Necessary and sufficient conditions for near-optimal harvesting control problem of stochastic age-dependent system

Applied Mathematics and Computation 221 (2013) 394–402

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Necessary and sufficient conditions for near-optimal harvesting control problem of stochastic age-dependent system Qimin Zhang, Dongmei Wei ⇑ School of Mathematics and Computer Science, NingXia University, YinChuan 750021, PR China

a r t i c l e

i n f o

Keywords: Stochastic age-dependent population system Near-optimal Ekeland’s variational principle Necessary and sufficient conditions

a b s t r a c t In this paper, we consider a near-optimal harvesting control problems for stochastic agestructured population system. We establish necessary and sufficient conditions for nearoptimality. These conditions are described by adjoint equations and a nearly maximum condition on the Hamiltonian. The proof of the main result is based on Ekeland’s variational principle and the adjoint processes with respect to the control variable. As is well known, optimal controls may fail to exist even in simple cases. This justifies the use of nearoptimal controls, which exist under minimal assumptions and are sufficient in most practical cases. At last, an example is given for illustrating our results. Ó 2013 Published by Elsevier Inc.

1. Introduction and Model Recently, the near-optimal control problem has become increasingly popular in stochastic differential equations, especially, because it provides an appealing framework for the analysis of control problem, such as Bahlali [1] studied necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs, Huang [2] researched near-optimal control problems for linear forward–backward stochastic system, and near-optimal control for stochastic recursive problems was discussed by Hui [3]. We consider the following stochastic age-dependent system:

8 @p @p t þ @r ¼ uðr; tÞp  lðr; tÞp þ f ðr; t; pÞ þ gðr; t; pÞ dw ; > dt @t > > > < pð0; tÞ ¼ bðtÞ R r2 mðr; tÞpðr; tÞdr; r1 > > pðr; 0Þ ¼ p0 ðrÞ; > > R : ðtÞ ¼ 0A pðr; tÞdr; p

ð1:1Þ

where p ¼ pðr; tÞ, Q :¼ ð0; AÞ  ½0; T; t 2 ð0; TÞ; r 2 ð0; AÞ; A is the life expectancy, and 0 < A < þ1; ½r1 ; r2  is the fertility interval. Though this paper: pðr; tÞ denotes the population density of age r at time t; lðr; tÞ denotes the mortality rate of age r at time t; mðr; tÞ is defined as the ratio of females of age r at time t; bðtÞ denotes the fertility rate of females of age r at time t; p0 ðrÞ is the initial age distribution;

⇑ Corresponding author. E-mail address: [email protected] (D. Wei). 0096-3003/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2013.06.048

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

395

ðtÞ is the total population at time t; p uðr; tÞ is the harvesting effort function, which is the control variable in the model and satisfies: uðr:tÞ 2 U ad :¼ fz 2 L2 ðQ Þ : 0 6 .1 6 z 6 .2 a.e. in Q g, where .1 ; .2 2 L2 ðQ Þ; t f ðr; t; pÞ þ gðr; t; pÞ dw denotes the stochastically perturbation, effecting of external environment for population system, dt such as earthquake, emigration and so on. For the system (1.1), there have many works studied the existence uniqueness and exponential stability of the solution convergence of numerical solution. Zhang studied the existence, uniqueness and exponential stability numerical solutions problems and convergence of numerical solutions [4–6]. For deterministic system (when g ¼ 0), there has been many works on the optimal control problem. For example, Rorres [7] investigated optimal age specific harvesting policy continuous-time population model. For the McKendrick model of population dynamics, Murphy [8] investigated maximum sustainable yield problem for an age-structured population. They obtained optimal solution which was attainable by a bimodal harvesting policy. Chan [9] studied optimal birth control of population systems of McKendrick type which was a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control. Anita [10] discussed optimal harvesting for a nonlinear age-dependent population dynamics. Barbu [11] concerned with the optimal control problem for a Gurtin-Mac-Camy type system describing the evolution of an age-structured population. Necessary optimality conditions were established in the form of an Euler–Lagrange system and existence of an optimal control. Anita [12] discussed analysis and control of age-dependent population dynamics. The optimality conditions for age-structured control system were given by Gustav [13]. As we all know, few results were obtained on the topic of near-optimal control problems of stochastic age-structure. The aim of this paper is to study near-optimality. More precisely speaking, the necessary as well as sufficient conditions of nearoptimality are established. These conditions are described in terms of an adjoint process, corresponding to the stochastic partial differential equations components and a nearly maximum condition on the Hamiltonian. Our main results are based on the Ekeland’s variational principle. This paper is an extension of [4]-[10]. This paper is organized as follows: In Section 2, we give the assumptions, notations, some basic definitions and some Lemmas. In Section 3 and Section 4, we establish necessary and sufficient conditions of near optimality. In Section 5, we provide an example to illustrate our results. 2. Preliminaries of the problem In this paper, let

V ¼ H1 ð½0; AÞ 



uju 2 L2 ð½0; AÞ;

