UNIQUENESS OF THE OPTIMAL CONTROL FOR A
LOTKA-VOLTERRA CONTROL PROBLEM WITH A
LARGE CROWDING EFFECT
J.L. GAMEZ AND J.A. MONTERO
Abstract. Uniqueness of the optimal control is obtained by assuming certain conditions on the crowding eect of the species. Moreover, an approximation procedure for the unique optimal control is developed.
1. Introduction
In this work we study an optimal control problem whose steady-state equation is a P.D.E. of Lotka-Volterra type:
;u(x) =u(x) a(x);f(x);b(x)u(x)] x2
u(x) = 0 x2@ (1.1)
where is a bounded and regular domain in IRN. The unknown u rep- resents the concentration of the biological species. The coecients which appear in (1:1) can be interpreted as follows: function ais the growth rate, function bmeans the crowding eect andf denotes a control on the process of production (see 14]). The space of admissible controls is
L 1
+() =fg 2L1() :g(x)0 a.e. in g:
Under the hypothesis
H]: ab2L
1(), essinfb>0,
the equation (1:1) has, for every admissible control f, a unique maximal nonnegative weak solution which will be denoted by uabf (see 2]).
The benet-payo functional may be given by
J
KL(f) =
Z
Ku
abf(x)f(x);Lf2(x)dx
which express the dierence between prot and cost. Here, K >0 denotes the sale price of the species, and L>0 is the cost of the control. To study the existence of maximum forJKLand some additional properties, we prefer to consider the re-scaled functional J = JKLL , this is
J(f) =
Z
(uabf(x)f(x);f2(x))dx
Dep. Analisis Matematico, Universidad de Granada. 18071-Granada, Spain. E-mail:
jlgamez@goliat.ugr.es - jmontero@goliat.ugr.es.
Received by the journal May 28, 1996. Accepted for publication November 8, 1996.
The authors have been supported in part by the \Direccion General de Ense~nanza Superior, Ministerio de Educacion y Ciencia" (Spain), under grant number PB95-1190 and by EEC contract, Human Capital and Mobility program, ERBCHRXCT940494.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
2 J.L.GAMEZ AND J.A. MONTERO
where the constant > 0 denotes the rate between the sale price of the biological species and the cost of the control.
We denote by (Pab) the problem of nding an admissible control, f, such that
J(f) = sup
g 2L 1
+ ()
J(g):
Such a control,f, will be called an \optimal control" of problem (Pab).
The existence of optimal controls under hypothesis H]has been proved in 4], where a necessary and sucient condition to obtain the positivity of the optimal benet is given.
We are interested in nding conditions which guarantee the uniqueness of the optimal control and its approximation.
In 5] it was obtained the uniqueness and an approach scheme for the optimal control was developed, in the case where , a, b are xed, and
> 0 is small enough. The result can also be applied for , a, xed, and ba suciently large positive constant, or b =b0, where 2IR is a a suciently large constant and function b0 satises H]. However, the case wherebis a nonconstant arbitrary function does not seem to be approachable by that method.
This last case is discussed here. We x , and a, and we prove unique- ness of the optimal control, provided that esssupb Messinfb, for some 1M <2, and essinfb is large enough (see Theorem 3.5).
Similar problems are studied in 8, 10, 11, 13]. In 11] the authors have studied a control problem with a steady-state equation similar to (1:1), but with Neumann boundary conditions. Their space of admissible controls is
C
=fg2L1() : 0g(x) a.e. in g
where 0 < < essinfa(x). In this particular situation they prove the existence of an optimal control f 2 C. Under suitable conditions, they describe the optimal control f in terms of the solution of an appropriate elliptic system (the optimality system). In 8] the periodic problem for the parabolic model is considered.
In section 2 we x some notation, and we also recall some results obtained in 4, 5]. In section 3 we obtain the optimality system, which is used to prove the uniqueness of the optimal control. In section 4 we provide an iterative scheme to approximate the unique solution of the optimality system.
2. Preliminary results
For a bounded functione2L1+(), we denotee= ess supe, and accord- ingly e= ess infe. From now on we consider the equation (1:1), assuming the hypothesis H].
For every q 2 L1(), we dene 1(q) to be the principal eigenvalue of the following eigenvalue problem:
;u(x) +q(x)u(x) =u(x) x2
u(x) = 0 x2@:
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It is known (see for instance 6]) that 1(q) satises the variational charac- terization
1(q) = inf
u2H 1
0 ()nf0g
Z
jruj
2+
Z
qu
2
Z
u
2
(2.1) where H01() is the usual Sobolev space. It is also known that 1(q) has algebraic multiplicity equal to one. Moreover, one can choose an associated eigenfunction, 1(q), such that 1(q) 2 C1() (the space of Holderian functions), 82(01),1(q) strictly positive in , and k1(q)kL1() = 1.
As a consequence of (2:1), it follows that1(q) has the following proper- ties:
i) Ifq1q2 2L1(), with q1(x) q2(x), a.e. in , then 1(q1)1(q2).
Moreover, if the setfx2 :q1(x)<q2(x)ghas positive measure, then
1(q1)<1(q2).
ii) 8q2L1(),8M 2IR,1(q+M) =1(q) +M. iii) 1(q) depends continuously onq 2L1().
We now recall some properties of the Schrodinger operator (; +q).
Lemma 2.1. Consider q2L1(), satisfying1(q)>0. Then:
i) For each f 2L2(), the linear problem
;u+qu=f in
u= 0 on @
admits a unique weak solution u 2 H01(). Moreover, if f 2 L1(), thenu2C1(),8 2(01).
ii) Letu1u2 2H1() be such that 82H01(), 0,
Z
ru
1
r+Z
qu
1
Z
ru
2
r+Z
qu
2
u
1 u
2 on @:
Then u1 u2 in . We remark here that the inequality u1 u2 on
@ means (u1;u2)+2H01().
iii) Consider f 2 L2(), f(x) 0 a.e. in , and p1p2 2 L1(), with
p
1(x)p2(x) a.e. in , and 1(p2) >0. We denote by !i (i= 12), the unique weak solution (it exists from i)) of problem
;!i+pi!i =f in
!
i= 0 on@: Then
!
1(x)!2(x) a.e. in :
Proof. Part i) can be found in 5]. Parts ii) and iii) are proved in 12].
We also point out some known results about problem (1:1):
Lemma 2.2. (see 2, 3, 7]) Assume hypothesisH]. Then, problem(1:1) has a nonnegative and nontrivial weak solution if and only if 1(;a+f) <0.
Moreover, in this case the nonnegative and nontrivial solution is unique. So, for any f 2 L1(), we can dene uabf to be the maximal nonnegative
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4 J.L.GAMEZ AND J.A. MONTERO
solution of equation (1:1). For 1(;a+f) < 0 one has uabf 2 C1(), and the following a priori bounds:
;
1(;a+f)
b
1(;a+f)(x)uabf(x) a;f
b
8x2: (2.2) Consequently, uabf 0 if 1(;a+f)0 and uabf is strictly positive in if1(;a+f)<0. Moreover, for abxed, the mapL1()!C1(),
f 7!u
abf is continuous.
Taking into account that every suciently large constant is an upper- solution of (1:1) (we refer to Amann 1] for the denition of lower- and upper-solutions), one infers that every bounded lower-solution, w, of (1:1) must satisfy
w(x)uabf(x) 8x2: We now recall some of the results obtained in 4, 5].
Theorem 2.3. ( 4], Existence of optimal control). Assume hypothesisH]. Then there exists an optimal control for problem (Pabf), i.e.
9f 2L 1
+() such that J(f) = sup
g 2L 1
+ ()
J(g):
Moreover if a0 then f 0, and if a>0, the optimal controlf satises
f a
b
a.e. in: (2.3)
Theorem 2.4. ( 4], Positivity of benet). Assume hypothesis H]. Then, sup
g 2L 1
+ ()
J(g)>0 () 1(;a)<0:
Remark 2.5. 1. Observe that we have not imposed any upper bound on the control space, but relation (2:3) implies that the space of \inter- esting" controls is (in fact) bounded.
2. Theorem 2.3 justies the hypothesis 1(;a) <0 in many of the next results. In particular, observe that1(;a)<0)a>0.
Lemma 2.6. ( 5, Lemma 3.2]). AssumeH]and1(;a)<0. Iff 2L1+() is an optimal control, then
f = 2 uabf(1;pabf)+ a.e. in where pabf is the unique solution of the linear problem
;pabf + (;a+f+ 2buabf)pabf =f in
p
abf = 0 on@ (2.4)
(Observe that 1(;a+f+ 2buabf)>1(;a+f+buabf) = 0).
3. Optimality system and uniqueness of the optimal control
In this section, under suitable conditions involving functionb, we are going to prove that any optimal control can be expressed in terms of a solution of a certain system, the optimality system (3:8)-(3:9).
We provide conditions to assure the uniqueness of solution for the opti- mality system and therefore, uniqueness of optimal control.
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We will x , a and , and we will look for conditions on function b to obtain the uniqueness of the optimal control. For the sake of simplicity, from now on, we denote by ubf the maximal nonnegative solution of (1:1), instead of uabf, and bypbf the solution of (2:4), instead ofpabf.
Our rst objective is to nd conditions on function b to guarantee, for any optimal control f 2 L1+(), the following estimation on the solution of (2:4):
p
bf
1 a.e. in :
In this case, we will be able to express any optimal control in terms of the solution of an appropriate system of P.D.E's (see Theorem 3:3 below).
To obtain this estimate we will use the statements ii)-iii) of Lemma 2:1.
In fact, as a consequence of ii) we deduce that pbf 0 in . To get an upper bound for pbf, it will be sucient to nd an upper bound for f and a lower bound for the expression ;a+f+ 2bubf.
Lemma 3.1. Consider ab satisfyingH],1(;a)<0 and
bMb for some 1M <2: (3.1) One may choose "2IR+ such that
"<
M
2 1(;a+ 2
M u
10): (3.2)
Then, there exists a positive constant b0 (which depends on ,a, and "), such that if bb0 andf is any optimal control, the functionpbf dened in (2:4) satises the estimates
0pbf Q
b
a.e. in where Q is the unique solution of
;Q+ (;a+M2 (u10;"))Q=a in
Q= 0 on @: (3.3)
Proof. As a consequence of the hypothesis 1(;a) <0 the function u10 is strictly positive in and therefore 1(;a+u10) = 0. So,
1(;a+ 2
M u
10)>0:
Due to the choice of " and the statement i) of Lemma 2:1, there exists a unique solutionQof (3:3). From the continuity of the mapL1()!C1(),
f 7!u
1f (see Lemma 2.2), there exists a positive constantb0 such that for
bb
0, if g2L1+() with g ab , then
u
1g u
10
;" a.e. in : (3.4) If f is an optimal control for (Pab), then by using that ubf is a lower- solution of (1:1) and thatk uk f =u1f 8k>0, it follows
2bubf = 2 b
b bu
bf
2
M bu
bf = 2
M u
1f
: (3.5)
Moreover since bb0, from (2:3) and (3:4) we obtain
;a+f + 2bubf ;a+ 2bubf ;a+ 2
M
(u10;") a.e. in : (3.6)
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6 J.L.GAMEZ AND J.A. MONTERO
The function Qb satises
;
Q
b
+
;a+ 2
M
(u10;")
Q
b
= a
b :
From ii) and iii) of Lemma 2:1 taking into account that ab f and the inequality (3:6), we conclude the proof of lemma 3.1.
Corollary 3.2. Assume H],1(;a) <0, (3:1) and
bb
1
max fb0kQk1g: (3.7)
Then for any optimal control f, the function pbf dened in (2:4) satises:
0pbf 1 a.e. in:
Theorem 3.3. (Optimality system). Let us assumeH],1(;a)<0, (3:1) and (3:7). Then, any optimal controlf 2L1+()can be expressed as
f = 2ubf(1;pbf)
where the pair (up)(ubfpbf)is a solution of the optimality system:
;u=u
a; b+2(1;p)]u
in
;p+p(;a+ 2bu) = 2u(1;p)2 in
u=p= 0 on @
(3.8) satisfying
0p1 u>0 a.e. in : (3.9) Proof. It is a direct consequence of previous results.
Lemma 3.4. Let us assume H], 1(;a) < 0, (3:1) and (3:7). Then any solution (vq) of system (3:8) under the conditions (3:9) satises
w(x)v(x) a
b
0q(x) Q(x)
b
a.e. in where w is the unique maximal nonnegative solution of
;w=w
a;
2w;bw
in
w= 0 on@ (3.10)
and Q is dened by(3:3).
Moreover, by choosing "as in (3:2), one has 2b(x)w(x) 2
M
(u10(x);") a.e. in :
Proof. Observe thatvis a nonnegative and nontrivial solution of an equation similar to (1:1), so, estimates of type (2:2) hold, obtaining 0 <v(x) a a.e. in . The function w is a bounded lower-solution for the equationb
satised by v, therefore
w(x)v(x) a
b
a.e. in : (3.11)
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By using the established notation for the solutions of (1:1), we have w =
u
b
2
w. Then, similarly to (3:5) and (3:4), 2bw= 2bub
2 w
2
M bu
b
2 w= 2
M u
1
2 w
2
M
(u10;") a.e. in : (3.12) To prove q Q
b
in , we denote h= 2v(1;q). Observe that q satises
;q+ (;a+h+ 2bv)q=h in
q= 0 on@:
Similarly to the proof of lemma 3:1, by using (3:11), (3:12), the denition of Q and lemma 2:1, the proof is complete.
The main result of this section is the uniqueness of the optimal control.
We will use the optimality system and the previous lemma to prove it.
Theorem 3.5. (Uniqueness of the optimal control). SupposeH],1(;a)<
0 and (3:1). Then problem (Pab) admits a unique optimal control, pro- vided that b is large enough.
Proof. By choosing">0 as in Lemma 3:1, we denote 1(;a+M2 (u10;
"))which, from (3:2)is strictly positive. Take
b 2
b
2
max
8
>
>
<
>
>
: a
2
2 + 2MkQk1
2 2 b21
9
>
>
=
>
>
(3.13) where Qis dened in Lemma 3:1, and b1 as in Corollary 3:2.
Fix = 1
2 + 2MkQk1
. In virtue of Lemma 3:4, it is easy to check that (up) is a solution of system (3:8)-(3:9) if and only if (ur)(up) is a solution of the following system,
;u;au+bu2+2(1;r )u2= 0 in
;r+r(;a+ 2bu);2u(1;r )2 = 0 in
u=r= 0 on @:
(3.14) with conditions,
0r(x) Q(x)
b
( 1
) w(x)u(x) a
b
a.e. in : (3.15) To show the uniqueness of (3:14), under conditions (3:15). Let us consider another solution (vs) of (3:14)-(3:15). Then
0 =;(u;v);a(u;v) +b(u2;v2)+
+2(u2;v2);2 (u2r;v2s) and 0 =;(r;s);a(r;s) + 2b(ur;vs);
;
2
u(1
;r)2;v(1
;s)2
:
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8 J.L.GAMEZ AND J.A. MONTERO
Similarly,
0 = ;(u;v);a(u;v) +b(u;v)(u+v)+
+2(u+v)(u;v);2 u2(r;s);2 s(u+v)(u;v) (3.16) and 0 = ;(r;s);a(r;s) + 2bu(r;s) + 2bs(u;v)+
+2 u
2
;(r+s)
(r;s);2 (u;v)
1
;s
2
:
(3.17) Multiplying (3:16) by (u;v), (3:17) by (r;s), integrating on , and adding both expressions,
0 =
Z
jr(u;v)j2;a(u;v)2+b(u+v)(u;v)2+ +2(u+v)(u;v)2(1;s); 2 u2(u;v)(r;s)
+ +
Z
jr(r;s)j2;a(r;s)2+ 2bu(r;s)2+ +(u;v)(r;s)
2bs;2
1
;s
2
!
+ +2 u
2
;(r+s)
(r;s)2
:
(3.18)
Observe that 2(u+v)(u;v)2(1;s) and 2 u;2 ;(r+s)(r;s)2 are nonnegative. Moreover, ifr(x)6=s(x) in a subset of with positive measure then, as a direct consequence of (3:15),
Z
2
2
;(r+s)
(r;s)2 >0:
Now, from (3:15) and (3:12), the functions (u+v)band 2busatisfy (u+v)b M2 (u10;")
2bu M2 (u10;")
in .The variational characterization of (recall (2:1)) implies
Z
jrqj
2+
Z
(;a+ 2
M
(u10;"))q2
Z
q
2
8 q2H 1
0():
Applying this inequality toq1 =u;vandq2 =r;sand taking into account previous considerations, from (3:18) it follows:
0
Z
(u;v)2+ (r;s)2+ +(u;v)(r;s)
2bs;2
1
;s
2
;
2 u2
!#
:
(3.19)
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Moreover, ifr(x)6=s(x) in a subset of with positive measure then previous inequality is strict. The choice of , (3:13) and (3:15) imply
2bs;2
1
;s
2
;
2 u2
2bkQk1
b
+ 22 +a2b22
1
2 + 2MkQk1
+a2b22
+ = 2 : Now, from (3:19),
Z
(u;v)2+ (r;s)2
Z
(u;v)(r;s)
2bs; 2
1
;s
2
;
2 u2
!
2
Z
ju;vjjr;sj:
This implies
Z
(ju;vj;jr;sj)2 0, with strict inequality if r(x)6=s(x) in a subset of with positive measure. Then it follows that r s, and therefore uv in .
Observe that the technique in 5] can be applied to prove uniqueness for
b = b0, where 2 IR is a suciently large constant and the function b0 satises H](and not necessarily (3:1)). This particular case induces us to think that hypothesis (3:1) could be improved, considering the possibility of
M 2. However, the proof above strongly requires (3:1).
4. Approximation of the optimal control
In this section, we consider the approximation problem to the optimal control. For it, we use the optimality system (3:8)-(3:9), because in virtue of Theorems 3:3, 3:5, the unique solution of that system describes the optimal control. We will dene an iterative scheme which, under more restrictives conditions on b, will converge to the unique solution of (3:8)-(3:9). The main diculty to prove this convergence is the absence of monotonicity of the terms appearing in system (3:8). Some of the ideas contained here can be found in 5, sect. 5] and 9, chap. V.].
For the sake of simplicity, we dene throughout this section the functions
BC D: IR2!IR as
B(xup) =u
a; b+2(1;p)]u
C(xup) =p(a;2bu)
D(xup) = 2u(1;p)2:
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10 J.L.GAMEZ AND J.A. MONTERO
Let us call u1w,u1 ab ,p10 andp1 Qb, wherewandQare dened in (3:10) and (3:3) respectively. We can nd ;>0 such that
B(xup) + ;u is %u %p
C(xup)+;2p is &u %p
D(xup)+ ;2p is %u %p
for any (xup) 2 0 ab ] 0kQk1b ] (here the symbol \%" means
\increasing with respect to", and \&" means \decreasing with respect to").
We construct inductively (observe that u1u1p1p1 have been dened before) the sequences fungfungfpngfpng as the unique solutions of
;un+1+ ;un+1=B(xunpn) + ;un in
u
n+1= 0 on @
;un+1+ ;un+1=B(xunpn) + ;un in
u
n+1= 0 on @
;pn+1+ ;pn+1=C(xunpn) + ;2pn+D(xunpn) + ;2pn in
p
n+1= 0 on @
;pn+1+ ;pn+1=C(xunpn) + ;2pn+D(xunpn) + ;2pn in
p
n+1= 0 on @:
We now collect the main properties of these sequences. The proof of the following assertions can be found in 5, Theorem 5.2] in a more general version.
Observe that the sequences fungfungfpngfpngdened above satisfy 1.
u
1 u
2
:::u
n u
n
u n;1
:::u 1 in
p
1 p
2
:::p
n p
n
p n;1
:::p 1 in and
u
n
%u
u
n
&u
p
n
%p
p
n
&p
pointwise in :
2. uupp 2W02q(),8q2(11). Moreover, these functions satisfy
;u =B(xup) in
;u =B(xup) in
;p=C(xup) +D(xup) in
;p=C(xup) +D(xup) in
u
=u =p =p = 0 on @:
(4.1) with conditions
wu
u
a
b
0pp Q
b
: (4.2)
3. If (up) is another solution of (3:8) with the property
u
1
uu 1
p
1
pp 1
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