Analysis of nonlinear fractional control systems in Banach spaces

Nonlinear Analysis 74 (2011) 5929–5942

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Analysis of nonlinear fractional control systems in Banach spaces✩ JinRong Wang a , Yong Zhou b,∗ a

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, PR China

b

Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China

article

info

Article history: Received 12 August 2010 Accepted 22 May 2011 Communicated by Ravi Agarwal MSC: 34G10 34G20 93C25 Keywords: Fractional control systems Mild solutions Weakly singular inequality Optimal controls Weakly compactness

abstract In this paper, we consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, p-mean continuity, weakly singular inequality and Leray–Schauder’s fixed point theorem with compact mapping, the sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we consider the nonlinear control system of fractional order such as Dq x(t ) = −Ax(t ) + f (t , x(t ))u(t ), x(0) = x0 , u(t ) ∈ U (t ), a.e.

C

a.e., q ∈ (0, 1),

(1)

where C Dq is the Caputo fractional derivative of order 0 < q < 1, −A : D(A) → X is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators {T (t ), t ≥ 0}, f : J × Xα → X is specified later, where J = [0, T ] is a closed, bounded interval in R+ = [0, ∞), Xα = D(Aα )(0 < α < 1) is a Banach space with the norm ‖x‖α = ‖Aα x‖ for x ∈ Xα and U (·) is the control constraint multifunction. The associated cost functions to be minimized over the family of admissible state control pairs (x, u) is given by

J ( x , u) =

∫

g (t , x(t ), u(t ))dt . J

During the past decades, fractional differential equations have attracted many authors (see for instance [1–4]). This is mostly because they efficiently describe many phenomena arising in engineering, physics, economy and science. We can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic (see for instance [5–10] and the references therein). ✩ Research supported by the National Natural Science Foundation of China (10971173), Tianyuan Special Funds of the National Natural Science Foundation of China (11026102) and Key Projects of Science and Technology Research in the Ministry of Education (211169). ∗ Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Y. Zhou).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.059

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J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

On the fractional differential equations in infinite dimensional spaces are attracted by many authors (see for instance [11–24], and the references therein). To study the theory of abstract differential equations with fractional derivatives in infinite dimensional spaces, the first step is how to introduced a suitable concept of mild solutions. Very recently, Hernández et al. [25] point that some recent papers of fractional differential equations in Banach spaces are incorrect and use another approach to treat abstract equations with fractional derivatives based on the well developed theory of resolvent operators for integral equations. Moreover, Zhou et al. [26,27] also introduced a new concept of a mild solution based on Laplace transform and probability density functions. When the fractional differential equations describe the performance index and system dynamics, an optimal control problem reduces to a fractional optimal control problem. The optimal control of a fractional distributed parameter system is an optimal control for which system dynamics are defined with fractional differential equations. There has been very little work in the area of optimal control problems on fractional distributed parameter systems (see [28,29]). An important question in optimal control theory is to determine whether or not there exists a pair (x0 , u0 ) satisfying the given constraints (1) (i.e., is admissible) and minimizes J (x, u) over all such admissible pairs. Cesari [30], Hou [31] and Papageorgiou [32] considered nonlinear infinite dimensional control systems and obtain some very interesting results. Very recently, Wang and Zhou [33] considered a class of fractional evolution equations and optimal controls and obtained some existence results under some conditions. In this paper, we examine a different class of nonlinear fractional control systems and prove the existence of solutions and existence of optimal controls. Firstly, a sufficient condition is given for the existence and uniqueness of mild solutions for system (1) by virtue of the fractional calculus, Hölder’s inequality, p-mean continuity, weakly singular inequality and Leray–Schauder’s fixed point theorem with compact mapping. Secondly, an existence result of optimal controls for the cost functional J (x, u) is presented by using the previous result of Castaing and Clauzure [34] on the lower semicontinuity of integral functionals. We extend the previous results of nonlinear systems to the fractional nonlinear systems in Banach spaces. Finally, an example of a fractional distributed parameter system is given to illustrate the effectiveness of the results obtained. 2. Preliminaries Let X , Y be two separable reflexive Banach spaces, where Y is the control space, Lb (X , Y ) be the space of bounded, linear operators from X into Y , Lb (X ) be the space of bounded, linear operators from X into itself. Suppose that −A : D(A) → X is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators {T (t ), t ≥ 0}. This means that there exists M > 1 such that ‖T (t )‖ ≤ M. We assume without loss of generality that 0 ∈ ρ(A). This allows us to define the fractional power Aα for 0 < α < 1, as a closed linear operator on its domain D(Aα ) with inverse A−α (see [35]). We have the following basic properties Aα . Theorem 2.1 ([35], pp. 69–75). (1) (2) (3) (4)

Xα = D(Aα ) is a Banach space with the norm ‖x‖α = ‖Aα x‖ for x ∈ Xα . T (t ) : X → Xα for each t > 0. Aα T (t )x = T (t )Aα x for each x ∈ Xα and t ≥ 0. For every t > 0, Aα T (t ) is bounded on X and there exists Mα > 0 and ν > 0 such that

‖Aα T (t )‖ ≤

Mα Mα −ν t e ≤ α . α t t

(5) A−α is a bounded linear operator in X with Xα = Im (A−α ). (6) If 0 < α ≤ β < 1, then D(Aβ ) ↩→ D(Aα ). Remark 2.2. Observe as in [36] that by Theorem 2.1(2) and (3), the restriction Tα (t ) of T (t ) to Xα is exactly the part of T (t ) in Xα . Let x ∈ Xα . Since ‖T (t )x‖α ≤ ‖Aα T (t )x‖ = ‖T (t )Aα x‖ ≤ ‖T (t )‖ ‖Aα x‖ = ‖T (t )‖ ‖x‖α , and as t decreases to 0, ‖T (t )x − x‖α = ‖Aα T (t )x − Aα x‖ = ‖T (t )Aα x − Aα x‖ → 0, for all x ∈ Xα , it follows that {T (t ), t ≥ 0} is a family of strongly continuous semigroup on Xα and ‖Tα (t )‖ ≤ ‖T (t )‖ ≤ M for all t ≤ 0. In what follows, we also use ‖f ‖Lm to denote the Lm (J , R+ ) norm of f whenever f ∈ Lm (J , R+ ) for some m with 1 ≤ m +

< ∞, ‖u‖LnY to denote the Ln (J , Y ) norm of u whenever u ∈ Ln (J , Y ) for some n with 1 < n ≤ ∞ and m1 + 1n = 1. For brevity, n m n we denote Lm + by L (J , R+ ) and LY by L (J , Y ). We set α ∈ (0, 1) and denote by Cα , the Banach space C (J , Xα ) endowed with supnorm given by ‖x‖∞ ≡ supt ∈J ‖x‖α , for x ∈ Cα . Let us recall the following known definitions. For more details, see [1]. Definition 2.3. The fractional integral of order γ with the lower limit zero for a function f is defined as γ

I f (t ) =

1

Γ (γ )

t

∫ 0

f (s) ds, (t − s)1−γ

t > 0, γ > 0,

provided the right side is point-wise defined on [0, ∞), where Γ (·) is the gamma function.

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

5931

Definition 2.4. The Riemann–Liouville derivative of order γ with the lower limit zero for a function f : [0, ∞) → R can be written as L

Dγ f (t ) =

1

dn

Γ (n − γ )

dt n

f ( s) ds, (t − s)γ +1−n

t

∫ 0

t > 0, n − 1 < γ < n.

Definition 2.5. The Caputo derivative of order γ for a function f : [0, ∞) → R can be written as

 γ

D f (t ) = D L

γ

f (t ) −

n −1 k − t

k!

k=0

 f

(k)

(0) ,

t > 0, n − 1 < γ < n.

Remark 2.6. (1) If f (t ) ∈ C n [0, ∞), then Dγ f (t ) =

Γ (n − γ )

f (n) (s)

t

∫

1

0

(t − s)γ +1−n

ds = I n−γ f (n) (t ),

t > 0, n − 1 < γ < n.

(2) The Caputo derivative of a constant is equal to zero. (3) If f is an abstract function with values in X , then integrals which appear in Definitions 2.4 and 2.5 are taken in Bochner’s sense. We need our previous work [26, Lemma 3.1 and Definition 3.1] and the following definition of mild solutions. Definition 2.7. By the mild solutions of the system (1), we mean that the function x ∈ Cα which satisfies x(t ) = T (t )x0 +

∫

t

(t − s)q−1 S (t − s)f (s, x(s))u(s)ds,

for u ∈ LnY , t ∈ J ,

0

where T (t ) =

∞

∫

ξq (θ )T (t q θ )dθ ,

S (t ) = q

0

ξq (θ ) =

∞

∫

θ ξq (θ )T (t q θ )dθ , 0

1 1 −1− 1 q ϖ (θ − q ) ≥ 0, θ q q

ϖq (θ ) =

∞ 1 −

π

(−1)n−1 θ −qn−1

n =1

Γ (nq + 1) sin(nπ q), n!

θ ∈ (0, ∞),

ξq is a probability density function defined on (0, ∞), that is ∫ ∞ ξq (θ ) ≥ 0, θ ∈ (0, ∞) and ξq (θ )dθ = 1. 0

Remark 2.8. It is not difficult to verify that for v ∈ [0, 1], ∞

∫ 0

θ v ξq (θ )dθ =

∞

∫

θ −qv ϖq (θ )dθ =

0

Γ (1 + v) . Γ (1 + qv)

The following results will be used throughout this paper. Lemma 2.9 (Lemma 2.9, [33]). In the Banach space X , the operators T and S have the following properties: (1) For any fixed t ≥ 0, T (t ) and S (t ) are linear and bounded operators, i.e., for any x ∈ X ,

‖T (t )x‖ ≤ M ‖x‖ and ‖S (t )x‖ ≤

qM

Γ (1 + q)

‖x‖.

(2) {T (t ), t ≥ 0} and {S (t ), t ≥ 0} are strongly continuous. (3) For every t > 0, T (t ) and S (t ) are also compact operators. (4) For any x ∈ X , β ∈ (0, 1) and α ∈ (0, 1), we have AS (t )x = A1−β S (t )Aβ x, t ∈ J , Mα qΓ (2 − α) −α q ‖Aα S (t )‖ ≤ t , Γ (1 + q(1 − α))

0 < t ≤ T.

(5) For fixed t ≥ 0 and any x ∈ Xα , we have ‖T (t )x‖α ≤ M ‖x‖α and ‖S (t )x‖α ≤

qM

Γ (1 + q)

‖x ‖α .

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J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

(6) Tα (t ) and Sα (t ), t > 0 are uniformly continuous, that is for each fixed t > 0, and ϵ > 0, there exists a h > 0 such that ‖Tα (t + ϵ) − Tα (t )‖α < ε, for t + ϵ ≥ 0 and |ϵ| < h. ‖Sα (t + ϵ) − Sα (t )‖α < ε, for t + ϵ ≥ 0 and |ϵ| < h, where ∞

∫

Tα (t ) =

ξq (θ )Tα (t q θ )dθ ,

Sα (t ) = q

∞

∫

θ ξq (θ )Tα (t q θ )dθ . 0

0

Lemma 2.10 (Schauder Theorem, [37]). A bounded linear operator T defined on a Banach space X is completely continuous if and only if the adjoint operator T ∗ of T defined on the conjugate space X ∗ of X is completely continuous. Lemma 2.11. In the reflexive Banach space X , we have the following results. (7) For any fixed t ≥ 0 , T ∗ (t ) and S ∗ (t ) are linear and bounded operators. (8) {T ∗ (t ), t ≥ 0} and {S ∗ (t ), t ≥ 0} are strongly continuous. (9) For every t > 0 , T ∗ (t ) and S ∗ (t ) are compact operators, where T ∗ (t ) =

∞

∫

ξq (θ )T ∗ (t q θ )dθ ,

S ∗ (t ) = q

∞

∫

θ ξq (θ )T ∗ (t q θ )dθ . 0

0

Proof. By virtue of the fact that {T (t ), t ≥ 0} is a strongly continuous semigroup in the reflexive Banach space X , then the adjoint semigroup {T ∗ (t ), t ≥ 0} is also a strongly continuous semigroup in X ∗ = X with the infinitesimal generator A∗ . For (7), it is easy to see that T ∗ (t ) and S ∗ (t ) are linear operators because of T ∗ (t ) is linear. Since ‖T (t )‖ = ‖T ∗ (t )‖ for t ≥ 0, one can repeat the proof process of Lemma 3.2 of [26] to derive the results. For (8), by applying the same method used in Lemma 3.3 of [26] and using the strongly continuity of {T ∗ (t ), t ≥ 0}, one can complete the rest proof. For (9), we know that, for t > 0, T ∗ (t ) is compact from Lemma 2.10. Similar to the proof of Lemma 3.4 of [26], one can prove T ∗ (t ) and S ∗ (t ) are compact operators for every t > 0.  In order to obtain the uniqueness of the mild solutions, we need the following weakly singular inequality. It will play an essential role in the study of nonlinear problems on infinite dimensional spaces. Lemma 2.12 (Weakly Singular Inequality, Theorem 1 [38]). Let κ ∈ (0, 1], x ∈ C (R+ , R+ ) satisfies the following inequality x(t ) ≤ a(t ) + c (t )

∫

t

(t αˆ − sαˆ )β−1 sγˆ −1 F (s)[x(s)]κ ds, ˆ

t ≥ 0,

(2)

0

where a(t ), c (t ) ∈ C (R+ , R+ )are not  decreasing functions, F (t ) ∈ C (R+ , R+ ). Case1: if αˆ ∈ (0, 1], βˆ ∈ 21 , 1 and γˆ ≥ 32 − βˆ , then



ˆ ˆ B1 (t ) x(t ) ≤ 2β a(t ) exp (1 − β)

t

∫

 1 [F (s)] 1−βˆ ds ,

0

where

 βˆ ˆ 1)+γˆ −1+βˆ 1−βˆ α( ˆ β− 1 βˆ + γˆ − 1 2βˆ − 1 1−βˆ B1 (t ) = ; B , [c (t )] 1−βˆ t αˆ αˆ βˆ βˆ   1−2βˆ 2 Case 2: if αˆ ∈ (0, 1], βˆ ∈ 0, 21 and γˆ ≥ , then 1−βˆ 2   ∫ t 1+3βˆ 1+4βˆ ˆ β ˆ ˆ x(t ) ≤ 2 1+4β a(t ) exp B2 (t ) [F (s)] β ds , 1 + 4βˆ 0 

2



ˆ − βˆ γˆ (1 + 4β) 4βˆ 2 B , ˆ αˆ α( ˆ 1 + 3β) 1 + 3βˆ



where B2 (t ) =

2



 1+3βˆ βˆ

[c (t )]

1+4βˆ

βˆ

t

ˆ β− ˆ 1)+γˆ (1+4β)− ˆ βˆ α( ˆ 1+4β)( βˆ

Here, B[·, ·] denotes the Beta function. Lemma 2.13 (p-Mean Continuity, [39]). For each ψ ∈ Lp (J , X ) with 1 ≤ p < +∞, T

∫

‖ψ(t + h) − ψ(t )‖p dt = 0

lim

h→0

0

where ψ(s) = 0 for s does not belong to J.

.

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

5933

Lemma 2.14. A measurable function V : J → X is Bochner integrable if ‖V ‖ is Lebesgue integrable. ˘ The following well known Eberlein–Smulian theorem is an important result relating three different kinds of weak compactness in a Banach space. ˘ Lemma 2.15 (Eberlein–Smulian Theorem). The following conditions on a subset A of a Banach space X are equivalent: (1) A is weakly compact. (2) A is weakly sequentially compact. (3) A is weakly limit point compact. 3. Main results 3.1. Existence and uniqueness of mild solutions We make the following assumptions. [Hf ]: f : J × Xα → Lb (Y , X ) is a mapping such that: (i) for all x ∈ Xα , t → f (t , x) is measurable and ‖f (t , x)‖Lb (Y ,X ) ≤ φ(t ), a.e., where m q

n1 n q

φ ∈ L+1 ∩ L+1 ,

0 < q1 ≤ min {1 − m1 n(1 − q), 1 − m1 nα q, 1 + m(q − α q − 1)}

1 for some 1 ≤ m, m1 < ∞, 1 < n, n1 ≤ ∞ and m + 1n = 1, m11 + n11 = 1; (ii) for all t ∈ J and all x, z ∈ Xα , there exists a constant Lf > 0 such that

‖f (t , x) − f (t , z )‖Lb (Y ,X ) ≤ Lf ‖x − z ‖α . [HU]: t → U (t ) has nonempty, weakly compact and convex values, it is measurable, i.e., for all x ∈ Xα , t → d(x) = inf ‖x − z ‖ is measurable, z ∈U (t )

and that

|U (t )| = sup ‖z ‖ ∈ LnY ,

1 < n ≤ ∞.

z ∈U (t )

Theorem 3.1. Assume that the conditions [Hf ] and [HU ] are satisfied. If x0 ∈ Xα and one of the following two conditions:

(C1)

 1  1 − < q(1 − α) < 1, m

1   < q(1 − α) < 1, 2

(C2)

for some 1 ≤ m < 2;

 1 1  1 − < q(1 − α) < , m

 0 < q(1 − α) <

1 2

2

,

for some 2 ≤ m < ∞,

for some 1 ≤ m < 2,

for some 2 ≤ m < ∞;

are fulfilled. Then system (1) has an unique mild solution on J. Proof. Define the function F : Cα → Cα by

(Fx)(t ) = T (t )x0 +

t

∫

(t − s)q−1 S (t − s)f (s, x(s))u(s)ds. 0

It is not difficult to verify that Fx ∈ Cα . In fact, for 0 ≤ t1 < t2 ≤ T , by Lemma 2.9 and Hölder’s inequality, one can deduce

‖(Fx)(t1 ) − (Fx)(t2 )‖α ≤ ‖T (t1 )x0 − T (t2 )x0 ‖α +

t1

∫

(t1 − s)q−1 ‖S (t1 − s)f (s, x(s))u(s)

0

− S (t2 − s)f (s, x(s))u(s)‖α ds ∫ t1 + [(t1 − s)q−1 − (t2 − s)q−1 ]‖S (t2 − s)f (s, x(s))u(s)‖α ds 0 ∫ t2 + (t2 − s)q−1 ‖S (t2 − s)f (s, x(s))u(s)‖α ds ≤ ‖Tα (t1 ) − Tα (t2 )‖α ‖x0 ‖α t1 t1

∫ + 0

(t1 − s)q−1 ‖Aα [S (t1 − s) − S (t2 − s)]‖ ‖f (s, x(s))u(s)‖ds

5934

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942 t1

∫ +

[(t1 − s)q−1 − (t2 − s)q−1 ]‖Aα S (t2 − s)‖ ‖f (s, x(s))u(s)‖ds

0 t2

∫ +

(t2 − s)q−1 ‖Aα S (t2 − s)‖ ‖f (s, x(s))u(s)‖ds

t1

≤ I1 + I2 + I3 + I4 , where

I = ‖T (t ) − T (t )‖ ‖x ‖ , 1 α 2 α 0 α  ∫ tα1 1    q  (t1 − s) −1 ‖Aα [S (t1 − s) − S (t2 − s)]‖φ(s)‖u(s)‖Y ds, I2 =    0 ∫ t1 Mα qΓ (2 − α) [(t1 − s)q−1 − (t2 − s)q−1 ](t2 − s)−αq φ(s)‖u(s)‖Y ds, I =  3  Γ ( 1 + q ( 1 − α))  0  ∫ t2   I = Mα qΓ (2 − α)  (t2 − s)q−αq−1 φ(s)‖u(s)‖Y ds. 4 Γ (1 + q(1 − α)) t1

(3)

It is easy to see

∫ ∞    α q q  ‖Aα S (t1 − s) − Aα S (t2 − s)‖ =  q θ ξ (θ ) A [ T (( t − s ) θ ) − T (( t − s ) θ )] d θ q 1 2   0 ∫ ∞ ∫ t1     = qθ ξq (θ )Aα AT ((t − s)q θ )d(t − s)q θ dθ    0 t2 ∫ t1  ∫ ∞  α+1  A T ((t − s)q θ ) d(t − s)q θ dθ ≤ qθ ξq (θ ) 0

t2

∞

∫

qθ ξq (θ )

≤

∫

0

t1

Mα+1 [(t − s)q θ ]−(α+1) d(t − s)q θ

dθ

t2

∞

∫

θ ξq (θ )

≤ qMα+1 ≤ q2 Mα+1

t1

∫

0

q(t − s)q−1 θ[(t − s)q θ ]−(α+1) dt



dθ

t2

∞

∫

θ 1−α ξq (θ )

t1

∫

0

≤



 (t − s)q−1−q(α+1) dt dθ

t2

q Mα+1 Γ (2 − α)

[(t2 − s)−qα − (t1 − s)−qα ].

α Γ (1 + q(1 − α))

Then, I2 =

≤

qMα+1 Γ (2 − α)

α Γ (1 + q(1 − α)) qMα+1 Γ (2 − α)

t1

∫

(t1 − s)q−1 [(t2 − s)−qα − (t1 − s)−qα ]φ(s)‖u(s)‖Y ds

0 t1

∫

α Γ (1 + q(1 − α))

−q α

|(t2 − s)

−q α m

− (t1 − s)

 m1 ∫

t1

| ds

0

(t1 − s)

n(q−1)

 1n

[φ(s)] ‖u(s)‖

n Y ds

t1

 n1n

n

0

where t1

∫

(t1 − s)

n(q−1)

[φ(s)] ‖u(s)‖ n

n Y ds

 1n

t1

∫ ≤

0

m1 n(q−1)

(t1 − s)

[φ(s)]

m1 n

 m1 n ∫ 1

ds

0

‖u(s)‖

0

and t1

∫

(t1 − s)

m1 n(q−1)

[φ(s)]

m1 n

 m1 n ds

t1

∫

1

≤

0

(t1 − s)

m1 n(q−1) 1−q1

 1m−qn1 ∫ ds

0

≤

≤

 nq1n 1

ds

(t1 − s)

m1 n(q−1) 1−q1

 1m−qn1 1

ds

0



[φ(s)]

n1 n q1

0 t1

∫

t1

1

1 − q1 1 − q1 + m1 n(q − 1)

‖φ‖

n1 n q

L+ 1 m n(q−1) 1+ 11−q 1 t1

 1m−qn1 1

‖φ‖

n1 n q

L+ 1

.

n1 n Y ds

1

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

5935

Thus, I2 ≤

qMα+1 Γ (2 − α)

t1

∫

−q α

 m1

−q α m

|(t2 − s) − (t1 − s) | ds α Γ (1 + q(1 − α)) 0   1m−qn1 m n(q−1) 1 1+ 11−q 1 − q1 1 × t1 ‖φ‖ n1 n ‖u‖Ln1 n . q Y 1 − q1 + m1 n(q − 1) L+ 1

Similar to the estimation of the term I2 , we have I3 ≤

Mα qΓ (2 − α)

∫

t1

 m1

|(t2 − s) − (t1 − s) | ds Γ (1 + q(1 − α)) 0 1−q [ m nα q ] m n1 m nα q 1− 11−q 1 1 − q1 1− 11−q 1 1 − t |(t2 − t1 ) × | ‖φ‖ n1 n ‖u‖Ln1 n . 2 q Y 1 − q1 − m1 nα q L+ 1 q −1

q −1 m

On the other hand,

∫

t2

(t2 − s)

q−α q−1

φ(s)‖u(s)‖Y ds ≤

∫

t1

t2

m(q−α q−1)

(t2 − s)

[φ(s)] ds m

 m1 ∫

t1

∫

‖u(s)‖

n Y ds

 1n

t1

t2

≤

t2

m(q−α q−1)

(t2 − s)

[φ(s)] ds m

 m1 ∫

t1

T

‖u(s)‖

n Y ds

 1n

0

where t2

∫

m(q−α q−1)

(t2 − s)

[φ(s)] ds m

 m1

∫

t2

≤

t1

(t2 − s)

m(q−α q−1) 1−q1

 1−mq1 ∫ ds

t1

[ ≤

t2

m q1

[φ(s)] ds

 qm1

t1

1 − q1 1 − q1 + m(q − α q − 1)

(t2 − t1 )

1+

m(q−α q−1) 1−q1

] 1−mq1 ‖φ‖

m q

L+1

.

Then,

∫

t2

(t2 − s)

q−α q−1

φ(s)‖u(s)‖Y ds ≤

t1

[

1 − q1 1 − q1 + m(q − α q − 1)

(t2 − t1 )

1+

m(q−α q−1) 1−q1

] 1−mq1 ‖φ‖

m q

L+1

‖u‖LnY .

Thus, by Lemma 2.9(6) and Lemma 2.13, one can check that Ii → 0 as t2 → t1 , i = 1, 2, 3, 4. This implies that Fx ∈ Cα . Then we proceed in four steps. (S1) F is a continuous operator on Cα . In fact, for x1 , x2 ∈ Cα ,

‖(Fx1 )(t ) − (Fx2 )(t )‖α ≤

t

∫

(t − s)q−1 ‖S (t − s)[f (s, x1 (s))u(s) − f (s, x2 (s))u(s)]‖α ds 0 t

∫

(t − s)q−1 ‖Aα S (t − s)‖ ‖f (s, x1 (s)) − f (s, x2 (s))‖ ‖u(s)‖Y ds ∫ t Lf Mα qΓ (2 − α) ≤ (t − s)q−1−αq ‖x1 (s) − x2 (s)‖α ds Γ (1 + q(1 − α)) 0 ∫ t  Lf Mα qΓ (2 − α) ≤ (t − s)q−1−αq ‖u(s)‖Y ds ‖x1 − x2 ‖∞ Γ (1 + q(1 − α)) 0   1−mq1 m(q−α q−1) 1+ 1−q1 q1 Lf Mα qΓ (2 − α)  (1 − q1 )t  ≤ T m ‖u‖Ln ‖x1 − x2 ‖∞ . Y Γ (1 + q(1 − α)) 1 − q1 + m(q − α q − 1) ≤

0

Therefore, it can easily been shown that

‖Fx1 − Fx2 ‖∞

    1−mq1 m(q−α q−1)   1+  L M qΓ (2 − α)  1−q1 q1 (1 − q1 )T f α   m n ≤ T ‖ u‖ L ‖x1 − x2 ‖∞ , Y   Γ (1 + q(1 − α)) 1 − q1 + m(q − α q − 1) 

that is, F is continuous.

5936

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

(S2) F is compact. Let B is bounded subset of Cα . Then FB is a bounded subset of Cα . In fact, let x ∈ B, using Lemma 2.9(1) and (4), one can obtain

∫ t (t − s)q−1 ‖S (t − s)f (s, x(s))u(s)‖α ds ‖(Fx)(t )‖α ≤ ‖T (t )x0 ‖α + 0 ∫ t (t − s)q−1 ‖Aα S (t − s)‖ ‖f (s, x(s))u(s)‖ds ≤ M ‖x 0 ‖α + 0 ∫ t Mα qΓ (2 − α) ≤ M ‖x 0 ‖α + (t − s)q−αq−1 φ(s)‖u(s)‖Y ds Γ (1 + q(1 − α)) 0   1−mq1 m(q−α q−1) 1+ 1−q1 Mα qΓ (2 − α)  (1 − q1 )t  ≤ M ‖x 0 ‖α + ‖φ‖ qm ‖u‖LnY . Γ (1 + q(1 − α)) 1 − q1 + m(q − α q − 1) L+1 Thus,



‖Fx‖∞ ≤ M ‖x0 ‖α +

1+

m(q−α q−1) 1−q1

 1−mq1

Mα qΓ (2 − α)  (1 − q1 )T  Γ (1 + q(1 − α)) 1 − q1 + m(q − α q − 1)

‖φ‖

m q

L+1

‖u‖LnY ,

which implies that FB is bounded. Define Π = FB and Π (t ) = {(Fx)(t ) | x ∈ B} for t ∈ J. Clearly, Π (0) = {(Fx)(0) | x ∈ B} = {x0 } is compact, and hence, it is only necessary to consider t > 0. For each h ∈ (0, t ), t ∈ (0, T ], arbitrary δ > 0, define

  Πh,δ (t ) = (Fh,δ x)(t ) | x ∈ B where

∫ ∞ ∫ t −h (Fh,δ x)(t ) = T (hq δ) ξq (θ )T (t q θ − hq δ)x0 dθ + T (hq δ) (t − s)q−1 0  ∫ δ∞  q q × q θ ξq (θ )T ((t − s) θ − h δ)dθ f (s, x(s))u(s)ds δ

∫ = δ

∞

ξq (θ )T (t q θ )x0 dθ + q

t −h

∫ 0

∞

∫ δ

θ (t − s)q−1 ξq (θ )T ((t − s)q θ )f (s, x(s))u(s)dθ ds.

Then the sets {(Fh,δ x)(t ) | x ∈ B} are relatively compact in Xα since the operator T (hq δ), hq δ > 0 is compact in Xα . It comes from the following inequalities

∫  ‖(Fx)(t ) − (Fh,δ x)(t )‖α ≤  

δ

  ξq (θ )T (t θ )x0 dθ   0 α ∫ t ∫ δ    q−1 q  +q θ (t − s) ξq (θ )T ((t − s) θ )f (s, x(s))u(s)dθ ds  0 0 α ∫ ∫  t ∞  + q θ (t − s)q−1 ξq (θ )T ((t − s)q θ )f (s, x(s))u(s)dθ ds  0 δ  ∫ t −h ∫ ∞   q −1 q − θ (t − s) ξq (θ )T ((t − s) θ )f (s, x(s))u(s)dθ ds  0 δ α ∫ δ ≤ ξq (θ )‖T (t q θ )x0 ‖α dθ q

0

δ

∫ t∫ +q 0

∫

θ (t − s)q−1 ξq (θ )‖Aα T ((t − s)q θ )‖‖f (s, x(s))u(s)‖dθ ds

0 t

∫

∞

+q t −h

δ ∫ δ

≤ M ‖x 0 ‖α 0

θ (t − s)q−1 ξq (θ )‖Aα T ((t − s)q θ )‖‖f (s, x(s))u(s)‖dθ ds

ξq (θ )dθ + Mα q

δ

∫ t∫ 0

0

θ (t − s)q−1 ξq (θ )((t − s)q θ )−α φ(s)‖u(s)‖Y dθ ds

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

∫

t

+ Mα q

θ (t − s)q−1 ξq (θ )((t − s)q θ )−α φ(s)‖u(s)‖Y dθ ds

δ

t −h

5937

∞

∫

∫ t∫ δ ∫ δ θ 1−α (t − s)q−qα−1 ξq (θ )φ(s)‖u(s)‖Y dθ ds ξq (θ )dθ + Mα q ≤ M ‖x0 ‖α 0 0 0 ∫ t ∫ ∞ θ 1−α (t − s)q−qα−1 ξq (θ )φ(s)‖u(s)‖Y dθ ds + Mα q δ

t −h

δ

∫

∫

t

q−qα−1

( t − s) φ(s)‖u(s)‖Y ds ξq (θ )dθ + Mα q ≤ M ‖x0 ‖α 0 0 ∫ ∞ ∫ t θ 1−α ξq (θ )dθ (t − s)q−qα−1 φ(s)‖u(s)‖Y ds + Mα q t −h

∫

δ

≤ M ‖x0 ‖α 0

δ

∫ ×

δ

∫

θ 1−α ξq (θ )dθ

0

0

  1−mq1 m(q−α q−1) 1+ 1−q1 ( 1 − q ) t 1    ξq (θ )dθ + Mα q  ‖φ‖ qm ‖u‖LnY  1 − q1 + m(q − α q − 1) L+1 

Γ (2 − α) Γ (1 + q(1 − α))  1−mq1 m(q−α q−1)

θ 1−α ξq (θ )dθ + Mα q

0



1+

(1 − q1 )h   ×  1 − q1 + m(q − α q − 1)



1−q1

‖φ‖

m q

L+1

 ‖u‖LnY 

that

‖(Fx)(t ) − (Fh,δ x)(t )‖α ≤ M ‖x0 ‖α

∫ 0

δ

∫ ×

δ



1+

m(q−α q−1) 1−q1

 1−mq1

(1 − q1 )T   ξq (θ )dθ + Mα q  1 − q1 + m(q − α q − 1) Mα qΓ (2 − α) Γ (1 + q(1 − α))  1−mq1 m(q−α q−1)

 ‖φ‖

m q

L+1

 ‖u‖LnY 

θ 1−α ξq (θ )dθ +

0



1+

(1 − q1 )h   ×  1 − q1 + m(q − α q − 1)



1−q1

‖φ‖

m q L 1

+

 ‖u‖LnY  .

Therefore, Π (t ) = {(Fx)(t ) | x ∈ B} is relatively compact in Xα for all t ∈ (0, T ] and since it is compact at t = 0 we have the relatively compactness in Xα for all t ∈ J. Next, let us prove Π = FB is equicontinuous. For T > h ≥ 0, it is not difficult to obtain Mα qΓ (2 − α) Γ (1 + q(1 − α))  1−mq1 m(q−α q−1)

‖(Fx)(h) − (Fx)(0)‖α ≤ ‖Tα (h) − I ‖α ‖x0 ‖α + 

1+

(1 − q1 )h   ×  1 − q1 + m(q − α q − 1)



1−q1

‖φ‖

m q L 1

+

 ‖u‖LnY  ,

and for 0 < s < t1 < t2 ≤ T ,

‖(Fx)(t1 ) − (Fx)(t2 )‖α ≤ I1 + I2 + I3 + I4 , where Ii , i = 1, 2, 3, 4 is given by (3). From the above discussion, we know that Ii tend to 0 as t2 → t1 . In summary, we have proven that FB is relatively compact, for t ∈ J , Π = {Fx | x ∈ B} is a family of equicontinuous functions. Hence by the Arzela–Ascoli Theorem, F is compact. (S3) F has a fixed point in Cα . According to Leray–Schauder fixed point theory, it is sufficient to show that the set Σ = {x ∈ Cα | x = σ Fx, σ ∈ [0, 1]} is a bounded subset of Cα . In fact, let x ∈ Σ ,

‖x(t )‖α = ‖σ (Fx)(t )‖α ≤ ‖T (t )x0 ‖α +

∫ 0

t

(t − s)q−1 ‖S (t − s)f (s, x(s))u(s)‖α ds

5938

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942 t

∫

(t − s)q−1 ‖Aα S (t − s)‖ ‖f (s, x(s))u(s)‖ds ∫ t af Mα qΓ (2 − α) (t − s)q−αq−1 φ(s)‖u(s)‖Y ds ≤ M ‖x 0 ‖α + Γ (1 + q(1 − α)) 0    1−mq1 m(q−α q−1) 1+ 1−q1 ( 1 − q ) T af Mα qΓ (2 − α) 1    ×  ≤ M ‖x 0 ‖α + ‖φ‖ qm ‖u‖LnY  , Γ (1 + q(1 − α)) 1 − q1 + m(q − α q − 1) L+1 ≤ M ‖x 0 ‖α +

0

which implies that Σ is a bounded subset of Cα . By Leray–Schauder fixed point theory F has a fixed point Cα . Consequently, system (1) has at least one mild solution x on J. (S4) The mild solution x(·) is unique. Let y(·) be another mild solution of system (1) with the initial value y0 . From

‖x(t ) − y(t )‖α ≤ ‖T (t )(x0 − y0 )‖α +

t

∫

(t − s)q−1 ‖S (t − s)(f (s, x(s)) − f (s, y(s)))u(s)‖α ds, 0

we can deduce that

‖x(t ) − y(t )‖α ≤ M ‖x0 − y0 ‖α +

Lf Mα qΓ (2 − α)

Γ (1 + q(1 − α))

t

∫

(t − s)q−αq−1 ‖u(s)‖Y ‖x(s) − y(s)‖α ds. 0

In order to use the results of Lemma 2.12, we define αˆ = γˆ = κ = 1, βˆ = q(1 − α). Case 1: if

 1  1 − < q(1 − α) < 1, m

1   < q(1 − α) < 1,

for some 2 ≤ m < ∞,

for some 1 ≤ m < 2;

2

then q(1 − α) ∈ ( 12 , 1) and 1 ≥ 32 − q(1 − α). By Lemma 2.12(1), one can obtain

‖x(t ) − y(t )‖α ