) by�D2un, integrating with respect to x on .0, 1/, we deduce that

1

2

d

dtkDunk

2C a1kD3unk2C a2kD2unk

2 D�a�

D2u3n, D2un

�C�junj

p�1un, D2un

�� .Bw, D2un/.

Noticing that �D2u3

n, D2un

�D 3

�u2

nD2unC 6un.Dun/2D2un, D2un

�,

and �junj

p�1un, D2un

�D�

�pjunj

p�1Dun, Dun

�.

Hence

1

2

d

dtkDunk


2C akunD2unk2

D�2a

Z 1

0u2

n.D2un/

2dx � 6

Z 1

0un.Dun/

2D2undx �

Z 1

0pjunj

p�1.Dun/2dx �

Z 1

0D2unBwdx

� a

Z 1

0junD2unj

2dxC c

Z 1

0jDunj

4dxC "

Z 1

0junj

2p�2dxCa2

2

Z 1

0jD2unj

2dxC1

2a2kBwk2.

Using Nirenberg’s inequality, we immediately obtain

c

Z 1

0jDunj

4dx � c

�Z 1

0u2

nxxxdx

� 56�Z 1

0u2

ndx

� 76

�a1

4

Z 1

0jD3unj

2dxC c2,

and

"

Z 1

0junj

2p�2dx � c"

�Z 1

0jD3unj

2dx

� p�23�Z 1

0u2

ndx

� 5p�43

�a1

4

Z 1

0jD3unj

2dxC c3.

Summing up, we have

d

dtkDunk


2 � c2C c3C1

a1M2. (15)

A simple calculation shows that

kDunk2 � kDun0k

2C c2C c3CM2

a1. (16)

Using Sobolev’s embedding theorem, we immediately obtain that

supx2Œ0,1�

jun.x, t/j � c4. (17)

Step 3, we prove a uniform L2.0, T ; V/ bound on a sequence fung. It then follows from (13) and (14) that

d

dtkunk

2C 2a1kD2unk2C 2a2kDunk

2 � c21 CM2.

Therefore Z T

0jD2unk

2dt �1

2a1

�c2

1 CM2�Ckun0k

2 D c5, (18)

and Z T

0kDunk

2dt �1

2a2

�c2

1 CM2�Ckun0k

2 D c6. (19)

Adding (14), (18), and (19) together gives

kunkL2.0,T ;V/ � c.

Then, the uniform L2.0, T ; V/ bound on a sequence fung is proved.

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8



Step 4, we prove a uniform L2.0, T ; V�/ bound on a sequence fun,tg. Noticing that

.D2un, �/D .un, D2�/� kunkkD2�k � kunkk�kV ,�D2u3

n, ��D�

u3n, D2�

�� sup

x2Œ0,1�junj

2 � .un, D2�/� supx2Œ0,1�

junj2 � kunkk�kV ,

�junj

p�1un, �� sup

x2Œ0,1�junj

p�1 � kunkk�k � supx2Œ0,1�

junjp�1 � kunkk�kV ,

.Bw, �/� kBwkk�k � kBwkk�kV .

Therefore

kun,tkV� CkD4unkV� � c

kBwkC kunkC sup

x2Œ0,1�junj

2kunkC supx2Œ0,1�

junjp�1kunk

!� c

�MC c1C c2

4c1C cp�14 c1

�. (20)

Then, we immediately conculde

kun,tkL2.0,T ;V�/ � c�

MC c1C c24c1C cp�1

4 c1

�T � c. (21)

Collecting the previous, we obtain

(1) For every t 2 Œ0, T�, the sequence fungn2N is bounded in L2.0, T ; V/, which is independent of the dimension of ansatz space n.(2) For every t 2 Œ0, T�, the sequence fun,tgn2N is bounded in L2.0, T ; V�/, which is independent of the dimension of ansatz space n.

