1
OPTIMAL CONTROL FOR THE DEGENERATE
ELLIPTIC LOGISTIC EQUATION
M. Delgado1, J. A. Mon e o2and A. Su´a ez1
1. Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Fac. Ma em´a icas, C/ Ta ia s/n
C. P. 41012, Uni . Se illa, Se illa, Spain
2. Dp o. An´alisis Ma em´a ico
C. P. 18071, Uni . G anada, G anada, Spain
e-mails: delgado@nume .us.es, jmon[email p o ec ed] and sua ez@nume .us.es
Abs ac
We conside he op imal con ol o he ha es ing o he di usi e degene a e ellip ic logis-
ic equa ion. Unde ce ain assump ions, we p o e he exis ence and uniqueness o an op imal
con ol. Mo eo e , he op imali y sys em and a cha ac e iza ion o he op imal con ol a e
also de i ed. Sub-supe solu ion me hod, singula eigen alue p oblem and di e en iabili y
wi h espec o he posi i e cone a e he echniques used o ge ou esul s.
Key Wo ds. Degene a e logis ic equa ion, Singula eigen alue p oblems, Op imal con ol.
AMS Classi ica ion. P ima y 49J20, 49K20, 92D25, Seconda y 35J65.
Running head. Op imal con ol o degene a e logis ic equa ion
2M. Delgado, J. A. Mon e o and A. Su´a ez
1 In oduc ion
This wo k conside s he op imal ha es ing con ol o a species whose s a e is go e ned by he
degene a e (nonlinea slow di usion) ellip ic logis ic equa ion, i.e.,









−∆wm= (a− )w−ew2in Ω,
w= 0 on ∂Ω,
(1.1)
whe e Ω is a bounded and egula domain o IRN,N≥1; m > 1; a, and ea e bounded
unc ions wi h some es ic ions ha will be de ailed below.
Equ. (1.1) was in oduced in popula ions dynamics by Gu in and MacCamy in [5] desc ibing
he beha iou o a single species inhabi ing in Ω and whose popula ion densi y is w(x). Since
he popula ion is subjec o homogeneous Di ichle bounda y condi ions, we a e assuming ha
Ω is ully su ounded by inhospi able a eas. In such model, he posi i e unc ion e(x) desc ibes
he limi ing e ec s o c owding in he species and a(x) ep esen s he g ow h a e o he species.
The unc ion (x) deno es he dis ibu ion o con ol ha es ing o he species. Since will be
conside ed non-nega i e, obse e ha leads by educ ing he g ow h a e. Finally, he ope a o
−∆ measu es he di usion, i.e., he mo ing a e o he species om high densi y egions o low
densi y a eas. In his case, m > 1 (nonlinea slow di usion) means ha he di usion is slowe
han in he linea case m= 1, which gi es ise o mo e ealis ic biological esul s, see [5].
To s udy (1.1), we make he change o a iables wm=uand ob ain









−∆u= (a− )uα−euβin Ω,
u= 0 on ∂Ω,
(1.2)
wi h α= 1/m and β= 2/m. Unde hypo hesis (H2) below, we p o e ha o each , he e
exis s a unique posi i e solu ion o (1.2), ha i will be deno ed by u . The op imal con ol
c i e ia is o maximize he payo unc ional
J( ) := ZΩ
(λu h( )−k( )),
whe e h∈C1(IR+; IR+), k∈C2(IR+; IR+) and λ > 0 will be conside ed as pa ame e . J
ep esen s he di e ence be ween economic e enue measu ed by RΩλu h( ) and he con ol
cos measu ed by RΩk( ). He e, λdesc ibes he quo ien be ween he p ice o he species and
Op imal con ol o degene a e logis ic equa ion 3
he cos o he con ol.
The special case (quad a ic unc ional)
h( ) = and k( ) = 2,
was in oduced in dynamics popula ion by Leung and S ojano ic in [10] (see also [3], [9] and
e e ences he ein).
An op imal con ol is a unc ion ∈ C, whe e Cis a sui able subse o L∞(Ω), such ha
J( ) = sup
g∈C
J(g).
In he case m= 1, i.e., α= 1 and β= 2, and h( ) = and k( ) = 2, his p oblem has been
s udied in de ail in [3], [10] and [11]. In ac , some esul s o his wo k ha e been mo i a ed by
[3]. In hese pape s, unde ce ain assump ions in he coe icien s o he p oblem, he au ho s
ob ained he exis ence and uniqueness o he op imal con ol, as well as a cha ac e iza ion o
he op imal con ol by means he solu ion o he op imali y sys em. To ob ain he esul s, he
au ho s used mainly he sub-supe solu ion me hod, he de i abili y o he maps 7→ u and
7→ J( ) and he exp essions o hei de i a i es.
When m > 1, i.e. α < 1, his de i abili y is a he di icul han in he case m= 1, because i
in ol es linea ellip ic and eigen alue p oblems wi h unbounded po en ials in a neighbou hood
o ∂Ω. These di icul ies ha e been sol en ed by using esul s o singula eigen alue p oblems
om [2] and [6], and some classical ones o K asnoselskii, see [7]. They le us deduce he
F ´eche de i abili y om he Gˆa eaux de i abili y wi h espec o he posi i e cone. Mo eo e ,
he in oduc ion o he unc ions hand kin he payo unc ional leads us o es ablish he
hypo heses o assu e he exis ence and uniqueness o he op imal con ol.
An ou line o his wo k is as ollows: in Sec ion 2 we in oduce some no a ions and we gi e
some esul s o he exis ence and uniqueness o he p incipal eigen alue and o solu ion o a linea
ellip ic p oblems wi h unbounded po en ials. In Sec ion 3 we show he exis ence and uniqueness
o posi i e solu ion o (1.2), collec ing a esul om [4]. Mo eo e , we s udy he de i abili y
o he map 7→ u gi ing an explici exp ession o ha . In Sec ion 4, we show ha o λ
su icien ly small he e exis s a unique op imal con ol. In he las Sec ion we cha ac e ize he
op imal con ol. This cha ac e iza ion p o ides us he op imali y sys em and ce ain egula i y
o he op imal con ol. I is well known ha his egula i y can sugges nume ic me hods o
app oxima e he op imal con ol, which a e no conside ed in his wo k.
4M. Delgado, J. A. Mon e o and A. Su´a ez
2 P elimina ies
In his pape we use he ollowing no a ion: Ω is a bounded domain in IRNwi h a smoo h
bounda y ∂Ω and γ∈(0,2) ixed. Fo any ∈L∞(Ω) we deno e
M:= ess sup L:= ess in ,
L∞
+(Ω) := { ∈L∞(Ω) : L≥0}L∞
−(Ω) := { ∈L∞(Ω) : M≤0}.
Mo eo e , we deno e by P he non-nega i e cone o C1
0(Ω), whose in e io is
in (P) := {u∈C1
0(Ω) : u > 0 in Ω, ∂u/∂n < 0 on ∂Ω}
whe e C1
0(Ω) = {u∈C1(Ω) : u= 0 on ∂Ω}and nis he ou wa d uni no mal a ∂Ω.
Finally, o any Ω0⊂Ω, σΩ0
1and ϕΩ0
1s and o he p incipal eigen alue and he co esponding
posi i e eigen unc ion o he ope a o −∆ and homogeneous bounda y Di ichle condi ion wi h
kϕΩ0
1k∞= 1. In pa icula , we w i e σ1:= σΩ
1and ϕ1:= ϕΩ
1.
Assume
(H1) M∈L∞
loc(Ω) e i ying M(x)dΩ(x)γ∈L∞(Ω),
whe e dΩ(x) := dis (x, ∂Ω).
Gi en σ∈IR and ∈L∞(Ω), we conside he ollowing p oblems









−∆u+M(x)u=σu in Ω,
u= 0 on ∂Ω,
(2.1)









−∆u+M(x)u= in Ω,
u= 0 on ∂Ω.
(2.2)
Rema k 2.1 Obse e ha we a e no assuming ha M∈L∞(Ω) and ha a weak solu ion o
(2.2) o an associa ed eigen unc ion o he eigen alue σo (2.1) a e well de ined by he Ha dy
inequali y, see o ins ance [8].
The nex esul ollows om [2] and [6]. We include i o he eade ’s con enience.
Theo em 2.2 Assume ha Msa is ies (H1). Then:
Op imal con ol o degene a e logis ic equa ion 5
a) The e exis s a unique p incipal eigen alue (i.e., a eal eigen alue wi h an associa ed eigen-
unc ion in in (P)), which is simple and we deno e i by σ1(−∆ + M). Mo eo e , i
sa is ies
σ1(−∆ + M) = in
u∈H1
0(Ω) {0}







