OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth. 2005; 26:229–254 Published online 23 May 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/oca.759

Numerical solution to the optimal birth feedback control of a population dynamics: viscosity solution approach Bao-Zhu Guo1,2,n,y and Bing Sun3,4 1

Institute of Systems Science, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, People’s Republic of China 2 School of Computational and Applied Mathematics, University of the Witwatersrand, Private 3, Wits 2050, Johannesburg, South Africa 3 Institute of Systems Science, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, People’s Republic of China 4 Graduate School of the Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

SUMMARY This paper is concerned with the optimal birth control of a McKendrick-type age-structured population dynamic system. We use the dynamic programming approach in our investigation. The Hamilton–Jacobi– Bellman equation satisﬁed by the value function is derived. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal birth feedback control is found explicitly through the value function. A ﬁnite diﬀerence scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is veriﬁed numerically by comparing with diﬀerent controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989. Copyright # 2005 John Wiley & Sons, Ltd. KEY WORDS:

population dynamics; Hamilton–Jacobi–Bellman equation; viscosity solution; ﬁnite diﬀerence scheme; optimal feedback control

1. INTRODUCTION Optimal control problem is one of the central problems in modern control theory and seeking optimal controllers, particularly control laws in feedback form is argued as the Holy Grail of the optimal control theory [1]. It is well-known that the seminal contributions in optimal control theory are the Pontryagin maximum principle, the Bellman dynamic programming method, and the Kalman optimal linear regulator theory, all developed in the 1960s. Numerous works have n

Correspondence to: Bao-Zhu Guo, Institute of Systems Science, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, People’s Republic of China. y E-mail: [email protected] Contract/grant sponsor: National Natural Science Foundation of China Contract/grant sponsor: National Research Foundation of South Africa

Copyright # 2005 John Wiley & Sons, Ltd.

Received 16 August 2004 Revised 1 April 2005 Accepted 4 April 2005

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been done for the Pontryagin maximum principle for both ﬁnite-dimensional systems [2] and inﬁnite-dimensional systems [3]. Unfortunately, the Pontryagin maximum principle provides only necessary conditions for the optimal control and it is usually not in feedback form. More seriously, ‘those sophisticated necessary conditions rarely give an insight into the structure of the optimal controls’ [4]. Nevertheless, due to the existence and uniqueness of the optimal control for most of practical problems, the Pontryagin maximum principle provides a possibility of seeking numerical solution for the optimal control at least in open-loop form. Basically, there are two ways of numerical methods solving optimal control problems through necessary conditions. It is generally believed that the indirect method that is mainly the multiple shooting method is the most powerful numerical method in seeking the optimal control through solving a two-point boundary-value problem obtained by the Pontryagin maximum principle. However, except for the complexity when the original problem involves inequality constraints of both state variables and controls, the big diﬃculty for shooting method is the ‘guess’ for the initial data to start the iterative numerical process. It demands that the user understands the essential of the problem well in physics, which is often not a trivial task [5]. For the direct method [6], the simpliﬁcation for the original problem leads to the fall of the reliability and accuracy, and when the degree of discretization and parameterization is very high, the work of computation stands out and the solving process gives rise to ‘curse of dimensionality’ [5]. Nevertheless, over the last two decades, much progress has been made on numerical methods for solving optimal control problems. Teo et al. [7] dealt with computational methods for several general classes of optimal control problems subject to constraints. A ﬁnite diﬀerence scheme was developed for solving a class of hyperbolic partial diﬀerential equations in Reference [8]. Teo et al. [9] presented a powerful computational method, where the switching time points can also be considered as decision variables. In addition, many softwares on the optimal control are available now, such as MISER 3.2 [10], which was used to compute optimal controls in Reference [11] where the performance of the computed optimal feedback control is compared with the open-loop optimal control with the same initial condition. For other developments in computing the optimal control, we refer to References [12–16]. In contrast to the Pontryagin maximum principle that deals with one extremal at a time, the dynamic programming method deals with, on the other hand, families of extremals. Once the Hamilton–Jacobi–Bellman (HJB for short) equation satisﬁed by the value function is established, the feedback control law can be found by solving this ﬁrst order non-linear partial diﬀerential equation. However, the new problem is raised immediately even from Pontryagin’s time that the HJB equation may have no classical solution no matter how smooth its coeﬃcients are. The fundamental turn comes when the viscosity solution of the HJB equation was introduced in the 1980’s [17]. The advantage of viscosity solution is that this kind of weak solution not only exists but also is unique for most of HJB equations. Once this mathematical basis for the dynamic programming approach was rigorously established, the seeking of optimal (feedback) control becomes possible [18]. For inﬁnite-dimensional optimal control problem by viscosity solution approach, we refer to References [19–23] and the references therein. Certainly, there are still other problems for seeking optimal control by solving viscosity solution of the HJB equation satisﬁed by the value function of the optimal control problem. One of the apparent problems is that in order to get the optimal feedback, we need not only the value function but also its gradient. The latter may not exist because the viscosity solution may not be diﬀerentiable. But for some big inertial systems, proper diﬀerence replacing the gradient of the value function still work for the numerical solution of the optimal feedback control. It Copyright # 2005 John Wiley & Sons, Ltd.

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was shown in Reference [24] that some simple diﬀerence scheme may produce numerically the viscosity solution. A recent interesting eﬀort in solving numerically the HJB equation can be found in References [25–27] for some ﬁnite-dimensional control problems. In this paper, we study the optimal control problem for a McKendrick-type age-structured population dynamics by viscosity solution approach. This is an inﬁnite-dimensional bilinear control problem. Many works have been done in the setting of partial diﬀerential equation models describing the population dynamics. In References [28,29], the optimal birth control problems of an age-structured population system of McKendrick-type were investigated and maximum principles for problems were established. The optimal control problem for a Gurtin– MacCamy-type system describing the evolution of an age-structured population was studied in Reference [30] and necessary optimality conditions were given. Anit-a [31] investigated an optimal harvesting problem for a non-linear age-dependent population system, and for some approximating problems, optimal controllers in feedback form were given by the dynamic programming method. Anit-a et al. [32] considered the optimal harvesting problem for the linear Lokta–McKendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. Necessary optimality conditions were presented and a numerical algorithm was developed to approximate the optimal control and optimal harvest. The model in this paper describes the population evolution of human beings with birth control, which was developed in Reference [33]. The HJB equation satisﬁed by the value function will be derived. It is shown that the value function is the viscosity solution of the HJB equation. The optimal birth feedback control is thus found through the value function and its gradient. A ﬁnite diﬀerence scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is veriﬁed numerically by comparing with diﬀerent controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989 [34]. The paper is organized as follows. In Section 2, the dynamic programming principle for the value function of the optimal control problem is established. Some other properties of the solution as well as the continuity of the value function are presented. Section 3 is devoted to show that the value function is just the viscosity solution of the corresponding HJB equation. In Section 4, the optimal feedback control is formulated by the value function under the smooth assumption. In the last section, Section 5, we formulate a ﬁnite diﬀerence scheme to the HJB equation of the population system with linear quadratic optimal control. The numerical solution of optimal feedback control is presented. The validity of the optimality of the obtained control is veriﬁed numerically. 2. PROBLEM FORMULATION AND DYNAMIC PROGRAMMING PRINCIPLE A McKendrick-type model of age-structured population dynamics developed in Reference [33] is a ﬁrst order partial diﬀerential equation with non-local boundary condition described by @pðr; tÞ @pðr; tÞ þ ¼ mðrÞpðr; tÞ; @t @r pðr; 0Þ ¼ p0 ðrÞ; pð0; tÞ ¼ bðtÞ

Z

05r5am ;

