Optimal Control for a Stationary Population

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Introduction
Overpopulation has always been the issue that Chinese government lays much emphasis on.It is also a serious crisis we are facing at present.The problems in basic necessity of life and employment as well as the shortage of natural resources brought by overpopulation have become great constraints in the development of Chinese economy.What's worse, it has brought society many problems as the overpopulation of rural labor forces have already caused.Since the policy of family planning was carried out in the 1970s, better control over the population has been achieved and the rate of population increase has decreased intensively.This will be more persuasive if it is analyzed from the perspective of statistics.It is necessary to analyze the population issue in the angle of mathematics, and that is, predicting and controlling the population by gaining population index from the number of population.
At the end of the 1970s and the beginning of the 1980s, Song Jian and other people began to analyze population issue from the perspective of mathematics.They established the partial differential equation model in the development of population, analyzed parameter equation like the mortality function and fertility function parameter equation, and answered the numerical value on the basis of population development equation.By doing so, it can be used to predict the number of population and various population indexes, such as birth rate, mortality, index of aging, number of the aged, number of labor force, index of labor force.All of these will bring Chinese population research into quantitative analysis and research from the pure qualitative analysis.
The optimal control on population system with age structure has been researched extensively in recent decades as many domestic and foreign scholars have done much work in this field.To our knowledge, many workers have devoted themselves in researching the optimal control since M. Brokata researched the optimal control with age structure and population dynamic system for the first time.Population system has formed into a huge system, spreading itself into the vast sea of documents.
The paper explains the optimal control of the dense population.Especially for the stationary population, this paper explains the meaning of the parameters and I obtain the optimal control by the theory of partial differential equations.I do some detailed study of a population equation, which is established on the emergence of the birth rate, mortality rates, gain factor (not depended on age) and other parameters.I prove the existence of the optimal control by the control convergence theorem.
The optimal control on population system with age structure has been researched extensively in recent decades as many domestic and foreign scholars have done much work in this field.To our knowledge, many workers have devoted themselves in researching the optimal control since M. Brokata researched the optimal control with age structure and population dynamic system for the first time.Population system has formed into a huge system, spreading itself into the vast sea of documents.
The paper explains the optimal control of the dense population.Especially for the stationary population, this paper explains the meaning of the parameters and I obtain the optimal control by the theory of partial differential equations.I do some detailed study of a population equation, which is established on the emergence of the birth rate, mortality rates, gain factor (not depended on age) and other parameters.I prove the existence of the optimal control by the control convergence theorem.

Derivation of Population Equation
The population issue is one that many people show their interest to.The ordinary differential equation model has often been used in the research of population development.The model only takes the total number of the population and its aggregate growth rate into consideration, and the age structure of population is not included.As a matter of fact, the age structure of population has a great effect on the change of the number of population.If you compare the youth-oriented society and the old age-oriented society, you will find great differences in the changing of population.In order to take the age structure into account, it is necessary to approach partial differential equation model.Here is the continuous model which considers the age of population.
Let x, t said the age, time, and function N(x, t) expressed the total number of population in the age of less than x at t.And let function N(x, t) on x, t is continuously differentiable, in which a m is the maximum age, N(t) is the total number of population at t. Obviously, N(x, t) has the following properties: (1) N(0, t) = 0; (2) To any a b ≥ a m , N(b, t) = N(t).
Then let p(x, t) = ∂N(x,t)  ∂x , call p(x, t) as the population density function.Due to function N(x, t) about x is not reduced and the above-mentioned properties (1) and (2), population density function p(x, t) has the following properties: (1) p(x, t) ≥ 0; (2) When b ≥ a m , p(b, t) = 0.
When the population density function p(x, t) is known, apparently, and the number of population is The mortality is different for people in different age, thus mortality function has a great influence on the population development.Regard M(x, Δx, t) is the average death toll of those whose age is between [x, x + Δx] at the unit time t, and the mortality function can be defined as:

p(y, t)dy
Set p(x, t) as continuous and differentiable function.When Δx, Δt is small to the full, in the t to t + Δt time, the number of death of the people whose age are between the x and the x + Δx is μ(x, t)p(x, t)ΔxΔt, And the age of the survived are between x + Δt and x + Δx + Δt at t + Δt.Thus it can be deduced that p(x, t)Δx − p(x + Δt, t + Δt)Δx = μ(x, t)p(x, t)ΔxΔt + α(Δx, Δt), In which, the α(Δx, Δt) is infinitesimal quantity at Δx → 0, Δt → 0. The above function can be rewritten into: Both sides are divided by ΔxΔt at the same time and make Δt → 0, and the following can be expressed like this: Equation ( 1) is partial differential equations of first order.In order to work out the population density function p(x, t), initial conditions and boundary conditions also should be given.Set the latest census time is t = 0, and p(x, 0) = p 0 (x) is known.Set p 0 (t) = ϕ(t) is the birth population at unit time t, and when ϕ(t) is known, the mathematical model for population is (2) The Equation ( 2) can be addressed as the continuous equation on population development.

The Solution of the Model
When predicting the future population situation in short term, we often address it in a simple way: assuming that the social conditions remains relatively stable over a period of time, the mortality function μ(x, t) = μ(x) has nothing to do with time and it is only a function of one variable regarding age as its variable quantity.The death function of the following years can be replaced by the mortality function of the starting year.This simple operation makes it easy to work out population development equation, which helps to predict population in short term in a highly accurate way.
A relatively stable society refers to a society remains unchanged in sex ratio of population, childbearing patterns of women, birth rate and mortality for a long time when social conditions remained relatively stable.
Here, set b(x) is birth rate, and it is only associated with age and has nothing to do with time t.Mark a(> 0) is the minimum age of child-bearing, then This is a boundary condition and the Equation (2) becomes (3) And it also assumes that the society is closed, and that is to say, the migration rate is f (a, t) = 0.
Here suppose the p 0 (x) meets the following consistency condition And the following equation is acceptable p 0 (a m ) = 0.
To get the solution easily, first make a substitution of unknown function, and the equation is homogeneous.Make q(x, t) = p(x, t)e x 0 μ(τ)dτ , (t ≥ 0, 0 ≤ x ≤ a m ), Vol. 4, No. 4;2012 Then Equation (3) can be evolved into q(x, 0) = q 0 (x) = p 0 (x)e x 0 μ(τ)dτ , 0 ≤ x ≤ a m ; q(0, t) = a m a B(y)q(y, t)dy, t ≥ 0. (5) In which, and compatible condition (4) can be turned into Here we use method of characteristic curves in solving Equation ( 6), and the characteristic equation is So the characteristic curve is χ(t) = t + x 0 − t 0 crossing the point (x 0 , t 0 ), and q(x, t) along characteristic lines differential is Namely it is constant along any feature line.The following is the discussion of the expression of solution in different regions.
In the region {(x, t); t ≤ x < a m , t ≥ 0}, the feature line crossing any point (x, t) must intersect the initial interval 0 ≤ x < a m in the decreasing direction of t.So it is In the region {(x, t); t − a ≤ x ≤ t, t ≥ 0, 0 ≤ x < a m }, the characteristic lines crossing any point (x, t) must intersect the point t − x on the zone [0, a] in t axis in the boundary region in the decreasing direction of t.Hence, from boundary condition q(x, t) = q(0, t − x) = a m a B(y)q(y, t − x)dy = a m a B(y)q 0 (y − t + x)dy.
This is because of 0 ≤ t − x ≤ a, a ≤ y ≤ a m , and q(y, t − x) under the integral sign can be expressed by Equation (7).
Likewise for any point (x, t) in the region {(x, t); t − 2a ≤ x ≤ t − a, t ≥ 0, 0 ≤ x < a m }, its solution can be expressed as q(x, t) = q(0, t − x) Repeat the above approach, and we can solve it in the whole area {(x, t); 0 ≤ x < a m , t ≥ 0}.The following recursion formula can be used: a m a B(y)q(y, t − x)dy, x < t; Then the solution of question (3) is p(x, t) = e − x 0 μ(τ)dτ q(x, t).Although the solution is obtained in divided area, the solution in the entire region is smooth and satisfies the equation due to the smooth of q 0 (x) and B(x).