ZΩ
|∇u|2+ZΩ
M(x)u2
ZΩ
u2







.
b) (S ong Maximum P inciple) σ1(−∆ + M)>0i and only i ∈W2,p(Ω) ∩C1(Ω), wi h
p > N such ha 6= 0,−∆ +M(x) ≥0in Ω, ≥0on ∂Ω, hen ∈in (P).
By he a ia ional cha ac e iza ion o σ1(−∆ + M), i ollows:
P oposi ion 2.3 a) (Mono onici y espec o he po en ial) Assume ha Mi,i= 1,2sa is y
(H1) and M1≤M2. Then
σ1(−∆ + M1)≤σ1(−∆ + M2).
b) (Con inui y espec o he po en ial) Assume ha Mn, M,n∈IN sa is y (H1) wi h
ZΩ
Mnϕ2→ZΩ
Mϕ2,as n→ ∞ and o all ϕ∈H1
0(Ω).(2.3)
Then,
σ1(−∆ + Mn)→σ1(−∆ + M)as n→ ∞.
The ollowing es ima e will play an impo an ole in he nex sec ions.
Lemma 2.4 Assume ha Mn, M,n∈IN sa is y (H1),σ1(−∆+M)>0and (2.3). Then, he e
exis a posi i e cons an C0<1(independien o n) and n0(C0)∈IN such ha
C0ZΩ
|∇u|2≤ZΩ
|∇u|2+ZΩ
Mnu2∀u∈H1
0(Ω),∀n≥n0.(2.4)
P oo : Since σ1(−∆ + KM)→σ1(−∆ + M)>0 as K↓1, he e exis s K0>1 such ha
σ1(−∆ + K0M)>0. Le C0be such ha K0= 1/(1 −C0).
To p o e (2.4) i is su icien o show ha σ1(−∆ + K0Mn)≥0 o n≥n0. Bu σ1(−∆ +
K0Mn)→σ1(−∆ + K0M)>0. 2
The ollowing esul shows ha (2.2) possesses a unique solu ion.

6M. Delgado, J. A. Mon e o and A. Su´a ez
Theo em 2.5 Assume ha Msa is ies (H1) and σ1(−∆+M)>0. Then, he e exis s a unique
solu ion u∈C1,κ(Ω), o some κ∈(0,1), o (2.2). Mo eo e , he e exis s a cons an K > 0
(independien o ) such ha
kukC1,κ(Ω) ≤Kk k∞.(2.5)
P oo : Fo ∈C1
0(Ω) we conside he p oblem









−∆u=−M(x) in Ω,
u= 0 on ∂Ω.
(2.6)
By P oposi ion 2.3 in [6], he e exis s a unique solu ion u∈C2(Ω)∩C1,κ(Ω), o some κ∈(0,1),
o (2.6) wi h
kukC1,κ(Ω) ≤K1k kC1(Ω).
De ine G1:C1
0(Ω) 7→ C1,κ
0(Ω), 7→ G1( ) he unique solu ion o (2.6). We ha e shown ha G1
is bounded.
Fo h∈L∞(Ω) we conside he p oblem









−∆u=h(x) in Ω,
u= 0 on ∂Ω.
(2.7)
I is well known ha ixed h∈L∞(Ω), he e exis s a unique solu ion u∈W2,p(Ω) o (2.7) o
all p > 1, and
kukC1,κ(Ω) ≤K1kukW2,p(Ω) ≤K2khk∞.
We can de ine he map G2:L∞(Ω) 7→ C1,κ
0(Ω), h7→ G2(h) he unique solu ion o (2.7). We
ha e go ha G2is bounded.
Now, i we de ine
H:C1
0(Ω) 7→ C1
0(Ω), H(u) := u−G1(u),
deno e by i:C1,κ
0(Ω) 7→ C1
0(Ω) he compac imbedding and we pose G:= H◦i:C1,κ
0(Ω) 7→
C1,κ
0(Ω), hen we can ew i e (2.2) as
G(u) = G2( )
Op imal con ol o degene a e logis ic equa ion 7
being Ga compac pe uba ion o he iden i y. Since σ1(−∆ + M)>0, Gis inyec i e. The
F edholm’s Theo em p o ides us he exis ence and uniqueness o solu ion u∈C1,κ
0(Ω) o (2.2)
sa is ying (2.5). 2
The nex esul is an easy consequence o Theo em 2.2 b).
Lemma 2.6 a) Assume ha Msa is ies (H1) and σ1(−∆ + M)>0. Conside i∈L∞(Ω),
i= 1,2wi h 1≤ 2and le ui,i= 1,2be he espec i e solu ions o (2.2). Then, u1≤u2.
b) Assume ha Mi,i= 1,2sa is y (H1) and M1≤M2wi h σ1(−∆ + M1)>0. Le ui,
i= 1,2be he espec i e solu ions o (2.2). Then, u2≤u1.
3 The degene a e logis ic equa ion
Conside









−∆u=buα−euβin Ω,
u= 0 on ∂Ω,
(3.1)
and assume ha
(H2) 0 < α < 1≤β, b ∈L∞
+(Ω) {0}, e ∈ A,
whe e
A:= { ∈L∞(Ω) : L>0}.
The nex esul has been p o ed in [4] when b, e ∈Cν(Ω), ν∈(0,1). The p oo is also alid in
his case.
Theo em 3.1 Assume (H2). The ollowing asse ions a e ue:
a) The e exis s a unique s ic ly posi i e solu ion ubo (3.1). Mo eo e , by ellip ic egula i y
ub∈W2,p(Ω), p > 1, and so ub∈C1,κ(Ω) ∩in (P), wi h 0< κ ≤1−N/p.
b) We ha e he ollowing a p io i bound,
kubk∞≤µbM
eL¶1/(β−α)
.(3.2)
8M. Delgado, J. A. Mon e o and A. Su´a ez
c) I bL>0, hen he e exis s ε0>0such ha o all ε≤ε0, i holds
εϕ1(x)≤ub(x)c.p.d. x∈Ω
whe e ε0>0sa is ies
bL−σ1ε1−α−eMεβ−α= 0.
d) I bL= 0, since bM>0 he e exis s a ball B:= B(x0, )such ha bL,B >0in B, whe e
bL,B is he essen ial in imum o bin B. Hence, εϕB
1≤ubc.p.d. in B o all ε≤ε1and
whe e ε1>0sa is ies
bL,B −σB
1ε1−α−eM,Bεβ−α= 0.
Rema k 3.2 By (H2),(3.1) sa is ies he s ong maximum p inciple and hen he e exis wo
posi i e cons an s k1, k2such ha
k1dΩ(x)≤ub(x)≤k2dΩ(x)∀x∈Ω.(3.3)
The ollowing esul plays an impo an ole along he wo k.
Theo em 3.3 Assume (H2). Then, he map b∈A⊂L∞(Ω) 7→ ub∈in (P)⊂C1
0(Ω) is
inc easing, con inuous and C1.
Fo he p oo o his esul we use he ollowing elemen a y lemma.
Lemma 3.4 a) Le α∈(0,1] and 0< 1< 2be. Then
α α−1
2( 2− 1)≤ α
2− α
1≤α α−1
1( 2− 1).
b) Le β∈[1,+∞)and 0≤ 1< 2be. Then
β β−1
1( 2− 1)≤ β
2− β
1≤β β−1
2( 2− 1).
P oo o Theo em 3.3: I ollows easily ha he map is inc easing. Fo he con inui y, le
bn, b ∈ A be such ha bn→bin L∞, hen (bn)M→bM. Hence, ixed δ > 0 he e exis s n0∈IN
such ha o n≥n0
kubnk∞≤µ(bn)M
eL¶1/(β−α)
≤µbM+δ
eL¶1/(β−α)
=C(independien o n),
and so, he sequence {ubn}is bounded in W2,p(Ω), p > 1. The e exis s a subsequence, elabeled
by n, such ha ubn→uin C1,κ(Ω), κ < 1−N/p. Mo eo e , uis a weak solu ion o (3.1). I
Op imal con ol o degene a e logis ic equa ion 9
emains o p o e ha u=ub. By he uniqueness o posi i e solu ion o (3.1), i su ices o p o e
ha u > 0. Since bM>0, he e exis x0∈Ω, 0>0, such ha (bn)≥(bn)L,B >0 c.p.d.
in B=B(x0, 0), o n≥n0. By Theo em 3.1 d), we ha e ha he e exis εn>0 such ha
εnϕB
1≤ubnc.p.d. in Bwhe e εnis such ha
(bn)L,B −σB
1ε1−α
n−eM,Bεβ−α
n= 0.
Since (bn)L,B →bL,B, i ollows ha εn→ε > 0 whe e εis such ha
bL,B −σB
1ε1−α−eM,Bεβ−α= 0,
and so εϕB
1≤uc.p.d. in Band hen u > 0.
Fo he de i abili y we use he Implici Func ion Theo em. Fixed p > N, we de ine he map
F:A × U ⊂ L∞(Ω) ×C1
0(Ω) 7→ Lp(Ω) whe e U:= W2,p(Ω) ∩in (P), as
F(b, u) := −∆u−buα+euβ.
Ais an open se in L∞(Ω) and i is well known, see [1], ha Uis also open in C1
0(Ω). I is clea
ha F(b0, ub0) = 0.
We show ha Fis C1, o which i is su icien o show i o he second componen . We calcula e
he Gˆa eaux de i a i e espec o his, which will be deno ed by DGF. Le (b, u)∈ A × U and
ξ∈C1
0(Ω) be, hen
DGF(b, u)ξ:= lim
ε→0
F(b, u +εξ)− F(b, u)
ε=−∆ξ−blim
ε→0
(u+εξ)α−uα
ε+elim
ε→0
(u+εξ)β−uβ
ε.
We claim ha :
(u+εξ)β−uβ
ε→βuβ−1ξand (u+εξ)α−uα
ε→αuα−1ξin Lp(Ω) as ε→0.(3.4)
Assume ε↓0. Using Lemma 3.4, o p o e (3.4) i is su icien o show ha
(u+εξ)β−1ξ→uβ−1ξand (u+εξ)α−1ξ→uα−1ξin Lp(Ω) as ε↓0.
The i s one is ue because β≥1. Fo he second one, we ha e
k[(u+εξ)α−1−uα−1]ξkp=k[(u+εξ)α−(u+εξ)uα−1]µξ
u+εξ ¶kp.(3.5)
Since u∈in (P), he e exis ε0>0 and k(ε) such ha u+εξ ∈in (P) o ε≤ε0and
k(ε) := in
x∈Ω
u(x) + εξ(x)
dΩ(x)>0.
16 M. Delgado, J. A. Mon e o and A. Su´a ez
P oo : Le ∈L∞
+(Ω) be. By (H5), he e exis s 0>0 such ha k( 0)/h( 0) = λK. We conside
g:= min{ , 0},
and we will p o e ha J(g)> J( ), whence he esul ollows.
By de ini ion, g≤ and hen ug≥u . I x0∈Ω is such ha (x0) = g(x0) hen
λug(x0)h(g(x0)) −k(g(x0)) ≥λu (x0)h( (x0)) −k( (x0)).
On he o he hand, i (x0)> g(x0) = 0>0, hen by (3.2)
λug(x0)h(g(x0)) −k(g(x0)) ≤λKh( 0)−k( 0) = 0,
and so by (H5), we ge
0≥λug(x0)h(g(x0))
g(x0)−k(g(x0))
g(x0)> λu (x0)h( (x0))
(x0)−k( (x0))
(x0).
Then,
J(g) = Z{ =g}
λh(g)ug−k(g) + Z{ >g}
λh(g)ug−k(g)≥Z{ =g}
λh( )u −k( )+
+Z{ >g}
(λh(g)
gug−k(g)
g)g > Z{ =g}
λh( )u −k( ) + Z{ >g}
λh( )u −k( ) = J( ).
2
Fo he uniqueness, we use he a gumen desc ibed in Sec ion 6 in [3]. Fi s ly, we p o e he
nex esul .
P oposi ion 4.4 Le J:D:= { ∈L∞(Ω) : (a− )∈ A} ⊂ L∞(Ω) 7→ IR be. Then Jis
F ´eche con inuously di e en iable and
J0( )(g) = ZΩ
(λh0( )u −λuα
P −k0( ))g, ∀ ∈ D,∀g∈L∞(Ω),(4.1)
whe e o any ∈ D,P ∈C1
0(Ω) is he unique solu ion o









−∆P +M (x)P =h( )in Ω,
P = 0 on ∂Ω,
(4.2)
being M := −α(a− )uα−1
+βeuβ−1
.

Op imal con ol o degene a e logis ic equa ion 17
To p o e his esul , we need some p e ious ones. Fo ∈ D and g∈L∞(Ω), le ξ ,g be he
unique solu ion o









−∆ξ+M (x)ξ=−guα
in Ω,
ξ= 0 on ∂Ω.
(4.3)
Obse e ha (4.2) and (4.3) ha e a unique solu ion because σ1(−∆ + M )>0 (see (3.10) and
(3.11)) and Theo em 2.5.
Lemma 4.5 The map ∈ D 7→ P ∈C1
0(Ω) is con inuous.
P oo : Fixed p > N, we conside he map G:D × (C1
0(Ω) ∩W2,p(Ω)) 7→ Lp(Ω) de ined by
G( , P) = −∆P+M P−h( ).
Obse e ha Gis con inuous. Indeed, he con inui y o he map 7→ M P ollows wi h a simila
a gumen o he one used in he p oo o Theo em 3.3 o show ha he map DGFis con inuous.
On he o he hand, i is clea ha G( , P ) = 0. Gi en ξ∈C1
0(Ω) ∩W2,p(Ω) is easy o p o e
ha D2G( , P )ξ=−∆ξ+M ξ. Mo eo e , as in (3.10), σ1(−∆ + M )>0 and so D2G( , P )
is non singula . The Implici Func ion Theo em comple es he p oo . 2
The nex esul is due by K asnoselskii, see [7], whe e we send o he de ini ions o he
ollowing concep s.
Lemma 4.6 Le Ebe a Banach space o de ed by a gene a ing posi i e cone P,Fa Banach
space and T:E7→ F. Assume ha he Gˆa eaux de i a i e o Twi h espec o P, deno ed by
DG,P T, exis s and i is con inuous in a neighbou hood o x0∈E. Then, he F ´eche de i a i e
coincides wi h he Gˆa eaux de i a i e and Tis C1nea x0.
Recall ha Pis gene a ing i E=P−P. I is well known, see P oposi ion 1.7 in [1], ha i
in (P)6=∅, hen Pis gene a ing.
P oo o P oposi ion 4.4: Fi s ly, we compu e he Gˆa eaux de i a i e espec o he cone,
deno ed by DG,P J. Le g∈L∞
+(Ω), ∈ D and ε > 0 be such ha +εg ∈ D. Using Lemma
3.5 and (4.3)
DG,P J( )g:= lim
ε↓0
J( +εg)−J( )
ε=ZΩ
λξ ,gh( ) + λh0( )u g−k0( )g.
18 M. Delgado, J. A. Mon e o and A. Su´a ez
By he equa ion ha sa is y ξ ,g yP (see (4.3) and (4.2)), i ollows ha
ZΩ
h( )ξ ,g +ZΩ
guα
P = 0,
and so,
DG,P J( )(g) = ZΩ
(λh0( )u −λuα
P −k0( ))g, ∀g∈L∞
+(Ω).
Le n→ ∈ D be in L∞and g∈L∞(Ω). Then, by Theo em 3.3 and Lemma 4.5 i ollows
sup
kgk∞≤1
|DG,P J( n)(g)−DG,P J( )(g)| ≤
≤sup
kgk∞≤1ZΩ
|λ(h0( n)u n−h0( )u )−λ(uα
nP n−uα
P )−(k0( n)−k0( ))g| → 0.
and so, DG,P Jis con inuous. Applying Lemma 4.6, he Gˆa eaux de i a i e coincides wi h he
F ´eche de i a i e and ha he map is C1.2
The nex esul shows ha some maps in ol ed in (4.1) a e Lipschi z con inuous.
Lemma 4.7 Assume (H3) −(H5). The e exis s Λ>0such ha o 0<λ<Λ he maps
∈[0, Tλ]7→ u , P , uα
P ∈L∞(Ω) a e Lipschi z con inuous.
P oo : Le , g ∈[0, Tλ] be, by he mono ony o he map 7→ u , i ollows ha
0< uTλ≤u , ug≤u0
o λsuch ha a−Tλ>0, ha is λ < λ0 o some λ0(see Rema k 4.3 a)). To he end o he
p oo we ake λ < λ0. By he Mean Value Theo em,
uα
−uα
g=αξα−1( , g)(u −ug), uβ
−uβ
g=βηβ−1( , g)(u −ug) wi h
0< uTλ≤min{u , ug} ≤ ξ( , g), η( , g)≤max{u , ug} ≤ u0.
(4.4)
Le w:= u −ugbe. Then, wsa is ies
(−∆ + N( , g))w= (g− )uα
g,in Ω, w= 0 on ∂Ω,
whe e N( , g) := −α(a− )ξα−1( , g) + βeηβ−1( , g). Using ≥0 and (4.4), i ollows ha
N( , g)≥ −αaξα−1( , g) + βeηβ−1( , g)≥ −αauα−1
Tλ+eβuβ−1
Tλ.
I is no ha d o show ha as λ↓0
ZΩ
(−αauα−1
Tλ+eβuβ−1
Tλ)ϕ2→ZΩ
(−αauα−1
0+eβuβ−1
0)ϕ2∀ϕ∈H1
0(Ω),
Op imal con ol o degene a e logis ic equa ion 19
and so, by P oposi ion 2.3 we ob ain ha
σ1(−∆ + N( , g)) ≥σ1(−∆−αauα−1
Tλ+eβuβ−1
Tλ)→σ1(−∆−αauα−1
0+eβuβ−1
0)>0
as λ↓0. Hence, he e exis s λ1>0 such ha
N( , g)≥ −αauα−1
Tλ1+eβuβ−1
Tλ1(4.5)
and
σ1(−∆ + N( , g)) ≥σ1(−∆−αauα−1
Tλ1+eβuβ−1
Tλ1)>0.(4.6)
Then, by (4.5), (4.6) and Lemma 2.6, we ha e ha w≤ψ1whe e ψ1is he unique solu ion o









−∆u+ (−αauα−1
Tλ1+eβuβ−1
Tλ1)u= (g− )uα
gin Ω,
u= 0 on ∂Ω.
(4.7)
In e changing and g, we ge ha −w≤ψ2whe e ψ2is he unique solu ion o (4.7) wi h second
membe ( −g)uα
. Then, aking in o accoun ha u possesses a p io i bound independien o
(see (3.2)) and Theo em 2.5, i ollows ha
ku −ugk∞=kwk∞≤max{kψ1k∞,kψ2k∞} ≤ max{kψ1kC1(Ω),kψ2kC1(Ω)} ≤ Ck −gk∞.(4.8)
This shows ha he map 7→ u is Lipschi z.
Be o e p o ing he Lipschi z cha ac e o he map ∈[0, Tλ]7→ P , we see ha
P ≤ P in Ω, (4.9)
whe e P ∈ C1
0(Ω), independien o . Indeed, le ∈[0, Tλ] be, hen M ≥ −αauα−1
Tλ+βeuβ−1
Tλ,
and so, using again Lemma 2.6 b), P ≤ P whe e Pis he unique solu ion o









−∆u+ (−αauα−1
Tλ1+eβuβ−1
Tλ1)u=Tin Ω,
u= 0 on ∂Ω,
whe e T:= max
∈[0,Tλ]max
x∈Ω
h( (x)).This implies (4.9).
We will p o e now ha he map is Lipschi z. Le , g ∈[0, Tλ] and z:= P −Pgbe. Then z
sa is ies
−∆z+M z=T( , g),in Ω, z= 0 on ∂Ω,
20 M. Delgado, J. A. Mon e o and A. Su´a ez
whe e
T( , g) = h( )−h(g) + Pg[α(a− )(uα−1
−uα−1
g)−βe(uβ−1
−uβ−1
g)] + α(g− )Pguα−1
g.
Applying again he Mean Value Theo em, we ge
uα−1
−uα−1
g= (α−1)ξα−2( , g)(u −ug), uβ−1
−uβ−1
g= (β−1)ηβ−2( , g)(u −ug)
0< uTλ≤min{u , ug} ≤ ξ( , g), η( , g)≤max{u , ug} ≤ u0.
(4.10)
Hence,
T( , g) = h( )−h(g) + Pg[α(α−1)(a− )ξα−2−β(β−1)eηβ−2](u −ug) + α(g− )Pguα−1
g.
By a simila a gumen o he used in he p oo o (4.8), we ob ain
kP −Pgk∞=kzk∞≤CkT( , g)k∞.(4.11)
Since P ∈ C1
0(Ω), and using (3.3), (3.7), (4.9) and (4.10), we ob ain
kα( −g)Pguα−1
gk∞≤Ck −gk∞kPguα−1
Tλ1k∞
≤Ck −gk∞kα−1
1kPdα−1
Ωk∞
≤Ck −gk∞kdα
Ωk∞kPkC1(Ω)
≤Ck −gk∞wi h Cindependien o and g.
On he o he hand, using (4.8), (4.9) and (4.10)
kα(α−1)(a− )Pgξα−2(u −ug)k∞≤CkPξα−2(u −ug)k∞
≤CkPξα−2max{|ψ1|,|ψ2|}k∞
≤CkPdα−2
Ωmax{|ψ1|,|ψ2|}k∞
≤CkPkC1(Ω)kdα
Ωk∞max{kψ1kC1(Ω),kψ2kC1(Ω)}
≤Ck −gk∞
wi h Cindependien o and g. Analogously i can be ea ed he e m −eβ(β−1)Pgηβ−2(u −
ug). Then, since his Lipschi z in [0, Tλ] and by (4.11), i ollows ha he map 7→ P is
Lipschi z.
Op imal con ol o degene a e logis ic equa ion 21
Le , g ∈[0, Tλ] be, we ha e
kuα
P −uα
gPgk∞≤ k(uα
−uα
g)P k∞+kuα
g(P −Pg)k∞.
By he Mean Value Theo em,
k(uα
−uα
g)P k∞=kαξα−1P (u −ug)k∞≤CkϕkC1(Ω)k −gk∞≤Ck −gk∞.
I is su icien o ake Λ := min{λ0, λ1}. This comple es he p oo . 2
Theo em 4.8 Assume (H3) −(H5). Then, he e exis s Λ0>0such ha i λ < Λ0 he e exis s
a unique op imal con ol.
P oo : Le ∈ C be an op imal con ol, hen by Lemma 4.2
∈I:= [0, Tλ]∞.
We ake λ < Λ ( he cons an ob ained in Lemma 4.7) and su icien ly small λsuch ha I⊂ C.
In I, con ex, he s ic ly conca e cha ac e o Jis equi alen o he mono ony o J0. Hence, by
(4.1), o , g ∈I, we ha e ha
(J0( )−J0(g))( −g) = ZΩ
[λ(h0( )u −h0(g)ug) + λ(uα
gPg−uα
P )−(k0( )−k0(g))]( −g)≤
≤ZΩ
(λL −k0)( −g)2<0,
aking λ < k0/L := Λ1, whe e L he Lipschi z cons an o he maps h0, 7→ u , 7→ P and
7→ uα
P (see Lemma 4.7). 2
5 Regula i y o he op imal con ol and op imali y sys em
In his sec ion we conside he special case h( ) = and k( ) = 2, which sa is y clea ly (H4)
and (H5). Mo eo e , in his case
Tλ=λK.
The ollowing esul p o ides us o a ca ac e iza ion o an op imal con ol. I ollows as Theo em
3.1 in [10], using now ou Lemma 3.5.
Lemma 5.1 Assume ∈ C and (H3). I is an op imal con ol, hen
=λ
2u (1 −uα−1
P )+.

22 M. Delgado, J. A. Mon e o and A. Su´a ez
The nex esul says us ha he op imal con ol is a H¨olde con inuous unc ion when λis
small and i le s us w i e he op imali y sys em.
P oposi ion 5.2 Assume (H3). The e exis s Λ1such ha i λ≤Λ1, hen P ≤u1−α
. So, i
is an op imal con ol, we ha e ha
=λ
2u (1 −uα−1
P ).(5.1)
P oo : Le be an op imal con ol. Fo λ < λ0:= aL/K, we ha e ha u ≥uλK>0. As in
Lemma 4.7, i ollows he exis ence o λ1such ha he e exis s a unique posi i e solu ion ψo









−∆ψ+ (−aαuα−1
λ1K+βeuβ−1
λ1K)ψ=Kin Ω,
ψ= 0 on ∂Ω.
By Lemma 2.6 and (3.2), i ollows ha
P ≤λψ o λ≤λ1. (5.2)
We de ine now
λ2:= in
x∈Ω
u1−α
λ1K
ψ≤in
x∈Ω
u1−α
ψ.
Obse e ha λ2>0. Indeed, since ψand uλ1Ka e posi i e unc ions, i ollows he exis ence o
a cons an k > 0 such ha
u1−α
λ1K
ψ> kd−α
Ω>0.
Taking Λ1:= min{λ0, λ1, λ2}and aking in o accoun (5.2) and he de ini ion o λ2, i ollows
P ≤u1−α
, and as a consequence o Lemma 5.1, we ob ain (5.1). 2
The ollowing esul is an easy consequence o he p e ious esul and i p o ides us wi h
he op imali y sys em.
Co olla y 5.3 Assume (H3) and λ≤Λ1. Then any op imal con ol may be exp essed as in
(5.1), whe e he pai (u , P ) := (u, P)sa is ies





















−∆u=uα(a−λ
2u+λ
2uαP−euβ−α)in Ω,
−∆P+ (−αauα−1+βeuβ−1)P=λ
2(u−uαP(1 + α) + αu2α−1P2)in Ω,
u=P= 0 on ∂Ω,
and u > 0.
Op imal con ol o degene a e logis ic equa ion 23
Acknowledgmen s. M. Delgado and A. Su´a ez hank o CICYT o Spain (MAR98-0486)
and J. A. Mon e o hanks o ”Jun a de Andaluc´ıa” (FQM116) and DGESIC (PB98-1343) by
he pa ial inancial suppo o he elabo a ion o his wo k.
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