 @u @u 2 L2 ð½0; AÞ; where are generalized partial derivatives : @x @x

V 0 ¼ H1 ð½0; AÞ is the dual space of V. We denote by j  j and k  k the norms in V and V 0 respectively, by h; i the duality product between V; V 0 , and by ð; Þ the scalar product in H. For an operator B 2 LðK; HÞ be the space of all bounded linear operators from K into H, we denote by kBk2 the Hilbert–Schmidt norm, i.e.

kBk22 ¼ trðBWBT Þ: Let ðX; F ; PÞ be a complete probability space with a filtrations fF t gtP0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Let wðÞ be a Wiener process defined on the complete probability space and taking its values in the separable Hilbert space K. Let ðF t ÞtP0 be the r-algebras generated by fws ; 0 6 s 6 tg, then wt is a martingale relative to ðF t ÞtP0 . Let C ¼ Cð½0; T; VÞ be the space of all continuous function from ½0; T into V with sup-norm kwkC ¼ sup06s6T jwðsÞj; LpV ¼ Lp ð½0; T; VÞ and LpH ¼ Lp ð½0; T; HÞ. Here, without loss of generality, the expected cost on the time interval ½0; T is

Jðuðr; tÞÞ ¼ E

Z

T 0

Z

A

 Z pðr; tÞdrdt ¼ E

0

T

ðtÞdt p



0

and the value function is defined as follows:

U¼

sup

Jðuðr; tÞÞ:

uðr;tÞ2U ad ½0;T

Since the objective of this paper is to study near-optimal rather than optimal controls of the system, we give the precise definition of near-optimality as given in [2]. Definition 1 (e-optimal [2]). For a given

e > 0; ue ð; Þ is called e-optimal if

jJðue ð; ÞÞ  Uj 6 nðeÞ holds for sufficiently small e, where n is a function of e satisfying nðeÞ ! 0 as e ! 0, The estimate nðeÞ is called an error bound. If nðeÞ ¼ C ed for some d > 0 independent of the constant C, then ue ð; Þ is called near-optimal with order ed .

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Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

Throughout this paper, we will write Lx for short of @Lða;b;xÞ , and assume the following: @x RA ðA1 Þ l 2 Llloc ðQ Þ; lðr; tÞ P 0; 0 lðr; t þ r  AÞdr ¼ þ1; ðr; tÞ 2 Q ; where lðr; tÞ is extended by zero on ð0; AÞ  ð1; 0Þ. ðA2 Þ f ðr; t; pÞ; gðr; t; pÞ; uðr; tÞ are continuous in [0,T], bounded and continuously differentiable with respect to pðr; tÞ. And there exists a constant K 1 > 0 such that

jf ða; b; xÞ  f ða; b; x0 Þj _ jgða; b; xÞ  gða; b; x0 Þj 6 K 1 jx  x0 j; ðA3 Þ 0 6 mðr; tÞ < 1; ðr; tÞ 2 Q , and mðr; tÞ  0, when r < r 1 or r > r 2 . ðA4 Þ 0 6 b0 6 bðtÞ 6 b0 ; 8t > 0; b0 and b0 are constants. ðA5 Þ p0 2 L1 0 ð0; AÞ; p0 ðrÞ P 0; 8r 2 ð0; AÞ. ðA6 Þ There are constants D0 ; D1 > 0 such that

ðjFðt; p; uÞ  Fðt; p; u0 Þj þ jF u ðt; p; uÞ  F u ðt; p; u0 ÞjÞ 6 D0 ju  u0 j; ðjgðt; p; uÞ  gðt; p; u0 Þj þ jg u ðt; p; uÞ  g u ðt; p; u0 ÞjÞ 6 D1 ju  u0 j: Throughout this paper, we defined

Fðr; t; p; uÞ :¼ uðr; tÞp  lðr; tÞp þ f ðr; t; pÞ and the Hamiltonian function is given as

  hw  Fi  kg ¼ p ðt; pÞ  huðr; tÞpðr; tÞ  lðr; tÞpðr; tÞ; wi þ ðf ðr; t; pÞ; wÞ  kg Hðr; t; p; u; e; kÞ :¼ p ðt; pÞ  ðuðr; tÞwðr; tÞ þ lðr; tÞwðr; tÞ  f ðr; t; pÞwÞw  kðr; tÞgðr; tÞ ¼ p   ðu þ l  f ðr; t; pÞÞw  kg: ¼p Under the assumption ðA2 Þ, there is a unique solution pðr; tÞ 2 L2F ð½0; A  ½0; T; HÞ which solves (2.1), where ðtÞ such that L2F ð½0; A  ½0; T; HÞ denotes the Hilbert space of F 1  adapted processes p

E

Z

t

ðsÞj2 ds < þ1: jp

0

For any uðr; tÞ 2 U ad with its corresponding state trajectory ðpðr; tÞÞ, we introduce the adjoint equation and the Hamiltonian function for our problem. The adjoint equations can be written as:

(

@wðr;tÞ @t

þ @wðr;tÞ ¼  @H ¼ ðlðr; tÞ þ u ðr; tÞ  fp ðr; t; pÞÞwðr; tÞ þ g p kðr; tÞ þ kðr; tÞdwt ; @p @r

wðr; TÞ ¼ wðA; tÞ ¼ 0;