04r4am

t>0 ð1Þ

a2

bðrÞpðr; tÞ dr;

t50

a1

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where pðr; tÞ denotes the age density distribution at time t and age r for a closed population. mðrÞ is the relative mortality of the population, which is a non-negative measurable function and satisﬁes Z r Z am mðrÞ dr51 for r5am and mðrÞ dr ¼ 1 ð2Þ 0

0

am is the highest age ever attained by individuals of the population. bðrÞ ¼R kðrÞhðrÞ: kðrÞ is the a ratio of females and hðrÞ is the fertility pattern of females satisfying a12 hðrÞ dr ¼ 1: It is reasonable to assume that bðrÞ is bounded and measurable in ½a1 ; a2 ; the fecundity period of females, 05a1 5a2 5am ; and bðrÞ ¼ 0 outside ½a1 ; a2 : p0 ðrÞ is the initial distribution. bðtÞ is the speciﬁc fertility rate of the females at time t; which is considered to be the birth control of the population in macro-level. Let H ¼ L2 ð0; am Þ be the state space with the usual inner product h ; i and the inner product induced norm jj  jj: For any t > s50; let U½s; t ¼ fbðtÞ 2 ½b0 ; b1   Rþ jt 2 ½s; t; bðtÞ is measurable on ½s; tg Given T > 0 and p0 2 H; the optimal control problem is to ﬁnd an optimal control bn ðÞ 2 U½0; T such that Jðbn Þ ¼ JðbÞ ¼

inf

bðÞ2U½0;T

Z

T

Z

0

JðbÞ

am

Lðpðr; tÞ; bðtÞ; r; tÞ dr dt þ

0

Z

ð3Þ

am

f0 ðr; pðr; TÞÞ dr 0

where pðr; tÞ is the solution to Equation (1) corresponding to bðÞ; and L; f0 satisfy Z am  Z am         4C1 þ C2 jjpjj 8ðt; bÞ 2 ½0; T  ½b0 ; b1 ; ; f ðr; pðrÞÞ dr LðpðrÞ; b; r; tÞ dr 0     0 0 Z am  Z am         ½ f0 ðr; p1 ðrÞÞ  f0 ðr; p2 ðrÞÞ dr;  ½Lðp1 ðrÞ; b; r; tÞ  Lðp2 ðrÞ; b; r; tÞ dr  0

0

4C3 jjp1  p2 jj Z

p2H

am

8ðt; bÞ 2 ½0; T  ½b0 ; b1 ; p1 ; p2 2 H

LðpðrÞ; b; r; tÞ dr is continuous in ðt; p; bÞ 2 Rþ  H  ½b0 ; b1 

ð4Þ

0

for some constants Ci ; i ¼ 1; 2; 3: Deﬁne the time-dependent operators Ab ðtÞ as follows: Ab ðtÞfðrÞ ¼ f0 ðrÞ  mðrÞfðrÞ 8f 2 DðAb ðtÞÞ  DðAb ðtÞÞ ¼

fðrÞ j f; Ab ðtÞf 2 H; fð0Þ ¼ bðtÞ

Z



a2

ð5Þ

bðrÞfðrÞ dr a1

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Then Equation (1) can be written as a ﬁrst order evolution equation in H: @pðr; tÞ ¼ Ab ðtÞpðr; tÞ @t

ð6Þ

pðr; 0Þ ¼ p0 ðrÞ From now on, we also use Ab to denote the operator Ab ðtÞ when bðtÞ  b independent of time t; and the semigroup generated by Ab will be denoted as Tb ðtÞ: It is seen that Ab ðtÞ ¼ AbðtÞ : Deﬁne a family of evolution operators Tðt; s; bÞ; 04s4t51 by Rr 8  mðrÞ dr > > ; r5t  s fðr  t þ sÞe rtþs > > > Z a2 < Rt Rr  mðrÞ dr  mðrÞ dr Tðt; s; bÞfðrÞ ¼ bðt  rÞ bðtÞfðt  t þ s þ rÞe ttþsþr dte 0 ; r5t  s > > a1 > > > : 8f 2 H; 04t  s4a1   ½ðtsÞ=a   Y 1 ts Tðt; s; bÞ ¼ T t; s þ Tðs þ na1 ; s þ ðn  1Þa1 ; bÞ; t  s > a1 ð7Þ a1 ; b a1 n¼1 where ½x denotes the maximal integer not exceeding x: Tðt; s; bÞ is uniquely determined by fbðtÞ j t 2 ½s; tg: The following Theorem 1 comes from Reference [35], which can be veriﬁed directly by a straightforward computation. Theorem 1 (i) For each bðÞ 2 U½s; t; Tðt; s; bÞ is a bounded linear operator on H with Tðs; s; bÞ ¼ I; Tðt; s; bÞ ¼ Tðt; t0 ; bÞTðt0 ; s; bÞ

804s4t0 4t51

where I stands for the identity operator on H: (ii) Tð; ; bÞf is (strongly) continuous for each f 2 H uniformly in bðÞ 2 U½s; t: (iii) Let Tb1 ðtÞ be the C0 -semigroup generated by Ab1 on H: Then [33] jjTðt; s; bÞjj4jjTb1 ðtÞjj4MeoðtsÞ

for some M > 1 and o 2 R

(iv) The family of evolution operators fTðt; s; bÞg deﬁned by (7) has the iterative relation 8 Rr  mðrÞdr > rtþs > fðr  t þ sÞe ; < Z Rr Tðt; s; bÞfðrÞ ¼ a2  mðrÞ dr > > 0 bðt  rÞ bðtÞTðt  r; s; bÞfðtÞ dte ; :

r5t  s ð8Þ r5t  s

a1

(v) If bðtÞ 2 C 1 ðs; 1Þ; then Tðt; s; bÞ is an evolution system, i.e. d Tðt; s; bÞp0 ¼ Ab ðtÞTðt; s; bÞp0 8p0 2 DðAb ð0ÞÞ dt Copyright # 2005 John Wiley & Sons, Ltd.

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B.-Z. GUO AND B. SUN

(vi) For any t > s50 and any bi ðÞ 2 L2 ðs; tÞ ði ¼ 1; 2Þ jjTðt; s; b1 Þp0  Tðt; s; b2 Þp0 jj4jjbjj jjp0 jj jjb1 ðÞ  b2 ðÞjjL2 ðs;tÞ

ð9Þ

Theorem 2 Let Anb be the adjoint operator of Ab in H: Then there exists an operator C on H that is bounded, linear, self-adjoint and positive deﬁnite such that for each b 2 ½b0 ; b1 ; Anb C is a bounded linear operator on H and pﬃﬃﬃﬃﬃﬃ sup jjAnb Cjj41 þ b1 jjbjj am ð10Þ b2½b0 ;b1 

Furthermore, the set Dn deﬁned by Dn ¼ DðAnb Þ ¼ ffðrÞ j fðrÞ; f0 ðrÞ  mðrÞfðrÞ 2 H; fðam Þ ¼ 0g

ð11Þ

is independent of b 2 ½b0 ; b1  and dense in H: Proof Let the operator D be deﬁned as DfðrÞ ¼ f0 ðrÞ  mðrÞfðrÞ ð12Þ DðDÞ ¼ ffðrÞ j f; Df 2 H; fðam Þ ¼ 0g Then   Rr d  mðrÞ dr fðrÞe 0 dr dr 0 Z am Rr Rr  mðrÞ dr  mðrÞ dr ¼ ½f0 ðrÞ  mðrÞfðrÞe 0 dr ¼ hDf; e 0 i

fð0Þ ¼

Z

am

0

It is easy to check that Anb fðrÞ ¼ f0 ðrÞ  mðrÞfðrÞ þ bbðrÞfð0Þ ¼ DfðrÞ  bbðrÞhDf; e



Rr 0

mðrÞ dr

i

ð13Þ

So, DðAnb Þ ¼ DðDÞ: Furthermore, let C ¼ ðI þ Dn DÞð1=2Þ : Then it follows from Theorem 13.13 of Reference [36] that C is a linear bounded self-adjoint positive operator and jjDCjj41: By (13), pﬃﬃﬃﬃﬃﬃ jjAnb Cjj41 þ b1 jjbjj am This is the desired result.