On the basis of the earlier discussion, we obtain u.x, t/ 2W.0, T ; V/. It is easy to check that W.0, T ; V/ is continuously embedded intoC.0, T ; U/, which denote the space of continuous functions. We conclude convergence of a subsequences, again denoted by fungweakinto W.0, T ; V/, weak-star in L1.0, T ; U/, and strong in L2.0, T ; U/ to functions u.x, t/ 2 W.0, T ; V/. Because the proof of uniqueness iseasy, we omit it.

Then, Theorem 2.1 is proved. �

Now, we shall discuss the relation among the norm of weak solution and initial value and control item. Theorem 2.2 in the followingensures that the norm of weak solution can be controlled by initial value and control item.

Theorem 2.2Suppose Bw 2 L2.0, T ; H/, u0 2 U, then there exists positive constants C0 and C00 such that

kuk2W.0,T ;V/ � C0

�ku0k

2U Ckwk2

L2.Q0/

�C C00, (22)

Proof 2.2Clearly, (22) means

kuk2L2.0,T ;V/ Ckutk

2L2.0,T ;V�/ � C0

�ku0k

2U CkBwk2

L2.0,T ;H/

�C C00. (23)

Passing to the limit in (13), we have

d

dtkuk2C 2a1kD2uk2C 2a2kDuk2 � kuk2CkBwk2. (24)

Using Gronwall’s inequality, we obtain

kuk2 � etku0k2CkBwk2 � eTku0k

2CkBwk2, t 2 Œ0, T�.

Then

kukL2.0,T ;H/ � eT Tku0k2CkBwk2

L2.0,T ;H/ � c7ku0k2CkBwk2

L2.0,T ;H/. (25)

Integrating (24) with respect to t on Œ0, T�, we deduce that

kD2unk2L2.0,T ;H/ �

1

2a1

�kuk2

L2.0,T ;H/ CkBwk2L2.0,T ;H/ Cku0k

2�� c8ku0k

2C c9kBwk2L2.0,T ;H/. (26)

and

kDunk2L2.0,T ;H/ �

1

2a2

�kuk2

L2.0,T ;H/ CkBwk2L2.0,T ;H/ Cku0k

2�� c10ku0k

2C c11kBwk2L2.0,T ;H/. (27)


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9


Passing to the limit in (15), a simple calculation shows that

kDuk2L2.0,T ;H/ � c, sup

x2Œ0,1�ju.x, t/j � c.

On the other hand, we have

.D2u, �/D .u, D2�/� kukkD2�k � kukk�kV ,

.D2u3, �/D .u3, D2�/� supx2Œ0,1�

juj2 � .u, D2�/� supx2Œ0,1�

juj2 � kukk�kV ,

.jujp�1u, �/� supx2Œ0,1�

jujp�1 � kukk�k � supx2Œ0,1�

jujp�1 � kukk�kV ,

.Bw, �/� kBwkk�k � kBwkk�kV .

Therefore

kutkV� CkD4ukV� � c

kBwkC kukC sup

x2Œ0,1�juj2kukC sup

x2Œ0,1�jujp�1kuk

!� c.MC c/.

Then, we immediately conculde

kunkL2.0,T ;V�/ � c.MC c/T � c. (28)

By (25)–(28) and the definition of extension operator B, we obtain (22). Therefore, Theorem 2.2 is proved. �

3. Optimal control problem

In this section, we consider the optimal control problem associated with problem (6)–(8) and prove the existence of optimal solutionbasing on J. L. Lions’ theory ([15]).

In the following, we suppose L2.Q0/ is a Hilbert space of control variables, we also suppose B 2 L.L2.Q0/, L2.0, T ; H// is the controllerand a control w 2 L2.Q0/, consider the following control system8<

:ut C a1D4u� a2D2u� aD2u3C G.u/D Bw,u.0, t/D u.1, t/D D2u.0, t/D D2u.1, t/D 0,u.0/D u0, x 2 .0, 1/.

