Research Article
Received 1 April 2012 Published online 31 May 2013 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2806
MOS subject classification: 49A22; 49K20
Optimal control problem of a generalized
Ginzburg–Landau model equation in
population problems
Xiaopeng Zhao
a
*
†
, Ning Duan
b
and Bo Liu
b
Communicated by G. Ding
In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equation
in population. The optimal control under boundary condition is given, the existence of optimal solution to the equation is
proved, 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
u
t
C a
1
D
4
u a
2
D
2
u aD
2
u
3
C G.u/ D 0, .x, t/ 2 .0, T /, (1)
where D D
@
@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/ Djuj
p1
u .0
p 8/ is a given nonlinear function that represents dynamic term or reaction term, a
1
, a
2
, 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 a
country, the population moving policy may still be the feasible measure for adjusting population state; therefore, when forecasting the
developing trend of the population in different regions, the population moving function is still an important factor necessary to be
considered.’ ([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 D
2
u.x, t/ D 0, x D 0, 1, (2)
and the initial condition
u.x,0/ D u
0
.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 the
famous generalized Ginzburg–Landau model equation (1) when they studied the growth and dispersal in populations. In [5], on the
basis 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] studied
the instability of the traveling waves of the equation (1). He proved that some traveling wave solutions are nonlinear unstable under H
2
perturbations. In [2, 3], Wang et al. investigated the time-periodic problem of the equation (1) for 1-dimensional case and 2-dimensional
case. They also proved the existence and uniqueness of time-periodic generalized solutions and classical solutions. We also noticed that
some 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. Two
methods are used for studying control problems in PDE: one is using a low model method and then changing to an ODE model [10];
a
School of Science, Jiangnan University, Wuxi 214122, China
b
College of Mathematics, Jilin University, Changchun, China
*Correspondence to: Xiaopeng Zhao, School of Science, Jiangnan University, Wuxi 214122, China.
†
E-mail: zxp032@126.com
Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 435–446
435