&

Theorem 3 Let Dn be deﬁned as (11) and let qðÞ 2 Dn : Then hqðÞ; pð; tÞi is diﬀerentiable almost everywhere on ½0; T and d hqðÞ; pð; tÞi ¼ hAnb ðtÞqðÞ; pð; tÞi; dt Copyright # 2005 John Wiley & Sons, Ltd.

t 2 ½0; T a:e:

ð14Þ

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VISCOSITY SOLUTION APPROACH

where pðr; tÞ ¼ Tðt; 0; bÞp0 ðrÞ for any p0 2 H and bðÞ 2 U½0; T: Anb ðtÞ ¼ AnbðtÞ is the adjoint operator of Ab ðtÞ: Proof Let t 2 ð0; TÞ: For simplicity, we only consider the right derivative. Let d > 0; t þ d5a1 : Then from (8), it has 8 Rr  mðrÞ dr > > ; r5d < pðr  d; tÞe td Z a2 Rt Rr pðr; t þ dÞ ¼  mðrÞ dr  mðrÞ dr > > bðtÞpðt  r þ d; tÞe trþd dte 0 ; r5d : bðd þ t  rÞ a1

So, hqðÞ; pð; t þ dÞ  pð; tÞi ¼

Z

am

qðrÞ½pðr  d; tÞe d

þ

Z



Rr td



d

qðrÞ bðd þ t  rÞ

Z

0

¼

Z

pðr; tÞ½qðr þ dÞe 0

Z

 pðr; tÞ dr

a2

bðtÞpðt  r þ d; tÞe

þ

Rt trþd

mðrÞ dr

dte



Rr

mðrÞ dr

0

R rþd r

mðrÞ dr

 qðrÞ dr 

Z

d

pðr; tÞqðrÞ dr 

Z

  pðr; tÞ dr

qðd þ t  rÞe



R dþtr 0

mðrÞ dr

Z

a2

bðtÞpðt þ r  t; tÞe

bðrÞ

am

qðrÞpðr; tÞ dr am d

0



t



dþt

Z



a1

am d



mðrÞ dr



Rt tþrt

mðrÞ dr

 dt dr

a1

d

qðrÞpðr; tÞ dr 0

¼

Z

am d

pðr; tÞ½qðr þ dÞe



R rþd r

mðrÞ dr

 qðrÞ dr 

Z

tþd

am

qðrÞpðr; tÞ dr am d

0

þ

Z

 qðd þ t  rÞe



R dþtr 0

mðrÞ dr

t

Z

a2

bðrÞ

bðtÞpðt þ r  t; tÞe



Rt tþrt

mðrÞ dr

 dt dr

a1

and hence

pð; t þ dÞ  pð; tÞ lim qðÞ; d#0 d Z am Z ¼ pðr; tÞ½q0 ðrÞ  mðrÞqðrÞ dr þ qð0ÞbðtÞ 0

a2

bðtÞpðt; tÞ dt

a1

¼ hAnb ðtÞqðÞ; pð; tÞi for all such t that the above limit exists, which are dense in ½0; T: Copyright # 2005 John Wiley & Sons, Ltd.

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By virtue of Theorem 3, we will consider pðr; tÞ ¼ Tðt; 0; bÞp0 ðrÞ as the (weak) solution of (1) in the sense of (14), which is obtained by integrating (1) along the characteristics. The value function Vðt; p0 Þ for the optimal control problem is deﬁned by Z T Z a m  Z am Vðt; p0 Þ ¼ inf Lðpðr; tÞ; bðtÞ; r; tÞ dr dt þ f0 ðr; pðr; TÞÞ dr ð15Þ bðÞ2U½t;T

0

t

0

where pðr; tÞ ¼ Tðt; t; bÞp0 ðrÞ is the solution of (1) corresponding to bðÞ 2 U½t; T: Theorem 4(Dynamic programming principle) For any 05d5T  t; Z tþd Z am  Vðt; p0 Þ ¼ inf Lðpðr; tÞ; bðtÞ; r; tÞ dr dt þ Vðt þ d; pð; t þ dÞÞ bðÞ2U½t;tþd

Z

VðT; p0 Þ ¼

0

t

ð16Þ

am

f0 ðr; p0 ðrÞÞ dr 0

where pðr; tÞ ¼ Tðt; t; bÞp0 ðrÞ: Proof Let WðtÞ ¼

Z

tþd

Z

am

inf

bðÞ2U½t;tþd

 Lðpðr; tÞ; bðtÞ; r; tÞ dr dt þ Vðt þ d; pð; t þ dÞÞ

0

t

# 2 U½t; T such that It is obvious that Vðt; p0 Þ4WðtÞ: For any given e > 0; there exists a bðÞ Z

T

Z

am

# Lð#pðr; tÞ; bðtÞ; r; tÞ dr dt þ

0

t

Z

am

f0 ðr; p# ðr; TÞÞ dr4Vðt; p0 Þ þ e

0

# 0 ðrÞ: Since where p# ðr; tÞ ¼ Tðt; t; bÞp Z T Z Vðt þ d; p# ð; t þ dÞÞ4 tþd

am

# Lð#pðr; tÞ; bðtÞ; r; tÞ dr dt þ

0

Z

am

f0 ðr; p# ðr; TÞÞ dr

0

it follows that Z

Z

tþd

am

WðtÞ4 t

Z

T

Z

4 t

# Lð#pðr; tÞ; bðtÞ; r; tÞ dr dt þ Vðt þ d; p# ð; t þ dÞÞ

0 am

# Lð#pðr; tÞ; bðtÞ; r; tÞ dr dt þ

0

Z

am

f0 ðr; p# ðr; TÞÞ dr4Vðt; p0 Þ þ e

0

By the arbitrariness of e; we obtain that WðtÞ4Vðt; p0 Þ: Therefore WðtÞ ¼ Vðt; p0 Þ: Copyright # 2005 John Wiley & Sons, Ltd.

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Theorem 5 With M and o as in Theorem 1 and constants Ci ; i ¼ 1; 2; 3 in (4), the following assertions are true: (i) Vðt; p0 Þ4ð1 þ TÞC1 þ MC2 ð1 þ o1 ÞeoT jjp0 jj for all t 2 ½0; T; p0 2 H: (ii) Vðt; p1 Þ  Vðt; p2 Þj4ð1 þ o1 ÞMC3 eoT jjp1  p2 jj for all t 2 ½0; T; p1 ; p2 2 H: (iii) For any ﬁxed p0 ; Vðt; p0 Þ is continuous in t: Proof For any bðÞ 2 U½t; T; Z

jVðt; p0 Þj4

T

ðC1 þ C2 jjTðt; t; bÞp0 jjÞ dt þ C1 þ C2 jjTðT; t; bÞp0 jj

t

M C2 eoT jjp0 jj þ C1 þ MC2 eoT jjp0 jj o   1 MC2 eoT jjp0 jj ¼ ð1 þ TÞC1 þ 1 þ o