(29)

Here in (29), it is assumed that u0 2 H10.0, 1/. By virtue of Theorem 2.1, we can define the solution map w ! u.w/ of L2.Q0/ into

W.0, T ; V/. The solution u is called the state of the control system (29). The observation of the state is assumed to be given by Cu. Here,C 2 L.W.0, T ; V/, S/ is an operator, which is called the observer, S is a real Hilbert space of observations. The cost function associatedwith the control system (29) is given by

J.u, w/D1

2kCu� zdk

2S C

ı

2kwk2

L2.Q0/, (30)

where zd 2 S is a desired state and ı > 0 is fixed.An optimal control problem about the problem (6)–(8) is

min J.u, w/, (31)

where .u, w/ satisfies (29).Let X DW.0, T ; V/� L2.Q0/ and Y D L2.0, T ; V/� H. We define an operator eD e.e1, e2/ : X! Y , where

e1 D GD .�2/�1�

ut C a1D4u� a2D2u� aD2u3C G.u/� Bw�

,

and

e2 D u.x, 0/� u0.

Here,�2 is an operator from V to V�. Then, we write (31) in following form

min J.u, w/ subject to e.u, w/D 0.

Now, we give the theorem on the existence of an optimal solution to problem (29) and prove it.

Theorem 3.1Suppose Bw 2 L2.0, T ; H/, u0 2 H1

0.0, 1/, then there exists an optimal control solution .u�, w�/ to the problem (29).

44

0



Proof 3.1Suppose .u, w/ satisfy the equation e.u, w/D 0. In view of (30), we deduce that

J.u, w/�ı

2kwk2

L2.Q0/.

By Theorem 2.2, we obtain

kukW.0,T ;V/!1 yields kwkL2.Q0/!1.

Therefore,

J.u, w/!1, when k.u, w/kX !1. (32)

As the norm is weakly lower semi-continuous, we achieve that J is weakly lower semi-continuous. Because for all .u, w/ 2 X ,J.u, w/� 0, there exists �� 0 defined by

�D inffJ.u, w/j.u, w/ 2 X , e.u, w/D 0g,

which means the existence of a minimizing sequence f.un, wn/gn2N in X such that

�D limn!1

J.un, wn/ and e.un, wn/D 0, 8n 2 N.

From (32), there exists an element .u�, w�/ 2 X such that when n!1,

un! u�, weakly, u 2W.0, T ; V/, (33)

wn! w�, weakly, w 2 L2.Q0/. (34)

Using (33), we obtain

limn!1

Z T

0

�un

t .x, t/� u�t , .t/�

V� ,V dtD 0, 8 2 L2.0, T ; V/. (35)

On the basis of the definition of W.0, T ; V/, we can derive that un ! u� strongly in L2.0, T ; L1/ as n!1. We can also deduce thatun! u� strongly in C.0, T ; U/when n!1.

Because sequence fungn2N converge weakly and fung is bounded in W.0, T ; V/, on the basis of the embedding theorem, we canobtain fungL2.0,T ;L1/ is also bounded.

Because un! u� strongly in L2.0, T ; L1/ as n!1, we know that ku�kL2.0,T ;L1/ is bounded too.Using (34) again, we derive that

ˇ̌̌ˇZ T

0

Z 1

0.Bw � Bw�/�dxdt

ˇ̌̌ˇ! 0, n!1, 8� 2 L2.0, T ; H/.

By (33), we deduce that

ˇ̌̌ˇZ T

0

Z 1

0.aD2.un/3 � a.D2.u�/3/�dxdt

ˇ̌̌ˇ

D a

ˇ̌̌ˇZ T

0

Z 1

0..un/3 � .u�/3/D2�dxdt

ˇ̌̌ˇ

� a

ˇ̌̌ˇZ T

0

�k.un/2C .u�/2C unu�kL1kun � u�kHkD2�kH

�dxdt

ˇ̌̌ˇ

� ak.un/2C .u�/2C unu�kL2.0,T ;L1/kun � u�kC.0,T ;H/k�kL2.0,T ;V/

! 0, n!1, 8� 2 L2.0, T ; V/,


44

1


and ˇ̌̌ˇZ T

0

Z 1

0.junjp�1un � ju�jp�1u�/�dxdt

ˇ̌̌ˇ

D

ˇ̌̌ˇZ T

0

Z 1

0.un � u�/.junjp�1C junjp�2u�C junjp�3ju�j2C � � � C ju�jp�1/�dxdt

ˇ̌̌ˇ

�

ˇ̌̌ˇ̌̌Z T

0

0@kun � u�kHk�kH

p�1XiD0

junjp�1�iju�ji

L1

1A dxdt

ˇ̌̌ˇ̌̌

� ıkun � u�kC.0,T ;H/

p�1XiD0

junjp�1�iju�ji

L2.0,T ;L1/

k�kL2.0,T ;H/

! 0, n!1, 8� 2 L2.0, T ; H/.

In view of the earlier discussion, we obtain

e1.u�, w�/D 0, 8n 2 N.

From u� 2W.0, T ; V/, we derive that u�.0/ 2 H.Because un! u� weakly in W.0, T ; V/, we can infer that un.0/! u�.0/weakly when n!1. Thus, we obtain

.un.0/� u�.0/, �/! 0, n!1, 8� 2 H,

which means e2.u�, w�/D 0. Therefore, we obtain

e.u�, w�/D 0, in Y .

So, there exists an optimal solution .u�, w�/ to problem (29).Then, Theorem 3.1 is proved. �

4. Optimality conditions

It is well known that the optimality conditions for w is given by the variational inequality

J0.u, w/.Nv �w/� 0, for all Nv 2 L2.Q0/, (36)

where J0.u, w/ denotes the Gateaux derivative of J.u, Nv/ at Nv D w.The following lemma is essential in deriving necessary optimality conditions.

Lemma 4.1The map Nv! u.Nv/ of L2.Q0/ into W.0, T ; V/ is weakly Gateaux differentiable at Nv D w and such the Gateaux derivative of u.Nv/ at Nv D win the direction Nv �w 2 L2.Q0/, say zDDu.w/.Nv �w/, is a unique weak solution of the following problem8̂<

:̂zt C a1D4z� a2D2z� 3aD2Œ.u.w//2z�C pju.Nv/jp�1zD B.Nv �w/, 0< t � T ,

z.0, t/D z.1, t/D D2z.0, t/D D2z.1, t/D 0,

z.0/D 0, x 2 .0, 1/.

(37)

Proof 4.1Let 0 � h � 1, uh and u be the solutions of (29) corresponding to wC h.Nv �w/ and w, respectively. Then, we prove the lemma in the

following two steps:Step 1, we prove uh! u strongly in C.0, T ; H/ as h! 0. Let qD uh � u, then

8̂̂<̂ˆ̂̂:

dqdt C a1D4q� a2D2q� a

�D2u3

h � D2u3�C .juhj

p�1uh � jujp�1u/

D hB.Nv �w/, 0< t � T ,

q.0, t/D q.1, t/D D2q.0, t/D D2q.1, t/D 0,

q.0/D 0, x 2 .0, 1/.

(38)

By Theorem 2.1, we obtain

ku.x, t/kL1 � c, kuh.x, t/kL1 � c.

44

2



Then, taking the scalar product of (38) with q, we have

1

2

d

dtkqk2C a1kD2qk2C a2kDqk2

D a�

D2�

u3h � u3

�, q�� .juhj

p�1uh � jujp�1u, q/C .hB.Nv �w/, q/

D a��

u2hC u2C uh

�q, D2q

��

0@q

p�1XiD0

juhjp�1�ijuji , q

1AC .hB.Nv �w/, q/

� a supx2Œ0,1�

ju2hC u2C uhuj � kqkkD2qkC sup

x2Œ0,1�

0@p�1X

iD0

juhjp�1�ijuji

1A � kqk2C .hB.Nv �w/, q/

�a1

2kD2qk2C ckqk2 C .hB.Nv �w/, q/.

Hence

d

dtkqk2 � ckqk2 C 2.hB.Nv �w/, q/� ckqk2 C h2kB.Nv �w/k2.

Using Gronwall’s inequality, it is easy to see that kqk2! 0 as h! 0. Then, uh! u strongly in C.0, T ; H/ as h! 0.Step 2, we prove that uh�u

h ! z strongly in W.0, T ; V/. Now, we rewrite (38) in the following form8̂̂ˆ̂<ˆ̂̂̂:

ddt

� uh�uh

�C a1D4

� uh�uh

�� a2D2

� uh�uh

�� a

�D2u3

h�D2u3

h

�C�juhj

p�1uh�jujp�1uh

�D B.v �w/, 0< t � T ,

uh�uh .0, t/D uh�u

h .1, t/D D2 uh�uh .0, t/D D2 uh�u

h .1, t/D 0,

uh�uh .0/D 0, x 2 .0, 1/.

We can easily verify that the earlier problem possesses a unique weak solution in W.0, T ; V/. On the other hand, it is easy to check thatthe linear problem (37) possesses a unique weak solution z 2W.0, T ; V/. Let r D uh�u

h � z, thus r satisfies8̂̂̂<̂ˆ̂̂̂:

ddt rC a1D4r � a2D2r � a

�D2 u3

h�u3

h � D2.3u2z/

�C�juhj

p�1uh�jujp�1uh � pjujp�1z

�D 0, 0< t � T ,

r.0, t/D r.1, t/D D2r.0, t/D D2r.1, t/D 0,

r.0/D 0, x 2 .0, 1/.

(39)

Taking the scalar product of (39) with r, we obtain

1

2

d

dtkrk2C a1kD2rk2C a2kDrk2 D a

u3

h � u3

h� 3u2z, D2r

!�

�juhj

p�1uh � jujp�1u

h� pjujp�1z, r

�. (40)

Noticing that u3

h � u3

h� 3u2z, D2r

!� k

u3h � u3

h� 3u2zkkD2rk D k3.uC �.uh � u//2

uh � u

h� 3u2zkkD2rk

�a1

4kD2rk2C ck3.uC �.uh � u//2

uh � u

h� 3u2zk2,

where � 2 .0, 1/. We have uh! u strongly in H as h! 0, then

k3.uC �.uh � u//2uh � u

h� 3u2zk2!k3u2

�uh � u

h� z�k2 � ckrk2, as h! 0.

Therefore u3

h � u3

h� 3u2z, D2r

!�

1

4kD2rk2C ckrk2.

On the other hand, we have

�

�juhj


h� pjujp�1z, r

�� kjuhj


h� pjujp�1zkkrk

D kp.uC �.uh � u//p�1 uh � u

h� pup�1zkkrk

�a1

4kD2rk2C ckp.uC �.uh � u//p�1 uh � u

h� pup�1zk2,


44

3


where � 2 .0, 1/. Noticing that uh! u strongly in H as h! 0, then

kp.uC �.uh � u//p�1 uh � u

h� pup�1zk2!kpup�1

�uh � u

h� z�k2 � ckrk2 as h! 0.

Therefore

�

�juhj


h� pjujp�1z, r

��

1

4kD2rk2C ckrk2.

Summing up, we obtain

d

dtkrk2 � ckrk2.

Using Gronwall’s inequality, it is easy to check that uh�uh is strongly convergent to z in W.0, T ; V/.

Then, Lemma 4.1 is proved. �

As in [15], we denoteD canonical isomorphism of S onto S�, where S� is the dual spaces of S. By calculating the Gateaux derivativeof (32) via Lemma 4.1, we see that the cost J.Nv/ is weakly Gateaux differentiable at w in the direction Nv �w. Then, (36) can be rewrittenas

.C�ƒ.Cu.w/� zd/, z/W.V/� ,W.V/ Cı

2.w, Nv �w/L2.Q0/

� 0, 8Nv 2 L2.Q0/, (41)

where z is the solution of (37).Now, we study the necessary conditions of optimality. To avoid the complexity of observation states, we consider the two types of

distributive and terminal value observations.

1. Case of C 2 L.L2.0, T ; V/; S/.In this case, C� 2 L.S�; L2.0, T ; V�// and (41) may be written asZ T

0.C�ƒ.Cu.w/� zd/, z/V� ,V dtC

ı

2.w, Nv �w/L2.Q0/

� 0, 8Nv 2 L2.Q0/. (42)

We introduce the adjoint state y.Nv/ by8̂<:̂� d

dt y.Nv/C a1D4y.Nv/� a2D2y � 3au.Nv/2D2yC pju.Nv/jp�1yD C�ƒ.Cu.Nv/� zd/, in .0, T/,

y.0, t/D y.1, t/D D2y.0, t/D D2y.1, t/D 0,

y.x, T ; Nv/D 0.

(43)

According to Theorem 2.1, the earlier problem admits a unique solution (after changing t into T � t).Multiplying both sides of (43) (with Nv D w) by z, using Lemma 4.1, we obtainZ T

0

��

d

dty.w/, z

�V� ,V

dtD

Z T

0

�y.w/,

d

dtz

�dt,

Z T

0.D4y.w/, z/V� ,V dtD

Z T

0.y.w/, D4z/dt,

Z T

0.D2y.w/, z/V� ,V dtD

Z T

0.y.w/, D2z/dt,

Z T

0.3u.w/2D2y.w/, z/V� ,V dtD

Z T

0.y.w/, 3D2..u.w//2z//dt,

and Z T

0.pju.w/jp�1y.w/, z/V� ,V dtD

Z T

0.y.w/, pju.w/jp�1z/dt.

Then, we obtain Z T

0.C�ƒ.Cu.w/� zd/, z/V� ,V dt

D

Z T

0

�y.w/, zt C a1D4z� a2D2z� 3aD2Œ.u.w//2z�C pju.Nv/jp�1z

�dt

D

Z T

0.y.w/, BNv � Bw/dtD .B�y.w/, Nv �w/.4

44



Hence, (42) may be written as

Z T

0

Z 1

0B�y.w/.Nv �w/dxdtC

ı

2.w, Nv �w/L2.Q0/

� 0, 8Nv 2 L2.Q0/. (44)

Therefore, we have proved the following theorem:

Theorem 4.1We assume that all conditions of Theorem 3.1 hold and C 2 L.L2.0, T ; V/; S/. The optimal control w is characterized by the system oftwo PDEs and an inequality: (29), (43), and (44).

2. Case of C 2 L.H; S/.In this case, we observe Cu.Nv/D Du.T ; Nv/, D 2 L.H; H/. The associated cost function is expressed as

J.u, Nv/D kDu.T ; Nv/� zk2S C

ı

2kNvk2

L2.Q0/, 8Nv 2 L2.Q0/. (45)

Then the optimal control w for (45) is characterized by

.Du.T ; w/� z, Du.T ; Nv/� Du.T ; w//V� ,V Cı

2.w, Nv �w/L2.Q0/

� 0, 8Nv 2 L2.Q0/. (46)

We introduce the adjoint state p.Nv/ by8̂<:̂� d

dt p.Nv/C a1D4y.Nv/� a2D2y � 3au.Nv/2D2yC pju.Nv/jp�1y D 0, in .0, T/,

y.0, t/D y.1, t/D D2y.0, t/D D2y.1, t/D 0,

y.T ; Nv/D D�.Du.T ; Nv/� zd/.

(47)

According to Theorem 2.1, the earlier problem admits a unique solution (after changing t into T � t).Let us set Nv D w in the earlier equations and scalar multiply both side of the first equation of (47) by u.Nv/� u.w/ and integrate from

0 to T . A simple calculation shows that (46) is equivalent to

Z T

0

Z 1

0B�y.w//.Nv �w/dxdtC

ı

2.w, Nv �w/L2.Q0/

� 0, 8Nv 2 L2.Q0/. (48)

Then, we have the following theorem:

Theorem 4.2We assume that all conditions of Theorem 3.1 hold. Let us suppose that D 2 L.H; H/. The optimal control w is characterized by thesystem of two PDEs and an inequality: (29), (47), and (48).

Acknowledgements

This work is supported by Graduate Innovation Fund of Jilin University (Project 20121059).

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Optimal control problem of a generalized Ginzburg-Landau model equation in population problems

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