Optimal control of harvesting for age-dependent predator-prey system

Available online at www.sciencedirect.com

MATHEMATICAL AND COMPUTER MODELLING

. , 1 1

sc,E

ELSEVIER

N C E f"f~l~ D, R ' : C r "

Mathematical and Computer Modelling 42 (2005) 573-584

www.elsevier.com/locate/mem

Optimal Control of Harvesting for Age-Dependent Predator-Prey System C H U N ZHAO* Faculty of Mathematics Science, Tianjin Normal University Tianjin 300074, P.R. China and

Faculty of Science, Xi'an Jiaotong University Xi'an 710049, P.R. China yczhaochun© 163. c o m

MIANSEN WANG School of Science, Xi'an Jiaotong University Xi'an 710049, P.R. China PING ZHAO Faculty of Science, Beijing Jiaotong University Beijing 100044, P.R. China

(Received January 2004; revised and accepted July 2004) A b s t r a c t - - I n this paper, we consider the optimal harvesting problems for age-dependent predatorprey system. The existence and uniqueness of solution for the system are proven using the Banach fixed-point theorem. The maximum principle is obtained. For small final time T, we give the existence and uniqueness of an optimal control. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Age-dependence, Predator-prey system, OptimM harvesting, Fixed-point theorem, Maximum principle.

1.

INTRODUCTION

The optimal control and harvesting of age-dependent single species have been widely investigated by m a n y authors. For the optimal harvesting problems of age-dependent single population dynamics we refer to the fundamental papers [1-51. The optimal control problems of multispecies have been also reported in the literature. Cafiada, Magal and Montero [6] study the optimal control of the harvesting for an elliptic system modelling two subpopulations of the same species. The optimal harvesting problem for a parabolic differential system modelling two subpopulations of the same species is investigated by Lenhart and Montero [7]. The existence and uniqueness of optimal control of boundary habitat hostility for interacting species are obtained by Lenhart, *Author to whom all correspondence should be addressed. The authors would like to thank the referees for their very helpful comments, which greatly improved the quality of this paper. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved.

doi:lO.1016/j.mcm.2004.07.019

Typeset by ~4A/~S-TEX

574

CHUN ZHAO et

aI.

Liang and Protopopescu [8]. Leung [9] considers the optimal control of the harvesting for an elliptic system of two interacting populations. Fister [10] studies a predator-prey model with Neumann boundary conditions, where the controls are the quotas of populations to be harvested. However, the optimal control problems of multispecies mentioned above are most concentrated on system ignoring the age structure. We take notice of no research work published yet on this topic. In this paper, we consider the following optimal harvesting problem, 2 (OH) sup i'u~ (a, t, x) Pi (a, t, x) - u~ P

for all u = (ul(a, t, x), u2(a, t, x)) C U, where the corresponding state variable (p~,p~) satisfies the state system,

Opl Opl (a,t,x)pl O---t-+ -~a - klApl = -#1 - A1 (a, t, x) P2 (t, x) Pl - ul (a, t, x) pl,

Op2

Op2

O--Y + -5-Ja - k 2 A p 2 =

in Q,

(a, t, z) p2

+ A2 (a, t, x) PiP2 - u2 (a, t, x) P2,

Pl (0, t, x) = ~0 A 131 (a, t, x, ]92 (t, x)) PI (a, t, x) da,

in Q,

(1)

in QT,

A

p2(O,t,x)= fo /32(a,t,x, P l ( t , x ) ) p 2 ( a , t , x ) d a ,

Opi (a, t, x) = O,

on E,

O~

; i (a, 0, x) = pi0

inQT,

(a, x/,

i = 1, 2,

in QA,

where Pi(t,x) = f : p ~ ( a , t , x ) d a in QT, i = 1,2. ~ is the outward unit normai and ~2 C R g (N = 2, 3) is a bounded domain with a smooth enough boundary 0f~. Q = (0, A) x (0, T) zft, QT = (0, T 1 x f~, QA = (0, A) x f~, E = (0, A / x (0, T) z 0ft. The positive constants kl and k2 are the diffusion rates for two species within f~, respectively. The function Pl represents the density of prey population and P2 the density of predator population. We assume that the two populations have the same life expectancy A, 0 < A < +oo. tzi is the average mortality of population pi./3i describes the average fertility of population p~. The coefficient Ai represents the interaction effect. The function Pi0 gives the initial density distribution of population Pi. The function ui is the harvesting rate of population p~. We assume that the harvesting rate (ul, u2) belongs to the following control set U = {Vl E L 2 (Q): O<_vl (a,t,x I <_ L1 a.e. in Q} x {v2 E L 2(Q) :0 _< v2 (a,t,z)<_L2 a.e. in Q } , where Li > 0 (i = 1, 2) are constants. The integral J (ul, u2) = ~ [ui (a, t, z) p~ (a, t, x) - ui (a, t, x)] da at dx i=1 represents the profit due to harvesting.

(2)

Optimal Control of Harvesting

575

Throughout this paper, we suppose the following hypotheses hold, (A1) t~i E L~oc([0, A ) x [0, T] x ft), #i(a,t,x) > #o(a,t) > 0, i = 1,2, where #0 E L~oc([0, A ) x [0, T]) and f A #o(a,t + a - - A ) d a = +ec. (A2) A~ E L°°(Q), and 0 < Ai(a,t,x) <_ M a.e. in Q, i = 1,2. M is constant. (A3) ~i(', ", ", s) E L~(Q), for all s E R +, and fl~(a, t, x, s) is twice continuously differentiable in s, such that o <_ ~ (a, t, x, s) <_ ~ ,

O/~i (a,t,x,s)

--5-J

+ O2/~i (a,t,x,s) Os 2

< Y/I

-

a.e. in Q x R +, i = 1, 2. /~/is constant. (A4) /~l(a, t, x, .) is nonincreasing a.e. (a, t, x) E Q and/32(a, t, x, .) is nondecreasing a.e. (a, t, x)

~Q. (Aa) Pio E L~(QA), pro(a, z) > 0 a.e. in QA, i = 1, 2. Denote by C (S; L 2 (a)) = {h: :~ -+ L 2 (ft): h continuous} and

A C ( S ; L 2(ft)) = { h :

S-+L 2(f~):h(ao+.,to+.):

(0, a ) - - + L 2(ft)

is absolutely continuous on any compact subinterval}, where (ao,to) e {0} x (0,T) U (0, A) x {0} and (ao + c~,to + a ) E {A} x (0, T) U (0, A) is the characteristic line of system (1), which can be given by S={(a,t)

E(0, A) x(0, T)" a - t = a 0 - t 0 } = { ( a 0 + s ,

t0+s):

x

{T}, S

s E ( 0 , a)}.

For the sake of convenience we introduce following the definition of the solution. DEFINITION 1.1. By the solution of system (1), we mean functions p~ (i = 1,2) E L2(Q), which belong to C(S; L2(ft)) A AC(S; L2(f~)) A L2(S; Hl(f~)) C~L~oc(S; H2(f~)) and satisfy

~Pl

OPl

O--T + ~

-

-

k l A p l = - P l (a,t,x)pl -- )'1 (a, t, x) P2 (t, x) Pl - ul (a, t, x) Pl,

in Q,

Op2 Opz Ot + ~ -- k2AP2 = -#2 (a, t, x) P2 4- A2 (a, t, x) PlP2 -- u2 (a, t, x) P2,

in Q,

j~

131(a,t,x, P2(t,x))pl (a,t,x) da,

in L 2 (fl),

lim P2 (e, t + e, x) = fo /32 (a, t, x, P1 (t, x))P2 (a, t, x) da,

in L 2 (fl),

lim Pi (a + e, e, x) = P~o (a, x),

in L 2 (~),

A

lim

pl(a,t ~-~,X)=

e----~O+

A E--.0 +

~--~0+

~ (a, t, x) = O, -P~ Ou

i=1,2,

onE.

576

CHUN

ZHAO

et al.

The purpose of this paper is to establish the necessary optimality conditions and the existence and uniqueness for the optimal control problem (OH). The results of this paper extend directly those in [2]. This paper is organized as follows. In Section 2, we obtain the existence and uniqueness of solution of system (1). Section 3 gives the necessary conditions of optimality. In Section 4, we prove the existence and uniqueness of an optimal control for problem (OH). 2. T H E EXISTENCE AND UNIQUENESS OF SOLUTION In this section, we discuss the well posedness of solution for system (1). THEOREM 2.1. For any (Ul,U2) C U system (1) has a unique nonnegative solution (p~,p~) E LZ(Q) x L2(Q) • L°°(Q) x L ~ ( Q ) , such that

o < p~ (a, t, x) < M1,

a.e. (a, t, x) • Q,

i = 1, 2,

where M1 > 0 is a constant independent of p~ and ui, i = 1, 2.

PROOF. Without loss of generality, we may assume that ul = 0, u2 = 0. For any given (hi, h2) • L2(Q) x L2(Q) n L°°(Q) x L~(Q), hi(a,t,x) >_ 0 (i = 1,2), denote by

Hi (t, x) =

/j

hi (a, t, x) da,

i = 1, 2.

By Theorem 4.1.3 in [2], the following system,

Opl + Opl _ k l A P l = - # l ( a , t , z ) p l - A x ( a , t , x ) I - I 2 ( t , x ) p l ,

o--i--g-J

inQ,

Op2 Op2 - k2Ap2 = - # 2 (a, t, x) P2 + A2 (a, t, x) PlP2, O-/ + ~-a

in Q,

/,

Pl (0, t, x) = / ~

A

/31 (a, t, x, H2 (t, x)) Pl (a, t, x) da,

in QT,

Jr)

(3) P2 (0, t, x) =

/32 (a, t, z, H1 (t, x)) P2 (a, t, x) da,

Opi (a, t, x) = O, 0~

Pi(a,0, x) = p i 0 ( a , x ) , has a unique nonnegative solution, (ph,ph) • n 2 (Q) × n 2 (Q) n n ~ (Q) × L ~ (Q) .

in QT,

on E,

i=1,2,

inQA,

Optimal Control of Harvesting

577

Let/51 C L2(Q) n Lee(Q) be the solution of the following system,

Opl Opl O----t+ ~ - klGpl = -#1 (a,t,x)pl,

Pl (0, t, x) =

in Q,

gl (a, t, x, O) Pl (a, t, x) da,

~Pl

0u (a, t, z) = 0,

in QT~

on ~, in QA.

Pl (a, 0, x) = Pl0 (a, x),

Using the comparison principle of population dynamics in [2], combining the Assumption (A4), we obtain that ph (a, t, z) _< Pl (a, t, z) for a.e. (a, t, x) E Q. Similarly, if hi (a, t, z) _< Pl (g, t, Z) for a.e. (a, t, x) E Q, then ph(a, t, x) <_ ~2(a, t, x) for a.e. (a, t, z) C Q, which/~2 E L2(Q)c~ L ~ ( Q ) i s the solution of the following system,

OP20t + ~OP2 _ k25p 2 = -#2 (a, t, z) P2 + A2 (a, t, x) P~/51,

P2 (0, t, x) =

/:(/0

f12 a, t, X,

fil (a, t, x) da

/

in Q,

P2 (a, t, x) da,

Op2 (a, t, z) = 0,

in QT,

on E,

0u

in QA.

P2 (a, O, x) = P2o (a, x), Let B={(hl,h~)EL

2 ( Q ; R 2) : 0 _< h~ (a, t, x) < / 5 i ( a , t , x ) , a.e. i n Q , i = 1 , 2 } .

Define the mapping A: L2+(Q) x L2+(Q) ---+L2+(Q) x L2+(Q), A (hi, h2) = (phl,ph),

Hi (t, x) ~-

/j

h i (a, t,x) da,

i : 1,2,

where (Pl,P2) h h is the solution of (3) corresponding to (hi, h2). Since 0 < phi(a , t, x) < Pi(a, t, x) a.e. in Q, for any (hi, h2) E/3, we have A(B) C ]3. Denote by norm on L2(Q) x L2(Q),

II(hl, h2)ll = (llh~ll 2 + IIh2112) 1 / 2 and for any constant r > 0 T

[Ihi[I ~ =

Irh~ (-,t, ')IIL~(QA)exp (--4rt) dr,

i = 1,2,

which is equivalent to the usual norm on L2(Q). With respect to this norm, A is a contraction on B. Indeed, for any (h~t, h~2) C B, system (3) has a unique nonnegative solution (Pil,P~2) C L2(Q) x L2(Q) A L ~ ( Q ) x L ~ ( Q ) , i = 1,2.

578

CHUN ZHAO et al.

Denote by (Wl, w2) = (Pll -P21,P12 -P2~). It is obvious that (wl, w2) is the solution of

Owl . Owl Ot ~--~-a - k l A w l = -t~w~-A1H12 (t,z)Wz-(H12 if, z) - H22 if, x)) ~1P21,

in Q,

C9W2 , OW2 ~ ^ Ot b ~ - a --t~2LAW2

in Q,

:

-l't2w2-Jc)t2PllW2JcA2P22W1'

A W 1 (0, t, X) = /

/31 (a, t, x, H12 (~, x)) w I (a, t, x) da A

+/

[A (a, t, x, His (t, ~)) - 91 (a, t, ~, H~2 (t, ~))] P21 da,

in QT, (4)

A

w2(O,t,x) =

fl2(a,t,x, H u (t,x))w2(a,t,x)da +

[32 (a, t, x, H n (t, x)) - 32 (a, t, x, H21 (t, x))] p22 aa,

Owi (a, t, x) = 0, 0L,

in QT,

on E,

in QA,

wi (a, 0, x) = 0,

where mj(t, x) = f0a h~j(a, t, x) aa, i, 5 = 1, 2. Multiplying (4)i by w~ (i = 1, 2) and integrating on (0, A) x (0, t) x Q, we have I1~1 (', t,.)][L:(QA) 2 2 + [[w2 (.,t, .) /IL~(e~)-< C ~t Ilhll (., s,

.) - h2~(.,s, ')IIL2(Q~) 2

ds

(5) +C

2 [[h12 (.,s,.)--h22(',S,')[lL~(QA)as,

where C > 0 is a constant independent of wi, hiy, i,j = 1, 2. It follows from (5) that liA (hll , hi2 ) - X (h21, h22)[12 = lizzIE ~ + [l~tl ~

= /T

(HWl (', t, ") IIL2(QA) 2 2 + []W2 (.,t, ')[IL2(QA)) exp ( - 4 C t ) dt T

<_C

t

/i/o(

libel(-, s, -) - h2~ (., s, ')tIL~(QA)

2 + 1]h12 (', s, ') - h22 (', s~ .)][L2(QA)) ds . e x p ( - 4 C t ) T ( ]lhll(',S,')--h21(',s,')llL2(Qn)

+llh12(.,s,.)--h22(.,s,.)llL2(QA)

_<~

Cexp(-4Ct) dtds

lib11 (., s, .) - h21 (., s,')ll~(qA)

2 + [th12 (., s, .) - h~2 (., s, ")[[L~(QA)) exp ( - 4 C s ) ds

1 = ~ I](hll, h12) - (h21, h22)112 •

dt

Optimal Control of Harvesting

579

Hence, A is a contraction on the space (B, I["[[) and via Banach's fixed-point theorem, we conclude the existence of a unique fixed point (p0, p0) E B for A, which is the solution of system (1). Let M1 = max{[]pl[[LO~(Q), [[/~2[ILO~(Q)}. We have

O<_p?(a,t,x)<_M1

a.e. (a,t,x) EQ,

i=1,2,

for all (Ul,U2) E U. This completes the proof.

3. T H E O P T I M A L I T Y

CONDITIONS

Before stating our main results, we prove the following lemmas, which is useful in proving our results. LEMMA 3.1. Let (p~, p~) and (p~, p~) be the solutions of system (I) corresponding to the controls (Ul, u2) and (ul, ~t2) E U, respectively. Then, 2

2

~=1

i=1

where C is a constant independent of p~, py, ui, fq, i = 1, 2. PROOF. Denote by (Yl, Y~) = (P1 - P~,P~ - P~). Then (Yl, Y2) is the solution of 0yl ~_0yl __ ]~I~Yl =--#lYl--,~lP~ (P~ ( t , x ) - P ~

Ot

Oa

(t,x))

--,~lP~ (t, X) Yl--UlYl--(Ul --~1)P~,

Oy2 ~a 0~ ~-

in Q,

A u

-k2AY2 =-/~2Y2+ 2P2Y1 ~2 + A2PlY2 - u2y2 - (u2 - ~22)p~,

in Q,

A

yl (O,t,x) =

fll (a,t,x,P~ (t,x))yl (a,t,x) da +

~l(a,t,x,P~(t,x))--fll (a,t,x,P~(t,x))]p~(a,t,x)da,

(6) in

~T

A

y2 (o,t,x)

=f0 9: (a,t,x,P (t,x))y2 (a,t,x) da A

+foo ~2 (a,t,x,P~ (t,x))-fl2 (a,t,x,P~ (t,x))] p~ (a,t,x) da,

=0,

yi (a,O,x) --0, where pu(t, x) = f : p•(a, t, x) da, P~(t,x) = f : p ~ ( a , t , x ) da, i = 1,2.

in QT,

on E,

in QA,

CHUN ZHAO et al.

580

Multiplying (6)~ by y~ (i = 1, 2), integrating on (0, A) x (0, t) x ~, we obtain 2

2

t

Ily~ (.,t, ")llc:(+~)-<

IIL:(,z~)

i=1

ds

i=1 2

t

A

J~o ~o

+M1E i=1

J~lu~ (a, s,x) - fq (a, s,x)l lYi (a,s,x)l dxdads,

where M1 is given in Theorem 2.1, C > 0 is a constant independent of y~, ui, and g/, i = 1, 2. This implies that

Ily~ (', s, ")IIL~(QA) 2 as

Ily~ (.,t, .)11L"(QA) ~ --< ( 1 + C ) i=1

= 2

t

A

+M~-~"fo fo fa lu4(a's'x)-ft~(a's'x)[2dxdads" i=1

By Bellman's 1emma, we deduce 2

2

i=1

i=l

for any t E (0, T). Integrating the last inequality on (0, T), we obtain 2

2

E ItY~IIL~(Q) 2 - 2 . < TM21e(I+c)TE Nui - ~d]/~ i=1

i=1

This completes the proof. Similarly, the following Lemma holds. LEMMA 3.2. Let (q~, q~) and (q~, q~) be the solutions of the following adjoint system (7) corresponding to the controls (Ul, u2) and (~2~, g2) E U, respectively,

8ql cOql O---t-+ -~a + klGql = #1ql - fll (a, t, x, P~ (t, x)) ql (0, t, ×

if0 a

Ofl2

Z) -- q2 (0, t, X)

(a, t, X, P~ (t, x)) p~ (a, t, x) da

+AlqlP~(t,x)+ut(l+ql)-a2p~q2,

inQ,

0q2 + Oq2 ~ + ~2/~q2 = ,~q~ - & (a, t, x, P ? (t, ~)) q2 (0, t, ~) - q~ (0, t, ~)

o~-

x

foa -~s 05~ (a,t,x,P~

-k u2 (1 + q~) +

aq.__j_/(a,t, x) ----0, 0~

rA

on

E,

qi (a, T, x) = O,

in QA,

qi (A, t, x) = O,

in QT,

u~ a

w~erc P ? ( t , x ) = Jo p~t , t , x ) d a ,

i = 1,2.

(t,x))pl (a,t,x) da - A2q2pl

(Alp~ql) (a, t, x) da,

(7) in Q,

Optimal Control of Harvesting

581

Then, 2

2

--

i=1

i=1

uil]L2(O)

where C is a constant independent of q~, @, u~, ui, i = 1, 2.

By Lemma 3.1, we have that the following lemma holds. LEMMA 3.3. Suppose that (u~, u~) C U is an optima1 centre1 for problem (OH) and denoted by (Pl~t*, P~ *) the solution of system (1) corresponding to (u~, u~) Then, for any (vl, v2) E L~(Q) x L°°(Q), such that (u~ + zvl, u~ + ¢v2) E U and for any e > 0 small enough, the following Bruit holds P~ + ~ - P ~

,P2

-P2

inL2(Q) x L 2 ( Q ) ,

--*(31,z2),

a s s ~ O +,

where (p~*+ev, P2u*+ev) is the solution of system (1) corresponding to (u*1 + zvl, u~ + ev2) and (zl, z2) satisfies Ozl Ozl 0--[ + ~ - k l A z l = -~1zl - A l p ~ ' Z 2 ( t , x ) - A 1 P ~ " (t,x) z l - u ~ z l -vlp~*, c9z2

in Q,

0z2

Ot + -~a - k 2 A z 2 = - # ~ z 2

in Q,

+ A2p~*zl + A2p~*z2 - u~z2 - v2p~*,

zl (O,t,x) = ~0 A ~ l ( a , t , x , P ~ * ( t , x ) ) z z ( a , t , x ) d a

+ 32 (t, x)

in Qr,

(8)

)

z2(O,t,x)=

/3~ a,t,x, P f (t,x) z2(a,t,x) da in QT,

~Z i

0--~ (a, t, x) = O,

on E,

z~ (a, O, z) = O,

in QA,

where Zi(t,x) = f : zi(a,t,x)da,

P S ( t , x ) = f:p~*(a,t,x)da, i = 1,2. Concerning the optimality conditions, the following results hold.

THEOREM 3.4. If u* = (ul, u~) is an optimal control for problem (OH) and (p~*, p~*) is the solution of system (1) corresponding to (u~, u~), then

u~=min{l(l+qi)+p~*,Li}, where (ql,

i---- 1,2,

q2) is the solution of adjoint system (7) corresponding to (u~, u~).

582

CIfUN ZHAO

et aL

PROOF. The existence and uniqueness of solution to system (7) and (8) can be proven by the fixed-point theorem. Since (u~, u~) is an optimal control for problem (OH), for any given (vt, v2) E L°°(Q) x L°°(Q), such that (u~ + evl,u~ + sv2) E U and for any ~ > 0 small enough, we get E/Q[

] dadtdx>

* u*-

i=1

(u~+~vi) -u*+sv

dadtdx,

'=

which implies *

£/Q

*

2

*P~+~V-P~ z dadtdx+~-~"/Qvi(P~*+~'-2u~-~vO

i=1

_

(9)

i=1

By Lemma 3.3, passing to the limit as s ~ 0 + in (9), we have 2

:CJQ

÷

i=1

<_O.

Multiplying (7)i by z~ (i = 1, 2) and integrating over Q, we obtain after some calculation that 2

2

~'~,~u~z~dadtdx=~-~/ev. ~p~~"q~dadtdx. • i=l

Q

(11)

i=1

It follows from (10) and (11) that

2~l/qV~[(l +qdp~"-

dadtdx <_O.

By standard control analysis, we conclude ui* = m i n { 2 ( 1 +qi) + Pi~* ,Li } ,

i=1,2.

This completes the proof. 4. E X I S T E N C E AND UNIQUENESS OPTIMAL CONTROL

OF

In this section, we prove the existence and uniqueness of optimal control. THEOREM 4.1. f i T is small enough, then problem (OH) has a unique optimal control in U. PROOF. For any given (ul, u2), (vl, v2) E U and ¢ E (O, 1), define H(s) = J(s(ul,u2) + (1 -

~)(v~,~)). We shall prove that H'(¢) is strictly monotone. Indeed, denote by (p~, p~) and (p~+5, p~+~) the state corresponding to controls e(ul, u2) + (1 - z)(vl, v2) and (s + 5)(Ux, u2) + (1 - (s + 5))(Vl, v2), respectively. By (2), we have 2

=

o

/

~+5 (ViAv (£_7~)(~ti_Vi))_pi (Vi~_£(Ui__~i))] da dt dx

i=1

- limE~ 5--~0 i = 1

(vi+(e+5)(ui-v~))2-(vi+e(ui-vi))

2 da at dx

Optimal Control of Harvesting

583

2

= ~-~01im~-~ ~1 fQ L,P~[(-~+6_ p~) (v~ + (s + 5) (u~ - v~)) + p~5 (u~ - v~)] da dt dx i=1 2

- lim ~--~. xa iQ [2~ (vi -F £(ui -- vi)) (ui -- Vi) -~-~2 (ui -- yi)2 ] dadtdx 570 2

i=1

i=l

y

a¢/ 2

- ~ l iQ 2 (vi + e (ui - v' ) ) (u~ - v') da dt where (z~, z~) is the solution of (8) corresponding to the control (vl + ~(ul - vl), v2 + e(u2 - v2)). By the same argument as in Theorem 3.4, we obtain 2

P

H' (c) = E / ~ ( u i - vi)[p~ (1 + q ~ ) - 2(vi + ¢ ( u i - vi))] dadtdx, i 1 J~ where (q~, q~) is the solution of (7) corresponding to the control (vl + s(ul - Vl), v2 + z(u2 - v2)). Given ~, p E (0, 1) and ~ ~ p, we have 2

[H' (¢) - H' (p)] (c - p) = (¢ - p) E i=1

W h e n T C < 1 (see Lemmas 3.1 and 3.2). By L e m m a 3.1, L e m m a 3.2 and using Holder's inequality,we get [H' (,) - H ' (p)] (z - p) _< T ~ I ( ,

- p)' (llul - ",llL./q) + II-. - v.llL.
- 2 (¢-p)~ (11~- "~ll~./q/+ I1~ ~ where C1 is a constant. When TCl < 1, if I1~ -~IIL'(Q) #

"~IIL.(Q)),

l[~ -,~NL.(Q), then we have

[H' (z) - H' (p)] (s- p) _< -(~ - p)2 (11~ - .~llz.(q) -I1~

If I1~ - v~llL.(q) = I1~ - .~tlL.(<~), then

- v~llz.(Q)) ~ < o.

we obtain

[H' ( ¢ ) - H' (p)] ( ~ - p ) < 4 ( T x / ~ -

1 ) ( ~ - p)' Ilul

vlN~(Q)

0.

Hence, H~(¢) is strictly monotone. Consequently, J(u~, u2) is strictly concave in U. De,he the functional ~ : L~(Q) × L~(Q) -~ [-~, +~), (Ul,U2)=~ J(ul'u2)' [ -o~,

if (ul,u2) EU, if (ul,u~) ~ U.

It is clear that (b is concave in L2(Q) × L2(Q). It follows from Lamina 3.1 that ~ : L=(Q) × L~(Q) --~ [-0% +co) is upper semicontinuous. Since U is a closed, convex and bounded subset in L~(Q) × L2(Q) and J(Ul, u2) is strictly concave in U, by Theorem Al.l.2 in [2], J(Ul, u2) attains

584

CHUN ZtIAO et al.

the unique m a x i m u m in U, which implies problem (OH) has a unique the o p t i m a l control in U. This completes the proof.

We now give the following two remarks to show the practicability of the optimality conditions given in Theorem 3.4. REMARK 1. If (1/2)(1 + qi)P~* > Li or (1/2)(1 + qi)P~* < 0, by T h e o r e m 3.4, t h e n we have

1 (1 + q~) p~" > Li, L~, if~ u~=

0,

if l ( l + q ~ ) p ~ .

i=1,2.

<0,

Namely, (u~, u~) is a bang-bang control. REMARK 2. Using s t a n d a r d arguments of the t h e o r y as in [2,11], we can o b t a i n the optimal feedback formula, which can be used for numerical c o m p u t a t i o n of the optimal control. We expect to go t h r o u g h this problem in a subsequent paper. Hence, t h e o p t i m a l i t y conditions given in T h e o r e m 3.4 have the practicability and can be used like the m a x i m u m condition in optimal

control.

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