Hindawi Publishing Corporation
Boundary Value Problems
Volume 2008, Article ID 192353, 21 pages
doi:10.1155/2008/192353
Research Article
Global Behaviors and Optimal Harvesting of
a Class of Impulsive Periodic Logistic
Single-Species System with Continuous Periodic
Control Strategy
JinRong Wang,1 X. Xiang,2 and W. Wei2
1
2
College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
College of Science, Guizhou University, Guiyang, Guizhou 550025, China
Correspondence should be addressed to JinRong Wang, wjr9668@126.com
Received 30 September 2008; Accepted 17 December 2008
Recommended by Wenming Zou
Global behaviors and optimal harvesting of a class of impulsive periodic logistic single-species
system with continuous periodic control strategy is investigated. Four new sufficient conditions
that guarantee the exponential stability of the impulsive evolution operator introduced by us
are given. By virtue of exponential stability of the impulsive evolution operator, we present the
existence, uniqueness and global asymptotical stability of periodic solutions. Further, the existence
result of periodic optimal controls for a Bolza problem is given. At last, an academic example is
given for demonstration.
Copyright q 2008 JinRong Wang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In population dynamics, the optimal management of renewable resources has been one of
the interesting research topics. The optimal exploitation of renewable resources, which has
direct effect on their sustainable development, has been paid much attention 1–3. However,
it is always hoped that we can achieve sustainability at a high level of productivity and good
economic profit, and this requires scientific and effective management of the resources.
Single-species resource management model, which is described by the impulsive
periodic logistic equations on finite-dimensional spaces, has been investigated extensively,
no matter how the harvesting occurs, continuously 1, 4 or impulsively 5–7. However, the
associated single-species resource management model on infinite-dimensional spaces has not
been investigated extensively.
Since the end of last century, many authors including Professors Nieto and Hernández
pay great attention on impulsive differential systems. We refer the readers to 8–22.
2
Boundary Value Problems
Particulary, Doctor Ahmed investigated optimal control problems 23, 24 for impulsive
systems on infinite-dimensional spaces. We also gave a series of results 25–34 for the firstorder second-order semilinear impulsive systems, integral-differential impulsive system,
strongly nonlinear impulsive systems and their optimal control problems. Recently, we
have investigated linear impulsive periodic system on infinite-dimensional spaces. Some
results 35–37 including the existence of periodic P C-mild solutions and alternative theorem,
criteria of Massera type, asymptotical stability and robustness against perturbation for a
linear impulsive periodic system are established.
Herein, we devote to studying global behaviors and optimal harvesting of the
generalized logistic single-species system with continuous periodic control strategy and
periodic impulsive perturbations:
∂
xt, y Ay, Dxt, y ft, y Ctut, y, u ∈ Uad , y ∈ Ω, t > 0, t / τk , k ∈ Z0 ,
∂t
xt, y 0, y ∈ ∂Ω, t > 0,
Δxt, y Δxt 0, y − Δxt, y Bk xt, y ck ,
y ∈ Ω, t τk , k ∈ Z0 .
1.1
On infinite-dimensional spaces, where xt, y denotes the population number of isolated
species at time t and location y, Ω ⊂ R2 is a bounded domain and ∂Ω ∈ C2 , operator
α
Ay, D |α|≤4 aα yD . The coefficients aα y, y ∈ Ω, t ≥ 0 are sufficiently smooth
2
α
functions of y in Ω, where α α1 , α2 , αi > 0, i 1, 2, |α| y1α y2α ,
i1 αi and y
α1 α2
α
y y1 , y2 ∈ Ω. Denoting Di ∂/∂yi i 1, 2, D D1 , D2 , then D D1 D2 . ft, y
is related to the periodic change of the resources maintaining the evolution of the population
and the periodic control policy u ∈ Uad , where Uad is a suitable admissible control set. Time
sequence 0 τ0 < τ1 < · · · < τk · · · and τk → ∞ as k → ∞, Δxτk , y denote mutation of the
isolate species at time τk where k ∈ Z0 .
Suppose X is a Banach space and Y is a separable reflexive Banach space. The objective
functional is given by
Ju
T0 0
Ω
lt, xt, y, u, ut, ydydt Ω
ΨxT0 , y, udy,
1.2
where l : 0, T0 × X × Y → R ∪ {∞} is Borel measurable, Ψ : X → R is continuous, and
nonnegative and x·, y, u denotes the T0 -periodic P C-mild solution of system 1.1 at location
y and corresponding to the control u ∈ Uad . The Bolza problem P is to find u0 ∈ Uad such
0 ≤ Ju
for all u ∈ Uad .
that Ju
Suppose that ft T0 , y ft, y, Ct T0 Ct, ut T0 , y ut, y, t ≥ 0 and T0 is
the least positive constant such that there are δ τk s in the interval 0, T0 , and τkδ min{τ ∈
{τk | τk1 > τk ; for all k ∈ Z }, Bkδ Bk , ckδ ck , k ∈ Z .
| τ ≥ τk T0 }, where D
D
0
0
The first equation of system 1.1 describes the variation of the population number x of the
species in periodically continuous controlled changing environment. The second equation of
system 1.1 shows that the species are isolated. The third equation of system 1.1 reflects
the possibility of impulsive effects on the population.
Let Ay, D satisfy some properties such as strongly elliptic in Ω and set DA such
as H 2 Ω∩H01 Ω. For every x ∈ DA define Ax Ay, Dx, A is the infinitesimal generator
JinRong Wang et al.
3
of a C0 -semigroup {T t, t ≥ 0} on the Banach space X such as H 2 Ω. Define x·y x·, y, f·y f·, y, and u·y u·, y then system 1.1 can be abstracted into the
following controlled system:
τk ,
ẋt Axt ft Ctut, u ∈ Uad , t /
Δxτk Bk x τk ck , t τk .
1.3
On the Banach space X, and the associated objective functional
Ju T0
lt, xt, u, utdt ΨxT0 , u,
1.4
0
where x·, u denotes the T0 -periodic P C-mild solution of system 1.3 corresponding to the
control u ∈ Uad . The Bolza problem P is to find u0 ∈ Uad such that Ju0 ≤ Ju for all
u ∈ Uad . The investigation of the system 1.3 cannot only be used to discuss the system 1.1,
but also provide a foundation for research of the optimal control problems for semilinear
impulsive periodic systems. The aim of this paper is to give some new sufficient conditions
which will guarantee the existence, uniqueness, and global asymptotical stability of periodic
P C-mild solutions for system 1.3 and study the optimal control problems arising in the
system 1.3.
The paper is organized as follows. In Section 2, the properties of the impulsive
evolution operator {S·, ·} are collected. Four new sufficient conditions that guarantee the
exponential stability of the {S·, ·} are given. In Section 3, the existence, uniqueness, and
global asymptotical stability of T0 -periodic P C-mild solution for system 1.3 is obtained.
In Section 4, the existence result of periodic optimal controls for the Bolza problem P is
presented. At last, an academic example is given to demonstrate our result.
2. Impulsive periodic evolution operator and it’s stability
Let X be a Banach space, £X denotes the space of linear operators on X; £b X denotes the
space of bounded linear operators on X. £b X is the Banach space with the usual supremum
{τ1 , . . . , τδ } ⊂ 0, T0 and define P C0, T0 ; X ≡ {x : 0, T0 → X | x is
norm. Denote D
x is continuous from left and has right-hand limits at t ∈ D}
and
continuous at t ∈ 0, T0 \ D,
1
P C 0, T0 ; X ≡ {x ∈ P C0, T0 ; X | ẋ ∈ P C0, T0 ; X}.
Set
xP C max
sup xt 0, sup xt − 0 ,
t∈0,T0 t∈0,T0 xP C1 xP C ẋP C .
2.1
It can be seen that endowed with the norm · P C · P C1 , P C0, T0 ; XP C1 0, T0 ; X is
a Banach space.
4
Boundary Value Problems
In order to investigate periodic solutions, we introduce the following two spaces:
p
LT0 0, ∞; X
≡
f : 0, ∞ −→ X | ft f t T0 ,
T
0
1/p
ft dt
< ∞ where 1 < p < ∞ ,
p
2.2
0
P CT0 0, ∞; X ≡ x ∈ P C0, ∞; X | xt x t T0 , t ≥ 0 .
Set
fLp T0
T
0
0
1/p
ft dt
p
,
xP CT0 max
sup xt 0, sup xt − 0 . 2.3
t∈0,T0 t∈0,T0 p
It can be seen that endowed with the norm ·Lp ·P CT0 , LT0 0, ∞; X P CT0 0, ∞; X
T0
is a Banach space.
We introduce assumption H1.
H1.1: A is the infinitesimal generator of a C0 -semigroup {T t, t ≥ 0} on X with domain
DA.
| τ ≥ τk T0 }, where D
{τk | τk1 >
H1.2: There exists δ such that τkδ min{τ ∈ D
τk ; ∀k ∈ Z0 }.
H1.3: For each k ∈ Z0 , Bk ∈ £b X, Bkδ Bk .
Under the assumption H1, consider
ẋt Axt,
t /
τk ,
Δxt Bk xt,
t τk ,
2.4
and the associated Cauchy problem
ẋt Axt, t ∈ 0, T0 \ D,
Δx τk Bk x τk , k 1, 2, . . . , δ,
2.5
x0 x.
For every x ∈ X, DA is an invariant subspace of Bk , using 38, Theorem 5.2.2,
page 144, step by step, one can verify that the Cauchy problem 2.5 has a unique classical
JinRong Wang et al.
5
solution x ∈ P C1 0, T0 ; X represented by xt St, 0x, where S·, · : Δ {t, θ | 0 ≤ θ ≤
t ≤ T0 } → £X given by
⎧
⎪
τk−1 ≤ θ ≤ t ≤ τk ,
⎪
⎪T t − θ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨T t − τk I Bk T τk − θ , τk−1 ≤ θ < τk < t ≤ τk1 ,
St, θ ⎪
⎪T t − τ
⎪
I Bj T τj − τj−1
I Bi T τi − θ ,
⎪
k
⎪
⎪
θ<τj <t
⎪
⎪
⎪
⎩
τi−1 ≤ θ < τi ≤ · · · < τk < t ≤ τk1 .
2.6
The operator {St, θ, t, θ ∈ Δ} is called impulsive evolution operator associated with
{T t, t ≥ 0} and {Bk ; τk }∞
k1 .
The following lemma on the properties of the impulsive evolution operator
{St, θ, t, θ ∈ Δ} associated with {T t, t ≥ 0} and {Bk ; τk }∞
k1 is widely used in this paper.
Lemma 2.1. Let assumption [H1] hold. The impulsive evolution operator {St, θ, t, θ ∈ Δ} has
the following properties.
1 For 0 ≤ θ ≤ t ≤ T0 , St, θ ∈ £b X, there exists a MT0 > 0 such that
sup0≤θ≤t≤T0 St, θ ≤ MT0 .
2 For 0 ≤ θ < r < t ≤ T0 , r / τk , St, θ St, rSr, θ.
3 For 0 ≤ θ ≤ t ≤ T0 , n ∈ Z , St nT0 , θ nT0 St, θ.
4 For 0 ≤ θ ≤ t ≤ T0 , n ∈ Z , St nT0 , 0 St, 0ST0 , 0n .
5 For 0 ≤ θ < t, there exits M ≥ 1, ω ∈ R such that
St, θ ≤ M exp ωt − θ .
ln M I Bn
2.7
θ≤τn <t
Proof. 1 By assumption H1.1, there exists a constant CT0 > 0 such that supt∈0,T0 T t CT0 < ∞. Using assumption H1.3, it is obvious that St, θ ∈ £b X, for 0 ≤ θ ≤ t ≤ T0 .
2 By the definition of C0 -semigroup and the construction of S·, ·, one can verify the result
immediately. 3 By assumptions H1.2, H1.3, and elementary computation, it is easy to
obtain the result. 4 For 0 ≤ θ ≤ t ≤ T0 , n ∈ Z , by virtue of 3 again and again, we arrive at
S t nT0 , 0 S t nT0 , T0 S T0 , 0 S t n − 1T0 , 0 S T0 , 0
2
S t n − 1T0 , T0 S T0 , 0 S T0 , 0 t n − 2T0 , 0 S T0 , 0
...
n
St, 0 S T0 , 0 .
2.8
6
Boundary Value Problems
5 Without loss of generality, for τi−1 ≤ θ < τi ≤ · · · < τk < t ≤ τk1 ,
· · · I Bi T τi − θ St, θ T t − τk I Bk T τk − τk−1
k
ωt−τk τ
−τ
I Bn Me n n−1 I Bi Meωτi −θ
≤ Me
ni1
≤ M exp ωt − θ 2.9
ln M I Bn
.
θ≤τn <t
This completes the proof.
In order to study the asymptotical properties of periodic solutions, it is necessary to
discuss the exponential stability of the impulsive evolution operator {St, θ, t ≥ θ ≥ 0}. We
first give the definition of exponential stable for {St, θ, t ≥ θ ≥ 0}.
Definition 2.2. {St, θ, t ≥ θ ≥ 0} is called exponentially stable if there exist K ≥ 0 and ν > 0
such that
St, θ ≤ Ke−νt−θ ,
t > θ ≥ 0.
2.10
Assumption H2: {T t, t ≥ 0} is exponentially stable, that is, there exist K0 > 0 and
ν0 > 0 such that
T t ≤ K0 e−ν0 t ,
t > 0.
2.11
An important criteria for exponential stability of a C0 -semigroup is collected here.
Lemma 2.3 see 38, Lemma 7.2.1. Let {T t, t ≥ 0} be a C0 -semigroup on X, and let A be its
infinitesimal generator. Then the following assertions are equivalent:
1 {T t, t ≥ 0} is exponentially stable.
2 For every x ∈ X there exits a positive constants γx < ∞ such that
∞
T txp dt < γx < ∞,
x ∈ X, t > 0, for some p, 1 ≤ p < ∞.
2.12
0
Next, four sufficient conditions that guarantee the exponential stability of impulsive evolution
operator {St, θ, t ≥ θ ≥ 0} are given.
Lemma 2.4. Assumptions [H1] and [H2] hold. There exists 0 < λ < ν0 such that
δ
K0 I Bk e−λT0 < 1.
k1
Then, {St, θ, t ≥ θ ≥ 0} is exponentially stable.
2.13
JinRong Wang et al.
7
Proof. Without loss of generality, for τi−1 ≤ θ < τi ≤ · · · < τk < t ≤ τk1 , we have
St, θ ≤ K0 e
−ν0 −λt−θ
−λt−θ
.
K0 I Bk e
2.14
θ<τk <t
Suppose t ∈ nT0 , n 1T0 and let b maxs∈0,T0 K0 I Bk e−λt−θ ≤
0<τk <s {K0 I
Bk }. Then,
K0 I Bk e−λnT0 beλθ
0≤τk <nT0
θ<τk <t
≤
δ
n
K0 I Bk e−λnT0 beλθ
2.15
k1
δ
K0 I Bk e−λT0
n
beλθ
k1
< beλθ .
Let K K0 beλθ > 0 and ν ν0 − λ > 0, then we obtain St, θ ≤ Ke−νt−θ , t > θ ≥ 0.
Lemma 2.5. Assume that assumption [H1] holds. Suppose
0 < μ1 inf
k1,2,...,δ
τk − τk−1 ≤ sup τk − τk−1 μ2 < ∞.
2.16
k1,2,...,δ
If there exists α > 0 such that
ω
1 ln MI Bk ≤ −α < 0,
μ
2.17
for k 1, 2, . . . , δ, where
μ
⎧
⎨μ 1 ,
α ω < 0,
⎩μ ,
2
α ω ≥ 0.
2.18
Then, {St, θ, t ≥ θ ≥ 0} is exponentially stable.
Proof. It comes from 2.17 that
ln MI Bk ≤ −μα ω < 0,
k 1, 2, . . . , δ.
2.19
8
Boundary Value Problems
Further,
ln MI Bk ≤ −
μα ω −μα ωNθ, t,
θ≤τk <t
2.20
θ≤τk <t
where Nθ, t is denoted the number of impulsive points in θ, t.
For τi−1 ≤ θ < τi ≤ · · · < τk < t ≤ τk1 , by 2.16, we obtain the following two
inequalities:
t − θ ≥ Nθ, t − 1μ1 ,
t − θ ≤ Nθ, t 1μ2 .
2.21
This implies
μ1 Nθ, t − 1 ≤ t − θ ≤ μ2 Nθ, t 1,
2.22
1
1
t − θ − 1 ≤ Nθ, t ≤
t − θ 1.
μ2
μ1
2.23
−μα ωNθ, t ≤ −α ωt − θ μ|α ω|.
2.24
that is,
Then,
Thus, we obtain
ωt − θ ln MI Bk ≤ −αt − θ μ|α ω|.
2.25
θ≤τk <t
By 5 of Lemma 2.1, let K Meμ|αω| > 0, ν α > 0, St, θ ≤ Ke−νt−θ , t > θ ≥ 0.
Lemma 2.6. Assume that assumption [H1] holds. The limit
N θ, θ T0
δ
lim
exists and is equal to
≡ p is finite.
T0 → ∞
T0
T0
2.26
Suppose there exists γ > 0 such that
ω p ln MI Bk ≤ −γ < 0,
k 1, 2, . . . , δ.
2.27
Then, {St, θ, t ≥ θ ≥ 0} is exponentially stable.
Proof. Let t, θ ∈ R with t > θ. It comes from
N θ, θ T0
δ
lim
≡p
T0 → ∞
T0
T0
2.28
JinRong Wang et al.
9
that there exits a h > 0 enough small such that
Nθ, t
− p < ph,
t−θ
2.29
that is,
1 − ht − θ <
Nθ, t
< 1 ht − θ.
p
2.30
From 2.27, we know that
θ<τk <t
1
Nθ, t
γ ω −
γ ω.
ln MI Bk ≤ −
p
p
θ<τ <t
2.31
k
Then, we have
⎧
1 h
⎪
⎪
t − θγ ω,
⎨−
Nθ, t
p
−
γ ω ≤
⎪
p
⎪− 1 − h t − θγ ω,
⎩
p
γ ω <0
γ ω ≥0
−γ ω − h|γ ω|t − θ.
2.32
Hence,
ω ht − θ ln MI Bk ≤ −γ − h1 |γ ω|t − θ.
2.33
θ<τk <t
Here, we only need to choose h > 0 small enough such that γ − h1 |γ ω| > 0, by 5 of
Lemma 2.1 again, let K M > 0, ν γ − h1 |γ ω| > 0, we have St, θ ≤ Ke−νt−θ , t >
θ ≥ 0.
Lemma 2.7. Assume that assumption [H1] holds. For some p, 1 ≤ p < ∞,
∞
0
St, θξp dt < ∞, ξ ∈ X, t > θ ≥ 0, θ is fixed,
I Bk is convergent.
2.34
θ≤τk <t
Imply the exponential stability of {St, θ, t ≥ θ ≥ 0}.
Proof. It comes from the continuity of t → T tξ, the inequality
St, θ ≤
ωt−θ
M
I Bk
e
2
θ≤τk <t
2.35
10
Boundary Value Problems
and the boundedness of Bk , θ≤τk <t I Bk are convergent, that limt → ∞ St, θξ 0 for every
ξ ∈ X and fixed θ ≥ 0. This shows that St, θξ is bounded for each ξ ∈ X and fixed θ ≥
0 and hence, by virtue of uniform boundedness principle, there exists a constant M2 ≥ 1
such that St, θ ≤ M2 for all t > θ ≥ 0. Let L denote the operator given by Lxt St, θx, x ∈ X and θ is fixed. Clearly, L is defined every where on X and by assumption it
maps X → Lp 0, ∞; X and it is a closed operator. Hence, by closed graph theorem, it is a
bounded linear operator from X to Lp 0, ∞; X. Thus, there exits a constant M3 > 0 such
that LxLp 0,∞;X ≤ M3 for all x ∈ and t > θ ≥ 0, θ is fixed.
Let 0 < κ < M2−1 , ξ ∈ X and t ≥ 0 and define τ ≡ τκ, ξ as
τ sup{t ≥ 0 : Ss, θξ ≥ κξ ∀0 ≤ θ < s ≤ t}.
2.36
Then,
τκξp ≤
τ
St, θξp dt ≤
0
∞
0
p
p
St, θξp dt LξLp 0,∞;X ≤ M3 ξ ,
2.37
and hence,
τ≤
M3
κ
p
≡ t0 .
2.38
Thus, for t > t0 θ /
τk ,
St, θξ ≤ St, t − τSt − τ, θξ ≤ M2 κξ ≡ βξ,
2.39
where β M2 κ < 1. Fix t1 N0 T0 > t0 θ. Then, for any t ∈ 0, ∞ we can write t − θ nt1 s
for some n ∈ N0 and s ∈ 0, t1 and we have
nN
St, θξ S nN0 T0 s, θ ξ Ss, θS T0 , 0 ξ 0 ≤ M2 enN0 ln β ξ M1 e−νt ξ,
2.40
where M1 M2 βs/t1 and ν −ln β/t1 . Since β < 1, this shows that our result.
3. Periodic solutions and global asymptotical stability
Consider the following controlled system:
.
x t Axt ft Ctut, u ∈ Uad , t / τk ,
Δx τk Bk x τk ck , t τk ,
3.1
JinRong Wang et al.
11
and the associated Cauchy problem
.
x t Axt ft Ctut, u ∈ Uad , t ∈ 0, T0 \ D,
Δx τk Bk x τk ck , k 1, 2, . . . , δ,
3.2
x0 x.
In addition to assumption H1, we make the following assumptions:
H3: f : 0, T0 → X is measurable and ft T0 ft for t ≥ 0.
H4: For each k ∈ Z , there exists δ ∈ N and ck ∈ X, ckδ ck .
H5: U· : 0, T0 → 2Y \ {Ø} has bounded, closed, and convex values and is graph
measurable, U· ⊆ Ω and Ω are bounded, where Y is a separable reflexive Banach
space.
H6: Operator C ∈ £∞ 0, T0 ; £Y, X and Ct T0 Ct, for t ≥ 0. Obviously, C :
Lp 0, T0 ; Y → Lp 0, T0 ; X1 < p < ∞.
Denote the set of admissible controls
Uad u· : 0, T0 −→ Y measurable | u t T0 ut, ut ∈ Ut a.e. for t ≥ 0 .
3.3
Obviously, Uad / Ø and Uad ⊆ Lp 0, T0 ; Y 1 < p < ∞, Uad is bounded, convex, and
closed.
We introduce P C-mild solution of Cauchy problem 3.2 and T0 -periodic P C-mild
solution of system 3.1.
Definition 3.1. A function x ∈ P C0, T0 ; X, for finite interval 0, T0 , is said to be a P C-mild
solution of the Cauchy problem 3.2 corresponding to the initial value x ∈ X and u ∈ Uad if
x is given by
x t, x, u St, 0x t
St, θfθ Cθuθdθ 0
S t, τk ck .
3.4
0≤τk <t
A function x ∈ P C0, ∞; X is said to be a T0 -periodic P C-mild solution of system
3.1 if it is a P C-mild solution of Cauchy problem 3.2 corresponding to some x and xt T0 , x, u xt, x, u for t ≥ 0.
Theorem 3.2. Assumptions [H1], [H3], [H4], [H5], and [H6] hold. Suppose {St, θ, t ≥ θ ≥ 0} is
exponentially stable, for every u ∈ Uad , system 3.1 has a unique T0 -periodic P C-mild solution:
xT0 t, x, u St, 0x t
0
St, θgu θdθ 0≤τk <t
S t, τk ck ≡ P gu , ck t,
3.5
12
Boundary Value Problems
p
where gu · f· C·u· ∈ LT0 0, ∞; X,
−1
x I − S T0 , 0
z,
z
T0
0
S T0 , θ gu θdθ S t, τk ck .
3.6
0≤τk <T0
Further,
p
P : LT0 0, ∞; X × X δ −→ P CT0 0, ∞; X
3.7
is a bounded linear operator and
P gu , ck P CT
0
0,∞;X
≤ B fLp C∞ uLp
T0
T0
δ ck
,
3.8
k1
where B KKQ 1 and Q I − ST0 , 0−1 .
Further, for arbitrary x0 ∈ X, the P C-mild solution x·, x0 , u of the Cauchy problem 3.2
corresponding to the initial value x0 ∈ X and control u ∈ Uad , satisfies the following inequality:
δ −νt
x t, x0 , u − xT t, x, u ≤ Be
fLp C∞ uLp ck ,
0
T0
T0
3.9
k1
where xT0 ·, x, u is the T0 -periodic P C-mild solution of system 3.1, B > 0 is not dependent on x0 ,
f, u, and ck . That is, x·, x0 , u can be approximated to the T0 -periodic P C-mild solution xT0 ·, x, u
according to exponential decreasing speed.
Proof. Consider the operator Q {S·, ·}, we have
∞
n0 ST0 , 0
n
. By 4 of Lemma 2.1 and the stability of
S T0 , 0 n S nT0 , 0 ≤ Ke−νnT0 −→ 0 as n −→ ∞.
3.10
∞
∞
∞
n
−νnT0
−νnT0
Thus, Q ≤
. Obviously, the series
is
n0 ST0 , 0 ≤
n0 Ke
n0 Ke
convergent, thus operator Q ∈ £b X. It comes from I − ST0 , 0Q QI − ST0 , 0 I
that Q I − ST0 , 0−1 ∈ £b X. It is well known that system 3.1 has a periodic P C-mild
solution if and only if xT0 x0. Since I − ST0 , 0 is invertible, we can uniquely solve
−1
x0 I − S T0 , 0
T
0
S T0 , θ gu θdθ 0
S t, τk ck .
3.11
0≤τk <T0
Let x I − ST0 , 0−1 z, where
z
T0
0
S T0 , θ gu θdθ S t, τk ck .
0≤τk <T0
3.12
JinRong Wang et al.
13
Note that
tT0
S t T0 , s Csusds t
3.13
St, sCsusds,
0
T0
it is not difficult to verify that the P C-mild solution of the Cauchy problem 3.2
corresponding to initial value x0 x given by
−1
xt, u St, 0 I − S T0 , 0
z
t
St, θgu θdθ 0
S t, τk ck
3.14
0≤τk <t
is just the unique T0 -periodic of system 3.1.
p
It is obvious that P : LT0 0, ∞; X × X δ → P CT0 0, ∞; X is linear. Next, verify the
estimation 3.8. In fact, for t ∈ 0, T0 ,
xT t, x, u ≤ St, 0x 0
t
0
S t, τ ck .
St, θgu θdθ k
3.15
0≤τk <t
On the other hand,
−1 T0 S T0 , θ gu θdθ S T0 , τ ck x ≤ I − S T0 , 0
k
0
T
≤ Q
0
Ke
0≤τk <T0
δ
−νT
−τ
ck gu θdθ Ke 0 k
−νT0 −θ 0
3.16
k1
≤ KQ fLp C∞ uLp
T0
T0
δ ck .
k1
Let B KKQ 1, next the estimation 3.8 is verified.
System 3.1 has a unique T0 -periodic P C-mild solution xT0 ·, x, u given by 3.5 and
3.6. The P C-mild solution x·, x0 , u of the Cauchy problem 3.2 corresponding to initial
value x0 and control u ∈ Uad can be given by 3.4. Then,
x t, x0 , u − xT t, x, u ≤ St, 0 x − x0 ≤ Ke−νt x0 x
0
3.17
δ −νt ≤ Ke
ck
.
x0 KQ fL1T C∞ uLPT 0
Let B max{K, K 2 Q} > 0, one can obtain 3.9 immediately.
0
k1
14
Boundary Value Problems
Definition 3.3. The T0 -periodic P C-mild solution xT0 ·, x, u of the system 3.1 is said to be
globally asymptotically stable in the sense that
lim x t, x0 , u − xT0 t, x, u 0,
3.18
t → ∞
where x·, x0 , u is any P C-mild solutions of the Cauchy problem 3.2 corresponding to
initial value x0 ∈ X and control u ∈ Uad .
By Theorem 3.2 and the stability of the impulsive evolution operator {S·, ·} in
Section 2, one can obtain the following results.
Corollary 3.4. Under the assumptions of Theorem 3.2, the system 3.1 has a unique T0 -periodic
P C-mild solution xT0 ·, x, u which is globally asymptotically stable.
4. Existence of periodic optimal harvesting policy
In this section, we discuss existence of periodic optimal harvesting policy, that is, periodic
optimal controls for optimal control problems arising in systems governed by linear
impulsive periodic system on Banach space.
By the T0 -periodic P C-mild solution expression of system 3.1 given in Theorem 3.2,
one can obtain the result.
Theorem 4.1. Under the assumptions of Theorem 3.2, the T0 -periodic P C-mild solution of system
3.1 continuously depends on the control on Lp 0, T0 ; Y , that is, let x1 x2 be T0 -periodic P Cmild solution of system 3.1 corresponding to u1 u2 ∈ Uad ⊆ Lp 0, T0 ; Y . There exists constant
> 0 such that
K
1
x − x2 u1 − u2 p
≤K
.
P C0,T0 ;X
L 0,T0 ;Y 4.1
Proof. Since x1 and x2 are the T0 -periodic P C-mild solution corresponding to u1 and u2 ∈ Uad ,
respectively, then we have
−1
xi t ≡ x t, ui St, 0 I − S T0 , 0
zi t
St, θ fθ Cθui θ dθ,
i 1, 2,
4.2
0
where
zi T0
S T0 , θ fθ Cθui θ dθ,
i 1, 2.
4.3
0
Further,
T0
1
x t − x2 t ≤ St, 0 I − S T0 , 0 −1 1
u1 − u2 p
≤K
,
L 0,T0 ;Y S T0 , θ C∞ u1 θ − u2 θdθ
0
MT0 MT0 Q 1C∞ . This completes the proof.
where K
4.4
JinRong Wang et al.
15
Lemma 4.2. Suppose C is a strong continuous operator. The operator Θ : Lp 0, T0 ; Y →
P C0, T0 ; X, given by
Θu· ·
S·, θCθuθdθ
4.5
0
is strongly continuous.
Proof. Without loss of generality, for τk−1 ≤ s < τk < t ≤ τk1 ,
Θut t
0
T t−
τk
I Bj
· T τj − τj−1 I Bi T τi − θ Csus ds.
4.6
θ<τj <t
By virtue of strong continuity of C, boundedness of Bk , supt∈0,T0 T t CT0 < ∞, Θ is
strongly continuous.
Let x·, u denote the T0 -periodic P C-mild solution of system 3.1 corresponding to
the control u ∈ Uad , we consider the Bolza problem P.
Find u0 ∈ Uad such that Ju0 ≤ Ju, for all u ∈ Uad , where
Ju T0
lt, xt, u, utdt Ψ x T0 , u .
4.7
0
We introduce the following assumption on l and Ψ.
Assumption H7.
H7.1 The functional l : 0, T0 × X × Y → R ∪ {∞} is Borel measurable.
H7.2 lt, ·, · is sequentially lower semicontinuous on X × Y for almost all t ∈ 0, T0 .
H7.3 lt, x, · is convex on Y for each x ∈ X and almost all t ∈ 0, T0 .
H7.4 There exist constants d ≥ 0, e > 0, ϕ is nonnegative and ϕ ∈ L1 0, T0 ; R such that
p
lt, x, u ≥ ϕt dx euY .
4.8
H7.5 The functional Ψ → R is continuous and nonnegative.
Now we can give the following results on existence of periodic optimal controls for
Bolza problem P.
Theorem 4.3. Suppose C is a strong continuous operator and assumption [H7] holds. Under the
assumptions of Theorem 3.2, the problem (P) has a unique solution.
Proof. If inf{Ju | u ∈ Uad } ∞, there is nothing to prove.
16
Boundary Value Problems
We assume that inf{Ju | u ∈ Uad } < ∞. By assumption H7, we have
T0
Ju ≥
ϕtdt d
0
T0
xtdt e
0
T0
0
p
utY dt Ψ x T0 , u ≥ −η > −∞,
4.9
where η > 0 is a constant. Hence ≥ −η > −∞.
By the definition of infimum there exists a sequence {un } ⊂ Uad ⊆ Lp 0, T0 , Y 1 <
p < ∞, such that limn → ∞ Jun .
Since {un } is bounded in Lp 0, T0 ; Y 1 < p < ∞, there exists a subsequence,
relabeled as {un }, and u0 ∈ U such that limn → ∞ un u0 weakly convergence in Lp 0, T0 ; Y ,
and Jun < ε. Because of Uad is the closed and convex set, thanks to the Mazur lemma,
u0 ∈ Uad . Suppose x·, un and x·, u0 are the T0 -periodic P C-mild solution of system 3.1
corresponding to un n 1, 2, . . . and u0 , respectively, then x·, un and x·, u0 can be given
by
xn t ≡ x t, un
t
−1
zn St, 0 I − S T0 , 0
0
x t ≡ x t, u
0
0
−1
z0 St, 0 I − S T0 , 0
t
St, θgun θdθ S t, τk ck ,
0≤τk <t
St, θgu0 θdθ 0
S t, τk ck ,
4.10
0≤τk <t
where
zn T0
0
z0 T0
S T0 , θ gun θdθ S t, τk ck ,
0≤τk <T0
S T0 , θ gu0 θdθ 0
4.11
S t, τk ck .
0≤τk <T0
Define
ηn t Θun t − Θu t 0
t
St, θCθ un θ − u0 θ dθ,
4.12
0
then by Lemma 4.2, we have
ηn −→ 0 in P C 0, T0 ; X with strongly convergence,
4.13
w
as un → u0 weakly convergence in Lp 0, T0 ; Y .
Next, we show that
xn −→ x0 in P C 0, T0 ; X with strongly convergence as n −→ ∞.
4.14
JinRong Wang et al.
17
In fact, for t ∈ 0, τ1 , we have
xn t − x0 t ≤ MT Q 1 ηn t
≤ C1 ηn C0,τ1 ;X .
0
C0,τ1 ;X
4.15
By elementary computation, we arrive at
xn τ − x0 τ ≤ B1 1 xn τ1 − x0 τ1 ≤ C ηn .
1
1
1
C0,τ1 ;X
4.16
Consider the time interval τ1 , τ2 , similarly we obtain
xn t − x0 t ≤ C2 ηn ,
Cτ1 ,τ2 ,X
xn τ − x0 τ ≤ C ηn .
2
2
2
Cτ1 ,τ2 ;X
4.17
k 1, 2, . . . , n, and the xn τk , x0 τk , prior to the jump at time
In general, given any τk ∈ D,
τk , we immediately follow the jump as
xn τk xn τk Bk xn τk ck ,
x0 τk x0 τk Bk x0 τk ck ,
4.18
the associated interval τk , τk1 , we also similarly obtain
xn t − x0 t ≤ Ck1 ηn ,
Cτk ,τk1 ;X
0
xn τ
≤ C ηn .
k1 − x τk1
k1
Cτk ,τk1 ;X
4.19
Step by step , we repeat the procedures till the time interval is exhausted. Let C max{C1 , C1 , C2 , C2 , . . ., Ck1 }, thus we obtain
xn − x0 P C0,T0 ;X
≤ Cηn P C0,T0 ;X ,
4.20
that is,
xn −→ x0 in P C 0, T0 ; X ,
4.21
with strongly convergence as n → ∞.
Since P C0, T0 ; X → L1 0, T0 ; X, using the assumption H7 and Balder’s
theorem, we can obtain
lim
T0
n→∞ 0
lt, xn t, un tdt Ψ xn T0 ≥
T0
l t, x0 t, u0 t dt Ψ x0 T0 ≥ . 4.22
0
This shows that J attains its minimum at u0 ∈ Uad . This completes the proof.
18
Boundary Value Problems
5. Example
Last, an academic example is given to illustrate our theory.
Let X U Y L2 0, 1 and consider the following population evolution equation
with impulses:
∂
∂
xt, y a xt, y kxt, y b sint, y ut, y,
∂t
∂y
3
1
u ∈ Uad ⊆ U, y ∈ 0, 1, t ∈ 0, 2kπ \
π, π, π, . . . , k ∈ Z0 ,
2
2
x0, y x2π, y 0, y ∈ 0, 1,
5.1
xt, 0 xt, 1 0, t > 0,
⎧
⎪
0.05 Ix τk , y , k 1,
⎪
⎪
⎨
Δx τk , y −0.05 Ix τk , y , k 2, y ∈ 0, 1, t > 0,
⎪
⎪
⎪
⎩0.05 Ixt, y, k 3,
where t denotes time, y denotes age, xt, y is called age density function, a and b are positive
constants, k is a bounded measurable function, that is, k ∈ L∞ 0, 1. k denotes the age-specific
death rate, b sint, y denotes the age density of migrants, and ut, y denotes the control. The
admissible control set Uad {u ∈ Y |uL2 0,T0 ;Y ≤ 1}.
A linear operator A defined on X by
Ax −a
dx
kx,
dy
∀x ∈ DA,
5.2
where the domain of A is given by
DA dx
∈ X,
x ∈ X : x is a absolutely continuous,
dy
x0 x2π 0 .
5.3
By the fact that the operator −d · /dy is an infinitesimal generator of a C0 -semigroup see
39, Example 2.21 and 38, Theorem 4.2.1, then A is an infinitesimal generator of a C0 semigroup since the operator kI is bounded.
Now let us consider the following operators family:
y
⎧
1
⎪
⎨exp
ksds · xy − at,
a y−at
T txy ⎪
⎩
0, otherwise.
if 0 ≤ t ≤
y
,
a
5.4
It is not difficult to verify that {T t, t ≥ 0} defines a C0 -semigroup and A is just the
infinitesimal generator of the C0 -semigroup {T t, t ≥ 0}. Since k ∈ L∞ 0, 1, then there exits
JinRong Wang et al.
19
a constant m > 0 such that |ks| ≤ m a.e. s ∈ 0, 1. For an arbitrary function x ∈ L2 0, 1, by
using the expression 5.4 of the semigroup {T t, t ≥ 0}, the following inequality holds:
∞
T tx2 dt ≤
y/a
0
0
≤
1/a
0
y
1
exp 2
mds ds · x2L2 0,1
a y−at
5.5
exp 2mtds · x2L2 0,1
exp 2ma−1 − 1
x2L2 0,1 .
≤
2m
Hence, Lemma 2.3 leads to the exponential stability of {T t, t ≥ 0}. That is, there exist K0 > 0
and ν0 > 0 such that T t ≤ K0 e−ν0 t , t > 0.
Let
Ju
2π 1
0
|xt, ξ|2 |ut, ξ|2 dξdt 0
1
|x2π, ξ|2 dξ.
5.6
0
Define x·y x·, y, sin·y sin·, y, B·u·y u·, y, B1 B3 0.05I,
B2 −0.05I. Thus system 5.1 can be rewritten as
3
1
ẋt Axt b sin t ut, u ∈ Uad , t ∈ 0, 2kπ \
π, π, π, . . . ,
2
2
i
i
Δx
π Bi x
π , i 1, 2, 3, . . . ,
2
2
k ∈ Z0 ,
5.7
with the cost function
Ju 2π
xt2 ut2 dt x2π2 .
5.8
0
By Lemma 2.4, for ν0 > λ > 3 ln K0 2 ln 1.05 ln 0.95, {St, θ, t ≥ θ ≥ 0} is
exponentially stable. Now, all the assumptions are met in Theorems 3.2 and 4.3, our results
can be used to system 5.1. Thus, system 5.1 has a unique 2π-periodic P C-mild solution
x2π ·, y, u ∈ P C2π 0 ∞; L2 0, 1 which is globally asymptotically stable and there exists
a periodic control u0 ∈ Uad such that Ju0 ≤ Ju for all u ∈ Uad .
The results show that the optimal population level is truly the periodic solution of the
considered system, and hence, it is globally asymptotically stable. Meanwhile, it implies that
we can achieve sustainability at a high level of productivity and good economic profit by
virtue of scientific, effective, and continuous management of the resources.
Acknowledgments
This work is supported by Natural Science Foundation of Guizhou Province Education
Department no. 2007008. This work is also supported by the undergraduate carve out
project of Department of Guiyang Science and Technology 2008, no. 15-2.
20
Boundary Value Problems
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