ðr; tÞ 2 Q ;

ð2:1Þ

note that the couple ðwðr; tÞ; kðr; tÞÞ is the adjoint process corresponding to the stochastic age-dependent system pðr; tÞ. It is a well known fact that under the assumption ðA2 Þ. The adjoint equation admits one and only one F t -adapted solution ðwðr; tÞ; kðr; tÞÞ. In an analogous way to the corresponding proof presented in [14–16], we may establish the following existence, uniqueness and stability conclusion: under the assumption ðA2 Þ, (2.1) has a unique conditions solution ðwðr; tÞ; kðr; tÞÞ. Moreover, since F p ; g p are bounded by, there exists a constant C > 0 independent of ðpðr; tÞ; uðr; tÞÞ, such that the solutions of the adjoint equation satisfy the following estimate:

! 2

E sup sup jwðr; tÞj 06t6T 06r6A

þE

Z

jkðr; tÞj2 ds < C:

ð2:2Þ

Q

The proof of (2.2) is similar to Lemma 3.1 in [3]. Let us recall Ekeland’s variational principle which will be used in the sequel. Lemma 1 (Ekeland’s principle [17]). Let ðV; dÞ be a complete metric space and F : V ! R [ fþ1g be a lower semicontinuous function, bounded from below. If for each e > 0, there exists ue 2 V such that Fðue Þ 6 inf u2V FðuÞ þ e. Then for any d > 0, there exists ud 2 V such that ðiÞ Fðud Þ 6 Fðue Þ, ðiiÞ dðud ; ue Þ 6 d, ðiiiÞ Fðud Þ 6 FðuÞ þ de dðu; ud Þ, for all u 2 V. For u; v 2 U ad , we define

dðu; v Þ ¼ dt  Pfðt; xÞ 2 ½0; T  X : uðr; t; xÞ – v ðr; t; xÞg

ð2:3Þ

where dt  q is the product measure of the Lebesgue measure dt with the probability measure q. It is well known that ðU ad ; dÞ is a complete metric space.

3. Necessary conditions of near-optimality This is the main result of this paper. In this section, we derive necessary conditions for a control to be near-optimal.

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

397

Lemma 2. For any 0 < a < 1 and 0 < q 6 2, there is a constant c1 ¼ c1 ða; qÞ > 0, such that for any uðr; tÞ; u0 ðr; tÞ 2 U ad , along with the corresponding trajectories pðr; tÞ and p0 ðr; tÞ, it holds that

  aq E sup jpðr; tÞ  p0 ðr; tÞjq 6 c1 dðuðr; tÞ; u0 ðr; tÞÞ 2 :

ð3:1Þ

06t6T

^’s formula to jp  p0 j2 , we obtain Proof. First, we assume q ¼ 2. Applying Ito

 Z t Z t @p @p0 þ  lðr; sÞðp  p0 Þ; p  p0 ds  2 hup  u0 p0 ; p  p0 ids þ 2 ðf ðr; s; pÞ  f ðr; s; p0 Þ; p @r @r 0 0 0 Z t Z t  p0 Þds þ 2 ðp  p0 ; gðr; s; pÞ  gðr; s; p0 ÞÞdws þ jjgðr; s; pÞ  gðr; s; p0 Þjj22 ds

jp  p0 j2 ¼ 2

Z t



0

0

 Z t Z t Z t Z t @ðp  p0 Þ ; p  p0 ds  2l0 ðp  p0 ; p  p0 Þds  2 hup  u0 p0 ; p  p0 i þ 2 ðf ðr; s; pÞ 6 2 @r 0 0 0 0 Z t Z t 0 0 0 0  f ðr; s; p Þ; p  p Þds þ 2 ðp  p ; ðgðr; s; pÞ  gðr; s; p ÞÞdws Þ þ jjgðr; s; pÞ  gðr; s; p0 Þjj22 ds: 0

ð3:2Þ

0

Considering the first term in the right side of (3.2), since

   2 Z A Z A @ðp  p0 Þ 1 1 2  ; p  p0 ¼  ðp  p0 Þdr ðp  p0 Þ ¼  jðp  p0 Þ jA0 ¼ bðsÞ mðr; sÞðp  p0 Þdr ; @r 2 2 0 0 by A5 and Holder inequality, we have

   2 Z A Z A Z A @ðp  p0 Þ 1 1 1 2 2  ; p  p0 ¼ bðsÞ mðr; tÞðp  p0 Þdr 6 b2 ðsÞ dr ðp  p0 Þ dr 6 Ab jp  p0 j2 : @r 2 2 2 0 0 0

ð3:3Þ

the third term in the right side of (3.2)

2

Z

t

hup  u0 p0 ; p  p0 i ¼ 2

0

Z 0

t

hup0  up; p  p0 i þ 2hðu0 p0  up0 Þjvu–u0 ; p  p0 i

6 4c11 jp  p0 j2 þ 4ðjp0 j2 þ ju0  uj2 þ 4jp  p0 j2 Þvu–u0 6 c12 jp  p0 j2 þ c13 dðu; u0 Þ:

ð3:4Þ

Now, turn to the second term of (3.2), by Burkholder-Davis-Gundy’s inequality, we have

"   Z t 1=2 # Z s 2 0 0 0 0 E sup ðp  p ; ðgðr; s; pÞ  gðr; s; p ÞÞdws Þ 6 3E sup jp  p j jjgðr; s; pÞ  gðr; s; p Þjj2 ds 06s6t

06s6t

0

0



 Z t  1 jjgðr; s; pÞ  gðr; s; p0 Þjj22 ds 6 E sup jp  p0 j2 þ K 1 E 4 06s6t 0   Z t 1 Ejjp  p0 jj2C ds: 6 E sup jp  p0 j2 þ K 1  K 22 4 06s6t 0

ð3:5Þ

for some positive constants K 1 ; K 2 > 0. With (3.2)–(3.5), (3.1) can be proved. h Lemma 3. For any 0 < a < 1 and 1 < n < 2 satisfying ð1 þ anÞn < 2, there is a constant c2 ¼ c2 ða; n; nÞ > 0, such that for any 0 uðr; tÞ; u0 ðr; tÞ 2 U ad , along with the corresponding trajectories pðr; tÞ; p0 ðr; tÞ and the solutions ðwðr; tÞ; kðr; tÞÞ; ðw0 ðr; tÞ; k ðr; tÞÞ of the corresponding adjoint equations, it holds that

E

Z

T

0

ann

fjwðr; tÞ  w0 ðr; tÞjn þ jkðr; tÞ  k ðr; tÞjn gdt 6 c2 dðuðr; tÞ; u0 ðr; tÞÞ 2 :

ð3:6Þ

0

 tÞÞ  ðwðr; tÞ  w0 ðr; tÞ; kðr; tÞ  k0 ðr; tÞÞ satisfies the following backward stochastic differential  tÞ; kðr; Proof. Noting that ðwðr; equation:

8 @ w @ w   > < @t þ @r ¼ ½lðr; tÞ þ uðr; tÞ  fp ðr; t; pÞw þ g p ðr; t; pÞkðr; tÞ þ ½ðuðr; tÞ þ fp ðr; t; pÞÞ 0 t ; ðu0 ðr; tÞ þ fp ðr; t; p0 ÞÞw0 ðr; tÞ þ ðg p ðr; t; pÞ  g p ðr; t; p0 ÞÞk ðr; tÞ þ kðr; tÞ dw dt > : wðr; TÞ ¼ wðA; tÞ ¼ 0; ðr; tÞ 2 Q; Let g be the solution of the following linear stochastic differential equation (SDE):

398

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

(

n1   tÞÞgdwt ;  tÞjn1 sgnðwðr;tÞÞgdt  dgðr; tÞ ¼ fðuðr; tÞp  lðr;tÞp þ fp ðr; t;pÞÞgðr;tÞ þ jwðr; sgnðkðr; þ fg p ðr; t;pÞgðr; tÞ þ jkðr;tÞj gðr; 0Þ ¼ 0: 



where sgnðaÞ  ðsgnða1 Þ; . . . ; sgnðan ÞÞ for any vector a  ða1 ; . . . ; an Þ Þ. Note that the existence and uniqueness of solutions to the above equation are verified by assumptions ðA2 Þ and the fact that

E

Z

T

 tÞjq0 1 sgnðkðr;  tÞÞj2 gdt < þ1;  tÞjq0 1 sgnðwðr;  tÞÞj2 þ jjkðr; fjjwðr;

0

then we have

E sup jgðr; tÞjq 6 c21 E

Z

06t6T

T

0

0

 tÞjqq q gdt 6 c21 E ðr; tÞjqq q þ jkðr; fjq

Z

0

T

0

0

 tÞjq gdt; ðr; tÞjq þ jkðr; fjq

ð3:7Þ

0

where q > 2 and 1q þ q10 ¼ 1. ^’s formula to Noting that the right hand side term of (3.7) is bounded due to (2.2). On the other hand, applying It o  tÞ  gðr; tÞ and taking expectations, we obtain wðr;

Z

T

 tÞ  ½jkðr;  tÞjn1 sgnðkðr;  tÞÞgdt  tÞ  ½jwðr;  tÞjn1 sgnðwðr;  tÞÞ þ kðr; fwðr; Z T   tÞ @ wðr;   gðr; tÞ ¼E  þ ðlðr; tÞ þ uðr; tÞ  fp ðr; t; pÞÞw @r

0 þ ðuðr; tÞ  lðr; tÞ þ fp ðr; t; pÞÞ  ðu0 ðr; tÞ  lðr; tÞ þ fp ðr; t; p0 ÞÞw0 ðr; tÞ Z t  0  tÞgðr; tÞdt: bðtÞmðr; tÞwðr; þ g p ðr; t; pÞ  g p ðr; t; p0 ÞÞk ðr; tÞgðr; tÞ dt  E

E

0

ð3:8Þ

0

We proceed to estimate the second term in the right hand side of (3.8). First,

 Z E

T



j ðuðr; tÞ þ fp ðr; t; pÞÞ  ðu ðr; tÞ þ fp ðr; t; p ÞÞ w0 ðr; tÞjp dt 0

0

1p  Z E

0

T

q

jgðr; tÞj

1q

dt

0

 Z T   Z  tÞjq dt  tÞjq þ jkðr; ½jwðr; E 6 c22 E 1 q0

0

  ðu0 ðr; tÞ þ fp ðr; t; p0 ÞÞjq jw0 ðr; tÞjq dt :

T

jðuðr; tÞ þ fp ðr; t; pÞÞ

ð3:9Þ

0

From Lemma 2 and using the fact that E sup06t6T jw0 ðr; tÞjq < 1 for any q, it can easily checked that

 Z E

T

jðuðr; tÞ þ fp ðr; t; pÞÞ  ðu0 ðr; tÞ þ fp ðr; t; p0 ÞÞw0 ðr; tÞjq dt

1q  Z E

0

T

jgðr; tÞjq

0

q10

abq

dt 6 c23 dðuðr; tÞ; u0 ðr; tÞÞ 2 :

ð3:10Þ

0

Using similar arguments developed above, we can easily prove that

 Z E

T

0

jðg p ðr; t; pÞ  g p ðr; t; p0 ÞÞk ðr; tÞjq dt

0

1q  Z E

T

0

jgðr; tÞjq

q10

abq

dt 6 C 24 dðuðr; tÞ; u0 ðr; tÞÞ 2 :

ð3:11Þ

0

It follows from (3.9) and (3.10) that

E

Z

T

0

 tÞgðr; tÞdt  tÞ þ ðg ðr; t; pÞ  g 0 ðr; t; p0 ÞÞkðr; ½ððuðr; tÞ  lðr; tÞ þ fp ðr; t; pÞÞ  ðu0 ðr; tÞ  lðr; tÞ þ fp0 ðr; t; p0 ÞÞÞwðr; p p Z

T

6 0

6

 tÞgðr; tÞdt  0 ðr; tÞ þ ðg ðr; t; pÞ  g 0 ðr; t; p0 ÞÞkðr; ½ðuðr; tÞ  lðr; tÞ þ fp ðr; t; pÞÞ  ðu0 ðr; tÞ  lðr; tÞ þ fp0 ðr; t; p0 ÞÞÞw p p

 Z E 0

 Z E

T

T

 tÞjq  0 ðr; tÞ þ ðg ðr; t; pÞ  g 0 ðr; t; p0 ÞÞkðr; jððuðr; tÞ  lðr; tÞ þ fp ðr; t; pÞÞ  ðu0 ðr; tÞ  lðr; tÞ þ fp0 ðr; t; p0 ÞÞw p p q0

jgðr; tÞj dt

q10

abq

dt 6 C 25 dðuðr; tÞ; u0 ðr; tÞÞ 2 :

1q

ð3:12Þ

0

Next, we proceed to estimate the first term in the right side of (3.8). By ðA2 Þ  ðA5 Þ and using the fact that wðr; tÞ is bounded, we obtain abq

 tÞ 6 C 26 jwðr;  tÞjq dt 6 C 27 dðuðr; tÞ; u0 ðr; tÞÞ 2 : ðlðr; tÞ þ uðr; tÞ  fp ðr; t; pÞÞ  ðlðr; tÞ  u0 ðr; tÞ þ fp0 ðr; t; p0 ÞÞÞwðr;

ð3:13Þ

We proceed to estimate the third term in the right side of (3.8). From Lemma 2 and assumption

E

Z 0

t

abq

 tÞgðr; tÞjq dt 6 C 28 dðuðr; tÞ; u0 ðr; tÞÞ 2 : jbðtÞmðr; tÞwðr;

ð3:14Þ

399

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

It follows from (3.11)–(3.13) that

E

Z

T

 tÞjn gdt 6 C 29 dðuðr; tÞ; u0 ðr; tÞÞa2bq :  tÞjn þ jkðr; fjwðr;

0

Now, we have completed the proof of Lemma 3.

h

Theorem 1. Assume that ðA6 Þ holds, there is a constant C 3 such that for any 0 6 b < 1; e > 0 and e-optimal, the necessary conditions

max E

Z

uðr;tÞ2U ad

T

e

Hðt; pe ðr; tÞ; ue ðr; tÞ; we ðr; tÞ; k ðtÞÞdt 6

0

Z

T

e

b

Hðt; pe ðr; tÞ; uðr; tÞ; we ðr; tÞ; k ðr; tÞÞdt þ C 3 e3

ð3:15Þ

0

e

hold, where ðwe ðr; tÞ; k ðr; tÞÞ is the solution of adjoint Eq. (2.1). Under the control ue ðr; tÞ, and H is the Hamiltonian function. Proof. Step 1: From assumptions ðA1 Þ and ðA2 Þ, it is easy to see that Jðuðr; tÞÞ is continuous U ad endowed with the 2 ~ e ðr; tÞ such methric defined by (2.2). Applying Ekeland’s variational principle with d ¼ e3 , there is an admissible control u that 2

~ e ðr; tÞÞ 6 e3 ; dðue ðr; tÞ; u

ð3:16Þ

eJðu ~e ðr; tÞÞ 6 eJðue ðr; tÞÞ; for any uðr; tÞ 2 U ad , where

eJðue ðr; tÞÞ ¼ Jðue ðr; tÞÞ þ e13 dðue ðr; tÞ; u ~ e ðr; tÞÞ: ~ e ðr; tÞ is optimal for the system with the new cost functions eJ. Let t 0 2 ½0; TÞ and uðr; tÞ 2 U ad be fixed. For This means that u any q > 0, define the spike variation uq 2 U ad ½0; T of



~q ð; Þ ¼ u

uðr; tÞ t 2 ½t 0 ; t0 þ q; ~ e ðr; tÞ otherwise; u

The fact that

eJðu ~e ðr; tÞÞ 6 eJðuq ðr; tÞÞ and

~ e ðr; tÞÞ 6 q: dðuq ðr; tÞ; u

ð3:17Þ

From (3.17), Lemma 1 and Toylor’s expansion, we derive 1

~ e ðr; tÞÞ ¼ E qe3 6 Jðuq ðr; tÞÞ  Jðu

T

ðr; t; pq ; uq Þ  p ðr; t; pe ; u ~ e Þdt ½p

0

Z

¼E

Z

T

ðr; t; pq ; uq Þ  p ðr; t; p ~e ; uq Þ þ ½p ðr; t; p ~e ; uq Þ  p ðr; t; p ~e ; u ~ e Þgdt f½p

0

(Z

T

6E

p ðr; t; p ~e ; uq Þðpq  p ~e Þdt þ p

Z

~tþq

) ðr; t; p ~e ; uÞ  p ðr; t; p ~e ; u ~ Þdt þ oðqÞ : ½p

ð3:18Þ

~t

0

Noticing assumption ðA2 Þ and Lemma 2, 1

qe3 6 E

Z

~tþq

ðr; t; p ~e ; uÞ  p ðr; t; p ~e ; u ~e Þdt þ E ½p

~t

Z

~tþq

~ e dt: ðu  u ~ e Þw p

ð3:19Þ

~t

Let q ! 0, we get 1 ~ e ; ðr; ~t; p ~e ; uÞ  p ðr; ~t; p ~e ; u ~e Þ þ E½pðu  u ~ e Þw e3 6 E½p

ð3:20Þ

i.e. 1 ~e Þ  Hðr; ~t; p ~e Þ: ~ e; k ~e; k ~e ; u; w ~e ; u ~; w e3 6 E½Hðr; ~t; p

~e ; u ~ e Þ. Let replace all the ðp ~e ; u ~ e Þ by ðpe ; ue Þ in (3.20) estimating the following difference Step 2: Necessary condition for ðp

E

Z 0

T

ðt; p ~e ; uq Þ  p ðt; p ~e ; u ~ e Þdt  E ½p

Z 0

T

ðt; pe ; uq Þ  p ðt; pe ; ue Þdt þ E ½p

Z 0

T

~ e dt  E ~ e Þw pðuq  u

Z 0

T

b

pðuq  ue Þwe dt 6 C 31 e3 :

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Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

Then, we have b

C 32 e3 6 E ¼E

Z Z

T

ðr; t; pe ; uq Þ  p ðr; t; pe ; ue Þdt þ E ½p

0 T

e

Hðt; pe ; uq ; we ; k Þdt  E

T

pðuq  ue Þwe dt

0

Z

0

Z

T

e

Hðt; pe ; ue ; we ; k Þdt:

0

So the Eq. (3.16) is proofed.

h

4. Sufficient conditions of near-optimality In this section, we will prove that the near-optimal condition on the Hamiltonian function is a sufficient condition for near-optimality. e

Theorem 2. Let ðpe ðr; tÞ; ue ðr; tÞÞ be near-optimal solution of the state equation, ðwe ðr; tÞ; k ðr; tÞÞ is the solution of the adjoint equations, corresponding to ðke ðr; tÞ; ue ðr; tÞÞ. e Assume that Hðt; :; we ðr; tÞ; k ðr; tÞÞ is concave for a:e:t 2 ½0; T, if for some e > 0,

E

Z

T

e

Hðt; ke ðr; tÞ; ue ðr; tÞ; we ðr; tÞ; k ðr; tÞÞdt P

sup

E

uðr;tÞ2U ad ½0;T

0

Z

T

e

Hðt; ke ðr; tÞ; uðr; tÞ; we ðr; tÞ; k ðr; tÞÞdt  ec :

ð4:1Þ

0

Then

Jðue ðr; tÞÞ 6

1

Jðuðr; tÞÞ þ De2 ;

inf

uðr;tÞ2U ad ½0;T

ð4:2Þ

where D > 0 is a constant, which is independent from e. Proof. Let fix

e > 0 and define a new metric d~ on U ad as follows

~ dðuðr; tÞ; ue ðr; tÞÞ ¼ E

Z

T

ye ðr; tÞjuðr; tÞ  ue ðr; tÞjdt;

ð4:3Þ

0

where

ye ðr; tÞ ¼ 1 þ jwe ðr; tÞj P 1:

ð4:4Þ

Define a function IðÞ on U ad ½0; T by:

Iðuðr; tÞÞ ¼ E

Z

T

e

Hðt; ke ðr; tÞ; uðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞdt:

0

A simple computation shows that

jIðuðr; tÞÞ  Iðu0 ðr; tÞÞj 6 D1 E

Z

T

ye ðr; tÞjuðr; tÞ  ue ðr; tÞjdt

0

~ which implies that IðÞ is continuous on U ab with respect to d. ~ e ðr; tÞ 2 U ad ½0; T such that By (4.1) and Ekeland’s Lemma, there exists u

~ e ðr; tÞ; u ~ e ðr; tÞÞ 6 e dðu and

E

Z

T

e pe ðr; tÞ; u ~ e ðr; tÞÞdt ¼ Hðt;

0

max

uðr;tÞ2U ad ½0;T

E

Z

T

e pe ðr; tÞ; uðr; tÞÞdt; Hðt;

ð4:5Þ

0

where

e pe ðr; tÞ; uðr; tÞÞ ¼ Hðt; pe ðr; tÞ; uðr; tÞ; ðwe ðr; tÞ; ke ðr; tÞÞÞ  e12 ye ðr; tÞjuðr; tÞ  u ~ e ðr; tÞj: Hðt;

ð4:6Þ

By the standard arguments, it can be proved that the integral maximum condition (4.6) implies a pointwise maximum condition, namely for a:e. t 2 ½0; T, and P  a:s. . .

e pe ðr; tÞ; u e pe ðr; tÞ; uðr; tÞÞ: ~ e ðr; tÞÞ ¼ max Hðt; Hðt;

ð4:7Þ

u2U ad

Using assumption ðA2 Þ, we can prove that e

1

~ e ; ðwe ðr; tÞ; k ðr; tÞÞÞ 6 D2 ye ðr; tÞjue ðr; tÞ  u ~ e ðr; tÞj þ e2 ye ðr; tÞ: Hu ðt; pe ðr; tÞ; u

ð4:8Þ

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

401

e

By the concavity of Hðt; p; u; we ; k Þ, we have e

e

Hðt; pðr; tÞ; uðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞ  Hðt; pe ðr; tÞ; ue ðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞ e

e

6 Hp ðt; pe ðr; tÞ; ue ðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞ þ Hu ðt; pe ðr; tÞ; ue ðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞ:

ð4:9Þ

By integrating both sides and noting and (4.8), then we can obtain

E

Z

T

e

e

fHðt; pðr; tÞ; uðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞ  Hðt; pe ; ue ; ðwe ðr; tÞ; k ðr; tÞÞÞgdt

0

6E

Z

T

e

1

Hp ðt; pe ðr; tÞ; ue ðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞÞðpðr; tÞ  pe ðr; tÞÞdt þ D3 e2 :

ð4:10Þ

0

On the other hand, by applying Itô’s formula respectively to we ðr; tÞðpðr; tÞ  pe ðr; tÞÞ and by taking expectations, we can obtain

E½we ðr; TÞðpðr; TÞ  pe ðr; TÞÞ þ E ¼E

Z

T

e

k ðr; tÞ½gðt; pðr; tÞÞ  gðt; pe ðr; tÞÞdt

0

Z

T

e

Hp ðt; pe ðr; tÞ; ue ðr; tÞ; ðwe ðr; tÞ; k ðr; tÞÞðpðr; tÞ  pe ðr; tÞÞÞdt

0

þE

Z

T

0

we ðr; tÞ½ðuðr; tÞ  lðr; tÞ þ fp ðr; tÞÞ  ðue ðr; tÞ  lðr; tÞ þ fpe ðr; tÞÞdt ¼ 0:

ð4:11Þ

Combining (4.10) with (4.11), using the fact ahead, then we can obtain

1 Jðue ðr; tÞÞ 6 Jðuðr; tÞÞ þ D4 e : 2 Since ðuðr; tÞÞ is arbitrary, so the desired result follows. Then, we can get what we want to prove:

Jðue ðr; tÞÞ 6 Jðuðr; tÞÞ:



5. Example Consider the following stochastic age-dependent system:

8 @p @p 1 þ ¼  ð1rÞ ðr; tÞ 2 ½0; 1  ½0; 1; > 2 p þ 2pt þ up  ptdwt ; > > @t @r > R < 1 pðr; tÞ ¼ t 2 0 pðr; tÞdr; ðr; tÞ 2 ð0; 1Þ  ½0; 1;  1 > > pðr; 0Þ ¼ exp  1r ; t 2 ½0; 1; > > : pð0; tÞ ¼ 0

ð5:1Þ

and the near-optimal control problem is

Jðuðr; tÞÞ ¼ E

Z

T 0

Z

A

 Z pðr; tÞdrdt ¼ E

0

T

 ðtÞdt : p

0

Here, wt is a real standard Brownian motion. Take T ¼ 1; A ¼ 1 in (5.1). We can set this problem in our formulation by taking H ¼ L2 ð½0; 1  ½0; 1Þ; V ¼ W 10 ð½0; 1Þ (a Sobolev space with elements satisfying the boundary conditions above),  1 2 1 lðr; tÞ ¼ ð1rÞ 2 , bðtÞ ¼ t , mðr; tÞ ¼ 1, f ðr; t; pÞ ¼ 2pt, and gðr; t; pÞ ¼ p, pðr; 0Þ ¼ exp  1r . Clearly, the operators f ðr; tÞ and gðr; tÞ and lðr; tÞ satisfy the assumption. To solve this problem, we write down the Hamiltonian function:

 þ 2pt  up  Hðr; t; p; u; w; kÞ ¼ p

p ð1  rÞ2

! w  kpt:

So the adjoint equation is

(

@wðr;tÞ @t

  dwt 1 þ @wðr;tÞ ¼ 2t  ð1rÞ 2 þ u wðr; tÞ  tkðr; tÞ þ kðr; tÞ dt ; @r

ð5:2Þ

wðr; TÞ ¼ 0; wðA; tÞ ¼ 0: Solving Eqs. (5.1) and (5.2), for any admissible control uðr; tÞ, if we have Lemmas 1 and 3, then (3.15) is valid, which is the necessary and sufficient conditions for near-optimality in this control problem.

402

Q. Zhang, D. Wei / Applied Mathematics and Computation 221 (2013) 394–402

The corresponding Hamiltonian

!

pe

e

e þ 2pe t  ue pe  Hðr; t; pe ; ue ; we ; k Þ ¼ p

e

we  k pe t

ð1  rÞ2

and e

e þ 2pe t  upe  Hðr; t; pe ; u; we ; k Þ ¼ p

!

pe ð1  rÞ2

e

we  k pe t:

If uðr; tÞ is e-optimal, the necessary condition for (5.1) is

E

Z

1

e þ 2pe t  ue pe  p

0

!

pe ð1  rÞ2

e

we  k pe t

! 6E

Z

1

e þ 2pe t  upe  p

0

pe ð1  rÞ2

!

e

we  k pe t

!  C ec ;

ð5:3Þ

where C is a constant. For example, if we let 1

ue ðr; tÞ ¼ 1  e2 ; where e > 0 is a sufficiently small parameter, then ue ðr; tÞ is a candidate e-optimal control. Moreover, because of pðr; tÞ is the function of uðr; tÞ , lðr; tÞ; f ðr; t; pÞ and gðr; t; pÞ satisfying the condition ðA6 Þ, we can conclude that

Jðue ðr; tÞÞ 6

sup

1

Jðuðr; tÞÞ þ e2 :

uðr;tÞ2U ad ½0;1

So we gain that (5.3) is the sufficient condition for (5.1), too. Acknowledgement The authors thank the referees for their very helpful suggestions which greatly improved this paper. The research was supported by the National Natural Science Foundation (No. 11061024, 11261043) China. References [1] K. Bahlali, N. Khelfallah, B. Mezerdi, Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs, System Control Letters 58 (2009) 857–864. [2] J.H. Huang, X. Li, G.C. Wang, Near-optimal control problems for linear forward–backward stochastic system, Automatica 46 (2010) 397–404. [3] E. Hui, J.H. Huang, X. Li, G.C. Wang, Near-optimal control for stochastic recursive problems, Systems Control Letters 60 (2011) 161–168. [4] Q.M. Zhang, W.A. Liu, Existence, uniqueness and exponential stability for stochastic age-dependent population, Applied Mathematics and Computation 154 (2004) 183–201. [5] Q.M. Zhang, H.C. Zhao, Numerical analysis for stochastic age-dependent population equations, Applied Mathematics and Computation 176 (2005) 210– 223. [6] Q.M. Zhang, C. Han, Convergence of numerical solutions to stochastic age-structured population system, Applied Mathematics and Computation 118 (2005) 134–146. [7] C. Rorres, W. Fair, Optimal age specific harvesting policy for continuous-time population model, in: T.A. Burton (Ed.), Modeling and Differential Equations in Biology, Dekker, New York, 1980. [8] L.F. Murphy, S.J. Smith, Maximum sustainable yield of a nonlinear population model with continuous age structure, Mathematical Biosciences 104 (1991) 259–270. [9] W.L. Chan, Optimal birth control of population dynamics II. Problem with free final time, phase contrast and mini–max cost, Journal of Mathematics and Application 146 (1990) 523–540. [10] S. Anita, Optimal harvesting for a nonlinear age-dependent population dynamics, Mathematical Analysis and Applications 226 (1998) 6–22. [11] V. Barbu, M. Iannelli, Optimal control of population dynamics, Optimal Theory Applied 102 (1999) 1–14. [12] S. Anita, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Academic Publishers, Dordrecht, 2000. [13] G. Feichtinger, Optimality conditions for age-structured control system, Mathematical Analysis and Applications 288 (2003) 47–68. [14] Y. Hu, S. Peng, Solution of forward-backward stochastic differential equations, Probability Theory and Related Fields 103 (1995) 273–283. [15] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability 28 (2000) 558–602. [16] K. Bahlali, E.H. Essaly, M. Hassani, Etienne Pardoux. Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient, Probability Theory 335 (2002) 757–762. [17] I. Ekeland, Non convex minimization problems, Bulletin of Australian Mathematical Society 1 (1979) 443–474.