4 C1 T þ

# 2 U½t; T such that This is (i). For any e > 0; take bðÞ Z T Z am Z # Vðt; p2 Þ5 Lð#p2 ðr; tÞ; bðtÞ; r; tÞ dr dt þ 0

t

am

f0 ðr; p# 2 ðr; TÞÞ dr  e

0

# 2 ðrÞ: Let p# 1 ðr; tÞ ¼ Tðt; t; bÞp # 1 ðrÞ: Then where p# 2 ðr; tÞ ¼ Tðt; t; bÞp Z T Z am Z am # Vðt; p1 Þ  Vðt; p2 Þ4 Lð#p1 ðr; tÞ; bðtÞ; r; tÞ dr dt þ f0 ðr; p# 1 ðr; TÞÞ dr t



0

Z

T

Z

0 am

# Lð#p2 ðr; tÞ; bðtÞ; r; tÞ dr dt 

0

t

Z

am

f0 ðr; p# 2 ðr; TÞÞ dr þ e

0

Z am    # #  ½Lð#p1 ðr; tÞ; bðtÞ; r; tÞ  Lð#p2 ðr; tÞ; bðtÞ; r; tÞ dr dt 4  0 t Z am    ½ f0 ðr; p# 1 ðr; TÞÞ  f0 ðr; p# 2 ðr; TÞÞ dr þ e þ  Z

T

0

Z 4 C3

T

jj#p1 ð; tÞ  p# 2 ð; tÞjj dt þ C3 jj#p1 ð; TÞ  p# 2 ð; TÞjj þ e

t

4

M C3 eoT jjp1  p2 jj þ C3 MeoT jjp1  p2 jj þ e ¼ ð1 þ o1 ÞMC3 eoT jjp1  p2 jj þ e o

Assertion (ii) is thus proved. Notice that in the last line above, we used the fact from (iii) of Theorem 1 that # jjp1  p2 jj4MeoðttÞ jjp1  p2 jj jj#p2 ð; tÞ  p# 2 ð; tÞjj4jjTðt; t; bÞjj Copyright # 2005 John Wiley & Sons, Ltd.

8t 2 ½t; T

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B.-Z. GUO AND B. SUN

% 2 U½t2 ; T such that Finally, for given p0 2 H and any t1 > t2 2 ½0; T; there exists a bðÞ Z T Z am Z am % Lðp% 2 ðr; tÞ; bðtÞ; r; tÞ dr dt þ f0 ðr; p% 2 ðr; TÞÞ dr  e Vðt2 ; p0 Þ5 0

t2

0

% 0 ðrÞ: Let p% 1 ðr; tÞ ¼ Tðt; t1 ; bÞp % 0 ðrÞ: Then where p% 2 ðr; tÞ ¼ Tðt; t2 ; bÞp Z T Z am Z am % Vðt1 ; p0 Þ  Vðt2 ; p0 Þ4 Lðp% 1 ðr; tÞ; bðtÞ; r; tÞ dr dt þ f0 ðr; p% 1 ðr; TÞÞ dr 0

t1



Z

T

Z

t1

Z

Z

T

Z

Z

am

0

t1

þ

am 0

t2

þ

am

0

t2

¼

0

Z

am

0

% Lðp% 2 ðr; tÞ; bðtÞ; r; tÞ dr dt 

Z

am 0

f0 ðr; p% 2 ðr; TÞÞ dr þ e

% Lðp% 2 ðr; tÞ; bðtÞ; r; tÞ dr dt % % ½Lðp% 1 ðr; tÞ; bðtÞ; r; tÞ  Lðp% 2 ðr; tÞ; bðtÞ; r; tÞ dr dt

½ f0 ðr; p% 1 ðr; TÞÞ  f0 ðr; p% 2 ðr; TÞÞ dr þ e

4 ðC1 þ C2 Me

oT

jjp0 jjÞðt1  t2 Þ þ C3

Z

T

% 0  Tðt; t2 ; bÞp % 0 jj dt jjTðt; t1 ; bÞp

t1

% 0  TðT; t2 ; bÞp % 0 jj þ e þ C3 jjTðT; t1 ; bÞp % 0  p0 jj þ e 4 ðC1 þ C2 MeoT jjp0 jjÞðt1  t2 Þ þ C3 ð1 þ TÞMeoT jjTðt1 ; t2 ; bÞp % 0  p0 jj ! 0: Now Hence, as t1  t2 ! 0; Vðt1 ; p0 Þ  Vðt2 ; p0 Þ ! 0; provided that jjTðt1 ; t2 ; bÞp assume t1  t2 5a1 : We have Rr Z am  mðrÞ dr % 0  p0 jj2 ¼ ½p0 ðr  t1 þ t2 Þe rt1 þt2  p0 ðrÞ2 dr jjTðt1 ; t2 ; bÞp t1 t2

Z

þ

t1 t2



% 1  rÞ bðt

Z

0

e



Z

a2

bðtÞp0 ðt  t1 þ t2 þ rÞ

a1

Rt tt1 þt2 þr

am

mðrÞ dr

dte



Rr 0

mðrÞ dr

2  p0 ðrÞ dr

½p0 ðr  t1 þ t2 Þ  p0 ðrÞ2 dr

42 t1 t2

þ2

Z

am

 R r  2  mðrÞ dr e rt1 þt2  1 p0 ðr  t1 þ t2 Þ dr

t1 t2

þ

2b21 jjbjjða2

þ2

Z

t1 t2

 a1 Þ1=2 jjp0 jj jt1  t2 j

jp0 ðrÞj2 dr ! 0

uniformly for b% as t1  t2 ! 0

0

So, (iii) is valid. The proof is complete. Copyright # 2005 John Wiley & Sons, Ltd.

& Optim. Control Appl. Meth. 2005; 26:229–254

VISCOSITY SOLUTION APPROACH

239

Remark 1 The proof of (iii) of Theorem 5 shows that for any ﬁxed p0 ; jjTðt1 ; t2 ; bÞp0  p0 jj tends to zero uniformly for bðÞ 2 U½t2 ; t1  as t1  t2 ! 0: We will use this fact frequently in the sequel.

3. HJB EQUATION AND VISCOSITY SOLUTION For brevity in notation, we rewrite (6) as dPðtÞ ¼ Ab ðtÞPðtÞ dt

ð17Þ

Pð0Þ ¼ P where PðtÞ ¼ pð; tÞ; P ¼ p0 ðÞ: The value function is rewritten as Z T Vðt; PÞ ¼ inf f ðt; PðtÞ; bðtÞÞ dt þ cðPðTÞÞ bðÞ2U½t;T

where

t

PðtÞ ¼ Tðt; t; bÞP f ðt; PðtÞ; bðtÞÞ ¼ cðPðTÞÞ ¼

Z

am

Lðpðr; tÞ; bðtÞ; r; tÞ dr 0

Z

am

f0 ðr; pðr; TÞÞ dr 0

Theorem 6 If Vðt; PÞ 2 C 1 ð½0; T  HÞ; then V satisﬁes the following HJB equation: Vt ðt; PÞ þ inf fhVp ðt; PÞ; Ab Pi þ f ðt; P; bÞg ¼ 0 b2½b0 ;b1 

VðT; PÞ ¼ cðPÞ 8t 2 ½0; T; P 2

\

DðAb Þ

ð18Þ

b2½b0 ;b1 

Proof For any 05d5T  t and b 2 ½b0 ; b1 ; the dynamic programming principle implies that Z tþd Vðt; PÞ4 f ðt; Tb ðt  tÞP; bÞ dt þ Vðt þ d; Tb ðt þ dÞPÞ

ð19Þ

t

If P 2 DðAb Þ; since Tb ðtÞP ¼ P; it follows that P  Tb ðt þ dÞP ¼ P  ½P þ dAb P þ 8ðdÞ ¼ dAb P þ 8ðdÞ Copyright # 2005 John Wiley & Sons, Ltd.

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B.-Z. GUO AND B. SUN

Hence, Vðt; PÞ  Vðt þ d; Tb ðt þ dÞPÞ ¼ Vt ðt; PÞd  hVp ðt; PÞ; Ab Pid þ 8ðdÞ from which (19) can be rewritten as 1 Vt ðt; PÞ  hVp ðt; PÞ; Ab Pi4 d

Z

tþd

f ðt; Tb ðt  tÞP; bÞ dt þ 8

t

ðdÞ d

Letting d ! 0; we obtain that Vt ðt; PÞ  hVp ðt; PÞ; Ab Pi4f ðt; P; bÞ So, Vt ðt; PÞ þ inf fhVp ðt; PÞ; Ab Pi þ f ðt; P; bÞg50 b2½b0 ;b1 

# 2 Uðt; t þ dÞ such that On the other hand, for any given e > 0; there exists a bðÞ Z tþd # # # þ dÞÞ  e f ðt; PðtÞ; bðtÞÞ dt þ Vðt þ d; Pðt Vðt; PÞ5

ð20Þ

t

# # where PðtÞ ¼ Tðt; t; bÞP: By (vi) of Theorem 1, we may assume without loss of generality that # 2 C 1 ½t; t þ d \ U½t; t þ d and bðtÞ # ¼ b for any given b 2 ½b ; b : If P 2 T bðÞ 1 2 b2½b0 ;b1  DðAb Þ; then # þ dÞ is diﬀerentiable with respect to d and Pðt d # Pðt þ dÞjd¼0 ¼ Ab# ðtÞP dd This together with (20) gives # Vt ðt; PÞ  hVp ðt; PÞ; Ab# ðtÞPi5f ðt; P; bðtÞÞ e i.e. # Vt ðt; PÞ þ hVp ðt; PÞ; AbðtÞ # Pi þ f ðt; P; bðtÞÞ4e So, Vt ðt; PÞ þ inf fhVp ðt; PÞ; Ab Pi þ f ðt; P; bÞg4e b2½b0 ;b1 

and hence Vt ðt; PÞ þ inf fhVp ðt; PÞ; Ab Pi þ f ðt; P; bÞg40 b2½b0 ;b1 

proving the theorem.

&

Remark 2 We shall denote D¼

\



0

DðAb Þ* fðrÞ; f ðrÞ þ mðrÞfðrÞ 2 H j fð0Þ ¼

b2½b0 ;b1 

Copyright # 2005 John Wiley & Sons, Ltd.

Z



a2

bðrÞfðrÞ dr ¼ 0

ð21Þ

a1

Optim. Control Appl. Meth. 2005; 26:229–254

VISCOSITY SOLUTION APPROACH

241

From now on, we always assume that Ab ðtÞ is dissipative for all b 2 ½b0 ; b1 : The assumption is reasonable since instead of (17), we may consider dQðtÞ ¼ ½Ab ðtÞ  CQðtÞ dt

ð22Þ

Qð0Þ ¼ P and then Ab ðtÞ  C with C ¼ 12 jjbjj2 is dissipative for each b 2 ½b0 ; b1  because RehAb ðtÞP; Pi412 jjbjj2 hP; Pi Since QðtÞ ¼ PðtÞeCt ; the value function becomes Z T  Vðt; PÞ ¼ inf f ðt; QðtÞeCt ; bðtÞÞ dt þ cðeCT QðTÞÞ bðÞ2U½t;T

t

where QðtÞ ¼ eCt Tðt; t; bÞP: First, we give a deﬁnition for the solution of the HJB equation (18) in the ‘viscosity sense’ [37]. Let O be an open set of H and set USCð½0; T   OÞ ¼ fupper-semicontinuous mappings u : ½0; T   O ! Rg LSCð½0; T   OÞ ¼ flower-semicontinuous mappings u : ½0; T   O ! Rg Deﬁnition 1 Uðt; PÞ 2 USCð½0; T  OÞ (respectively, Uðt; PÞ 2 LSCð½0; T   OÞÞ is a subsolution (respectively, supersolution) of (18) on ½0; T  O if for every test function F ¼ j þ g; j; jp 2 Cð½0; T  O; RÞ; g 2 C1 ðO; RÞ satisfying the condition (23) and the condition (24) below: 8 Rangeðjp Þ  Dn > > < the mapping ðt; PÞ ! hAnb jP ðt; PÞ; Pi > > : ð23Þ from ½0; T  O to R is equicontinuous in b 2 ½b0 ; b1  8 > there exists a h* : ½0; 1Þ ! R such that > > < h* is non-decreasing; h* 0 ð0Þ ¼ 0 and > > > : gðPÞ ¼ hðjjPjjÞ * 8P 2 H

ð24Þ

and for the local maximum point (respectively, the minimum point) ðt; PÞ of U  F (respectively, U þ F), we have jt ðt; PÞ þ inf fhAnb jp ðt; PÞ; Pi þ f ðt; P; bÞg50 b2½b0 ;b1 

Copyright # 2005 John Wiley & Sons, Ltd.

ð25Þ

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B.-Z. GUO AND B. SUN

(respectively, jt ðt; PÞ þ inf fhAnb jp ðt; PÞ; Pi þ f ðt; P; bÞg40Þ b2½b0 ;b1 

ð26Þ

Uðt; PÞ 2 Cð½0; T  OÞ is a viscosity solution of (18) if it is both a subsolution and a supersolution on ½0; T  O: Theorem 7 The value function Vðt; PÞ is a viscosity solution of the HJB equation (18). Proof Assume that ðt; PÞ is a local maximum point of V  F for the test function F: Then there exists a d > 0 such that Vðt; QÞ  Fðt; QÞ4Vðt; PÞ  Fðt; PÞ 8jt  tj; jjQ  Pjj5d i.e. Fðt; QÞ  Fðt; PÞ5Vðt; QÞ  Vðt; PÞ 8jt  tj; jjQ  Pjj5d By Remark 1, there exists a e > 0 such that for all b 2 ½b0 ; b1 ; Fðt þ d0 ; Tb ðd0 ÞPÞ  Fðt; PÞ5 Vðt þ d0 ; Tb ðd0 ÞPÞ  Vðt; PÞ 5

Z

tþd0

f ðt; Tb ðt  tÞP; bÞ dt

as 05d0 5e

t

The last inequality comes from Theorem 4. Notice that under conditions (23) and (24), we have

jðt þ d0 ; Tb ðd0 ÞQÞ  jðt; QÞ ¼

Z

d0 0

¼

Z

d0

jt ðt þ r; Tb ðrÞQÞ dr þ

0

Z 0

d jðt þ r; Tb ðrÞQÞ dr dr

d0

hAnb jp ðt þ r; Tb ðrÞQÞ; Tb ðrÞQi dr

ð27Þ

that holds for all t; d0 and all Q by ﬁrst considering those Q 2 DðAb Þ and then for Q 2 H via the density argument. Also (24) implies that gðTb ðd0 ÞQÞ  gðQÞ40 Copyright # 2005 John Wiley & Sons, Ltd.

ð28Þ

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VISCOSITY SOLUTION APPROACH

for all Q 2 H; and all b 2 ½b0 ; b1  by ﬁrst for Q 2 DðAb Þ and then via the density argument. Putting things all together, we have Z tþd0  f ðt; Tb ðt  tÞP; bÞ dt4 Fðt þ d0 ; Tb ðd0 ÞPÞ  Fðt; PÞ t

4 jðt þ d0 ; Tb ðd0 ÞPÞ  jðt; PÞ ¼

Z

d0

jt ðt þ t; Tb ðtÞPÞ dt 0

þ

Z

d0

hAnb jp ðt þ t; Tb ðtÞPÞ; Tb ðtÞPi dt

0

for all 05d0 5e: Dividing d0 on both sides above and letting d0 go to zero, we obtain jt ðt; PÞ þ hAnb jp ðt; PÞ; Pi þ f ðt; P; bÞ50 And hence (25) holds by the arbitrariness of b: Next, assume that ðt; PÞ is a local minimum point of V þ F for the test function F: Then there exists a e > 0 such that Fðt; PðtÞÞ þ Fðt; PÞ4Vðt; PðtÞÞ  Vðt; PÞ

ð29Þ

for all jt  tj5e and all admissible control bðÞ 2 U½t; t þ e; in which PðtÞ ¼ Tðt; t; bÞP: Let 05d0 5e: By Theorem 4,  Z tþd0  Vðt; PÞ ¼ inf f ðt; PðtÞ; bðtÞÞ dt þ Vðt þ d0 ; Pðt þ d0 ÞÞ bðÞ2U½t;tþd0 

t

Z 5

bðÞ2U½t;tþd0 



tþd0

inf

f ðt; PðtÞ; bðtÞÞ dt þ Vðt; PÞ  Fðt þ d0 ; Pðt þ d0 ÞÞ þ Fðt; PÞ

t

So, Z

f ðt; PðtÞ; bðtÞÞ dt  Fðt þ d0 ; Pðt þ d0 ÞÞ þ Fðt; PÞ 40

inf

bðÞ2U½t;tþd0 



tþd0

ð30Þ

t

holds for all 05d0 5e: As (27) and (28), we have Z d0 jðt þ d0 ; Pðt þ d0 ÞÞ  jðt; PÞ ¼ jt ðt þ t; Pðt þ tÞÞ dt 0

þ

Z

d0 0

hAnb ðtÞjp ðt þ t; Pðt þ tÞÞ; Pðt þ tÞi dt

and gðPðt þ d0 ÞÞ  gðPÞ40 for all bðÞ 2 U½t; t þ h: Then (30) implies that  Z tþd0  inf f ðt; PðtÞ; bðtÞÞ dt  jðt þ d0 ; Pðt þ d0 ÞÞ þ jðt; PÞ 40 bðÞ2U½t;tþd0 

t

Copyright # 2005 John Wiley & Sons, Ltd.

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B.-Z. GUO AND B. SUN

or Z

tþd0

f ðt; PðtÞ; bðtÞÞ dt 

inf

bðÞ2U½t;tþd0 

d0

jt ðt þ t; Pðt þ tÞÞ dt

0

t



Z

Z



d0 n

0

hAb ðtÞjp ðt þ t; Pðt þ tÞÞ; Pðt þ tÞi dt 40

ð31Þ

Suppose on the contrary that jt ðt; PÞ þ inf fhAnb jp ðt; PÞ; Pi þ f ðt; P; bÞg > 2W b2½b0 ;b1 

for some W > 0: Then for all b 2 ½b0 ; b1 ; jt ðt; PÞ  hAnb jp ðt; PÞ; Pi þ f ðt; P; bÞ > 2W By Remark 1, there exists a d 2 ð0; eÞ such that jt ðt; Pðt þ tÞÞ  hAnb ðtÞjp ðt; Pðt þ tÞÞ; Pðt þ tÞi þ f ðt þ t; Pðt þ tÞ; bðtÞÞ > W for all t5d and all bðÞ 2 U½t; t þ d: Contradicts to (31). So, V is a supersolution. The proof is complete. &

4. OPTIMAL FEEDBACK CONTROL A necessary condition as well as a characterization of optimal trajectory–control pairs can be formulated using the value function Vðt; PÞ: Theorem 8 Let Vðt; PÞ be the value function. Then for any trajectory–control pair ðPn ðtÞ; bn ðtÞÞ; Pn ðtÞ ¼ Tðt; 0; bn ÞP; the function Z T t ! Vðt; Pn ðtÞÞ  f ðt; Pn ðtÞ; bn ðtÞÞ dt ð32Þ t

is non-decreasing in ½0; T: Moreover, ðPn ðÞ; bn ðÞÞ is an optimal pair if and only if the above function is constant on ½0; T: Consequently, if Vðt; PÞ 2 C 1 ð½0; T  HÞ; bn ðÞ 2 C 1 ½0; T; Pn ð0Þ 2 DðAbn ð0Þ Þ; then ðPn ðÞ; bn ðÞÞ is an optimal pair if and only if Vt ðt; Pn ðtÞÞ þ hVp ðt; Pn ðtÞÞ; Abn ðtÞ Pn ðtÞi þ f ðt; Pn ðtÞ; bn ðtÞÞ ¼ 0 ð33Þ VðT; Pn ðTÞÞ ¼ cðPn ðTÞÞ

8t 2 ½0; T

Proof The ﬁrst assertion is a direct consequence of the dynamic programming principle. The last assertion comes from the diﬀerentiation of (32) with respect to time t and (v) of Theorem 1. Copyright # 2005 John Wiley & Sons, Ltd.

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245

As for the second assertion, if ðPn ðtÞ; bn ðtÞÞ is not optimal then there exists another trajectory– control pair ðPðtÞ; bðtÞÞ; PðtÞ ¼ Tðt; 0; bÞP; such that Z

T

f ðt; PðtÞ; bðtÞÞ dt þ cðPðTÞÞ5

Z

0

T

f ðt; Pn ðtÞ; bn ðtÞÞ dt þ cðPn ðTÞÞ 0

and hence Vð0; PÞ ¼ Vð0; Pn ð0ÞÞ ¼ Vð0; Pð0ÞÞ4

Z Z

T

f ðt; PðtÞ; bðtÞÞ dt þ cðPðTÞÞ

0 T

f ðt; Pn ðtÞ; bn ðtÞÞ dt þ cðPn ðTÞÞ

5 0

¼

Z

T

f ðt; Pn ðtÞ; bn ðtÞÞ dt þ VðT; Pn ðTÞÞ

0

i.e. Vð0; Pn ð0ÞÞ 

Z

T

f ðt; Pn ðtÞ; bn ðtÞÞ dt5VðT; Pn ðTÞÞ

0

RT

n

Thus, Vðt; Pn ðtÞÞ  t f ðt; Pn ðtÞ; b ðtÞÞ dt is not constant on ½0; T: On the other hand, if it is not constant on ½0; T then the ﬁrst assertion implies that Z

T

f ðt; Pn ðtÞ; bn ðtÞÞ dt þ cðPn ðTÞÞ

Vð0; PÞ5 0

RT Let e ¼ 0 f ðt; Pn ðtÞ; bn ðtÞÞ dt þ cðPn ðTÞÞ  Vð0; PÞ > 0: By the deﬁnition of the value function, there exists a ðPðtÞ; bðtÞÞ such that Z

T

f ðt; PðtÞ; bðtÞÞ dt þ cðPðTÞÞ  e

Vð0; PÞ > 0

Hence, Z

T

f ðt; PðtÞ; bðtÞÞ dt þ cðPðTÞÞ5 0

Z

T

f ðt; Pn ðtÞ; bn ðtÞÞ dt þ cðPn ðTÞÞ 0

which implies that ðPn ðtÞ; bn ðtÞÞ is not optimal, proving the required result.

&

The last assertion of Theorem 8 will lead to the classical veriﬁcation theorem, which plays an important role in testing for optimality of a given admissible pair, and more importantly, in constructing optimal feedback controls [38,39]. In fact, suppose the value function Vðt; PÞ is smooth. Since Vðt; PÞ is a viscosity solution of the HJB equation, we have Vt ðt; PÞ þ inf fhVp ðt; PÞ; Ab Pi þ f ðt; P; bÞg ¼ 0 b2½b0 ;b1 

Copyright # 2005 John Wiley & Sons, Ltd.

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B.-Z. GUO AND B. SUN

then the condition of optimality (33) is equivalent to hVp ðt; Pn ðtÞÞ; Abn ðtÞ Pn ðtÞi þ f ðt; Pn ðtÞ; bn ðtÞÞ ¼ inf fhVp ðt; Pn ðtÞÞ; Ab Pn ðtÞi þ f ðt; Pn ðtÞ; bÞg b2½b0 ;b1 

:¼ HðPn ðtÞ; Vp ðt; Pn ðtÞÞÞ

for t 2 ½0; T a:e:

ð34Þ

Now, let SðZÞ :¼ arg min fhVz ðt; ZÞ; Ab Zi þ f ðt; Z; bÞg b2½b0 ;b1 

¼ fb 2 ½b0 ; b1  j HðZ; Vz ðt; ZÞÞ ¼ hVz ðt; ZÞ; Ab Zi þ f ðt; Z; bÞg

ð35Þ

Then (34) says that bn ðÞ 2 U½0; T is an optimal control for the initial state p0 if and only if bn ðtÞ 2 SðPn ðtÞÞ

for t 2 ½0; T a:e:

ð36Þ

where Pn ðtÞ ¼ Tðt; 0; bn Þp0 : The formula for ﬁnding the optimal feedback control is (36). It should be indicated that (36) is true only when the value function is diﬀerentiable. However, this is not true usually in the sense of analyticity. Nevertheless, it is still an important formula to construct numerically the optimal feedback control, which will be tested in the next section.

5. FINITE DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF OPTIMAL FEEDBACK CONTROL In this section, we will limit ourselves to the following linear quadratic optimal control problem: Jðbn Þ ¼

inf

bðÞ2U½0;T

JðbÞ

1 ¼ inf bðÞ2U½0;T 2

Z

T

% 2 dt þ ½bðtÞ  bðtÞ

0

Z

am

0

2

½pðr; TÞ  p% ðrÞ dr

 ð37Þ

% denotes the mean critical fertility rate of females in an ideal society and p% ðrÞ denotes where bðtÞ the stable age density distribution of the population with the zero increasing rate. Corresponding to (3) and (18), we have 1 % 2; Lðpðr; tÞ; bðtÞ; r; tÞ ¼ ½bðtÞ  bðtÞ 2 f ðt; pð; tÞ; bðtÞÞ ¼

am % 2; ½bðtÞ  bðtÞ 2

Copyright # 2005 John Wiley & Sons, Ltd.

1 f0 ðr; pðr; TÞÞ ¼ ½pðr; TÞ  p% ðrÞ2 2 Z am 1 ½pðr; TÞ  p% ðrÞ2 dr cðpð; TÞÞ ¼ 2 0 Optim. Control Appl. Meth. 2005; 26:229–254

247

VISCOSITY SOLUTION APPROACH

and the HJB equation can be rewritten as n o am % 2 ¼0 ½b  bðtÞ hVp ðt; pÞ; Ab pi þ Vt ðt; pÞ þ inf b2½b0 ;b1  2 Z am \ 1 ½pðrÞ  p% ðrÞ2 dr 8t 2 ½0; T; p 2 DðAb Þ VðT; pÞ ¼ 2 0 b2½b ;b  0

ð38Þ

1

Now we come to solve the above equation numerically. To do so, let tj ¼ T þ jDt; j ¼ 0; 1; 2; . . . ; N in which Dt ¼ T=N and N is an integer. For given e > 0; we use the following approximation: hVp ðt; pÞ; Ab pi ¼

    Ab p jjAb pjj Ab p jjAb pjj  V t; p þ e Vp ðt; pÞ; e  Vðt; pÞ jjAb pjj e jjAb pjj e

For the initial state p0 let pi ¼ pi1 þ ðAb pi1 =jjAb pi1 jjÞ e; i ¼ 1; 2; . . . ; M: Approximate (38) by the diﬀerence equation: j Vijþ1  Vij Vij  Vi1 am j % j 2 þ jjAb j pi jj þ ½b  b  ¼ 0 i Dt e 2 i ( ) jþ1 Vijþ1  Vi1 am jþ1 jþ1 2 % bi 2 arg inf þ jjAb pi jj þ ½b  b  b2½b0 ;b1  e 2

ð39Þ

for i ¼ 1; 2; . . . ; M and j ¼ 0; 1; 2; . . . ; N; where Vij  Vðtj ; pi Þ: Referring to Li and Feng [40], the following suﬃcient condition for the stability of the diﬀerence scheme will be assumed: jDtj max jjAb pi jj41 e 14i4M

ð40Þ

Set aij ¼ ðDt=eÞjjAb j pi jj: We give the algorithm for the numerical solution of (38) as follows. i

Algorithm of solving the HJB equation. Step 1: Initialization. Set Vi0 ¼ VðT; pi Þ ¼ cðpi Þ; b0i 2 arg inf



b2½b0 ;b1 

pi ¼ pi1 þ

cðpi Þ  cðpi1 Þ am jjAb pi jj þ ½b  b% 0 2 e 2

:¼ arg inf fHðb; pÞg; b2½b0 ;b1 

Copyright # 2005 John Wiley & Sons, Ltd.

Ab pi1 e jjAb pi1 jj

i ¼ 1; 2; . . . ; M



ð41Þ

Optim. Control Appl. Meth. 2005; 26:229–254

248

B.-Z. GUO AND B. SUN

Here the formula for b0i comes from

n

bðt; pðÞÞ 2 arg inf

b2½b0 ;b1 

hVp ðt; pÞ; Ab pi þ

o am % 2 ½b  bðtÞ 2

ð42Þ

Step 2: Iteration. By (39), j Vijþ1 ¼ ð1  aij ÞVij þ aji Vi1 

( bjþ1 i

2 arg inf

b2½b0 ;b1 

am j % j 2 ðb  b Þ Dt 2 i

jþ1 Vijþ1  Vi1 am jjAb pi jj þ ½b  b% jþ1 2 e 2

)

for i ¼ 1; 2; . . . ; M and j ¼ 0; 1; 2; . . . ; N  1: It is seen that 8 b1 if Hðb1 ; pÞ ¼ minfHðb1 ; pÞ; Hðb0 ; pÞ; Hðb% 0 ; pÞg > > < b0i ¼ b0 if Hðb0 ; pÞ ¼ minfHðb1 ; pÞ; Hðb0 ; pÞ; Hðb% 0 ; pÞg > > : *0 b if Hðb* 0 ; pÞ ¼ minfHðb1 ; pÞ; Hðb0 ; pÞ; Hðb% 0 ; pÞg

ð43Þ

ð44Þ

for i ¼ 1; 2; . . . ; M: The similar formula can be also obtained for bji ; i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N  1: From (42), the optimal feedback control is n o am % 2 ½b  bðtÞ bnp0 ðt; pn ð; tÞÞ 2 arg inf hVp ðt; pn ð; tÞÞ; Ab pn ð; tÞi þ ð45Þ b2½b0 ;b1  2 in which pn ð; tÞ is the optimal trajectory of the system. Because it involves the optimal trajectory in (45), we need to ﬁnd the solution of (1). Here we utilize the diﬀerence scheme to compute the approximate solution of the initial value problem (1) (see Reference [33]). Adopting the equally spaced grid method, we have pi;j  pi1;j pi;j  pi;j1 þ ¼ mi pi;j ; Ds Ds pi;0 ¼ p0 ðiDsÞ; p0;j ¼ bj Ds

a2 X

14j4J; 14i42K

04i42K bi pi; j ;

ð46Þ

14j4J

i¼a1

in which bi ¼ bðiDsÞ; bj ¼ bð jDsÞ; i ¼ a1 ; . . . ; a2 ; 2KDs ¼ am ; JDs ¼ T: It follows that pi; j ¼

pi1; j þ pi; j1 ; 2 þ mi Ds

pi;0 ¼ p0 ðiDsÞ; p0;j ¼ bj

a2 X

14j4J; 14i42K

04i42K

bi pi;j Ds;

ð47Þ

14j4J

i¼a1

Copyright # 2005 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2005; 26:229–254

249

VISCOSITY SOLUTION APPROACH

Next we give steps of ﬁnding the optimal feedback control in detail. It focuses on the solving the diﬀerence scheme (47) to obtain the optimal trajectory. Moreover, in the process, the algorithm of solving the HJB equation will be called frequently to get the corresponding feedback control function. Steps of ﬁnding the optimal feedback control. Step 1: Call the algorithm of solving the HJB equation to get the feedback control function bðt; p0 Þ: Substitute bð0; p0 Þ into (47) to get the optimal trajectory pn ð; s1 Þ; s1 ¼ Ds: Step 2: Take pn ð; s1 Þ as the initial data p0 as in the ﬁrst step, and call the algorithm of solving the HJB equation again to get the feedback control bðs1 ; pn ð; s1 ÞÞ: Substitute bðs1 ; pn ð; s1 ÞÞ into (47) to get the optimal trajectory pn ð; s2 Þ; s2 ¼ 2Ds: Step 3: Repeating the processes above, we get all feedback control functions bðsj ; pn ð; sj ÞÞ; sj ¼ jDs; j ¼ 0; 1; . . . ; J; that is to say, bnp0 ðt; pn ð; tÞÞ ¼ fbð0; p0 ðÞÞ; bðs1 ; pn ð; s1 ÞÞ; bðs2 ; pn ð; s2 ÞÞ; . . . ; bðT; pn ð; TÞÞg

ð48Þ

which is the optimal feedback control. Now we are in a position to ﬁnd the numerical solution of linear quadratic optimal control for the Chinese population based on the scheme (41), (43) and (47). The initial data are extracted from the Chinese population sampling census in 1989 [34], which are plotted by MATLAB 6.1 as Figure 4 (age-structure of females); Figure 5 (the relative mortality); Figure 6 (the age-structure of the total population). The age-structure of an ideal society taken from Reference [33] are listed in Table I, which is used to get p% ðrÞ by multiplying the proportion in the table with the total ideal population Nsum ¼ 1 400 000: The fertility pattern hðrÞ is approximated by the Gamma density distribution curve in statistics [33] in which the peak value of the fertility age is assumed to be 24. Other parameters are listed as follows: T ¼ 25;

b0 ¼ 1;

a1 ¼ 15;

a2 ¼ 49; am ¼ 99;

b1 ¼ 3;

% ¼2 bðtÞ e ¼ 0:01

Table I. The age structure of an ideal society. Age, r

Proportion

Age, r

Proportion

0 1–5 6–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45

0.013 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.064 0.064

46–50 51–55 56–60 61–65 66–70 71–75 76–80 81–85 > 85

0.063 0.062 0.061 0.057 0.053 0.047 0.037 0.024 0.003

Copyright # 2005 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2005; 26:229–254

250

B.-Z. GUO AND B. SUN

In addition, due to the limitation of computer memory, the data in Figure 4 and Figure 6 and Nsum are divided by 10 000. All computations are performed in Visual C++ 6.0 and numerical results are plotted by MATLAB 6.1. Figure 1 is the value function Vðt; pn ð; tÞÞ and Figure 2 is the optimal feedback control bnp0 ðt; pn ð; tÞÞ: To end the paper, we have to check the optimality of the numerical solution of optimal feedback control. This is done by comparing the cost functional Jðbnp0 ðt; pn ð; tÞÞÞ of obtained optimal control–trajectory pair with that of arbitrarily chosen control bp0 ðtÞ; Jðbp0 ðtÞÞ under the

value function : v(t,p*(.,t))

22

x 10 5

v

4 3 2 1 0 0

5

10

15

t

20

25

Figure 1. The value function Vðt; pn ð; tÞÞ:

optimal feedback control : β*p0(t,p*(.,t)) 3 2.8 2.6 2.4 β

2.2 2 1.8 1.6 1.4 1.2 1 0

5

10

15

20

25

t

Figure 2. The optimal feedback control bnp0 ðt; pn ð; tÞÞ: Copyright # 2005 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2005; 26:229–254

VISCOSITY SOLUTION APPROACH

251

Jðbnp0 ðt; pn ð; tÞÞÞ4Jðbp0 ðtÞÞ

ð49Þ

same initial condition, that is,

Referring to Figure 3, we compute these costs Jðbi Þ ði ¼ 1; 2; . . . ; 7Þ for seven diﬀerent controls bi 2 U½0; T; respectively. The cost value Jðbnp0 ðt; pn ð; tÞÞÞ is also computed. These results are listed in Table II. It is seen from Table II that for the optimal feedback control bnp0 ðt; pn ð; tÞÞ; Jðbnp0 ðt; pn ð; tÞÞÞ ¼ 11641607717:814667; which is evidently less than other costs Jðbi Þ; i ¼ 1; 2; . . . ; 7: In other words, we do get the numerical solution of optimal birth feedback control for the Chinese population from 1989 to 2014.

β1(t)

β2(t)

3

3

2

2

1

1 0

5

10

15

20

25

0

5

10

β3(t) 3

2

2

1

1 5

10

15

20

25

0

5

10

β5(t) 3

2

2

1

1 5

10

15

20

25

0

5

7

3

3

2

2

1

1 5

10

15

20

25

10

15

20

25

β*p0 (t,p* (.,t))

β (t)

0

25

β6(t)

3

0

20

β4(t)

3

0

15

15

20

25

0

5

10

15

20

25 n

Figure 3. Seven diﬀerent arbitrarily chosen control bi and the optimal feedback control bp0 ðt; pn ð; tÞÞ: Copyright # 2005 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2005; 26:229–254

252

B.-Z. GUO AND B. SUN

Table II. Diﬀerent b and its corresponding cost JðbÞ: b b1 ðtÞ b2 ðtÞ b3 ðtÞ b4 ðtÞ

JðbÞ

b

JðbÞ

11641608005.975365 11641608005.975365 11641607901.862444 11641608172.639233

b5 ðtÞ b6 ðtÞ b7 ðtÞ bnp0 ðt; pn ð; tÞÞ

11641608172.639233 11641608172.639233 11641608172.639233 11641607717.814667

female number

15000

10000

5000

0 0

10

20

30

40

50 60 age

70

80

90 100

Figure 4. The age-structure of females of Chinese population in 1989.

relative mortality

0.5

µ(r)

0.4 0.3 0.2 0.1 0 0

10

20

30

40

50 r

60

70

80

90 100

Figure 5. The relative mortality of Chinese population in 1989. Copyright # 2005 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2005; 26:229–254

253

VISCOSITY SOLUTION APPROACH

4

Initial density distribution

x 10 3 2.5

p0(r)

2 1.5 1 0.5 0 0

10

20

30

40

50 r

60

70

80

90

100

Figure 6. The age-structure of Chinese population in 1989.

ACKNOWLEDGEMENTS

The authors would like to thank an anonymous reviewer who indicates many recent developments in numerical solutions to optimal control problems, particularly, the software MISER of Reference [11] may be better to be used for performance comparisons. This work was supported by the National Natural Science Foundation of China and the National Research Foundation of South Africa.

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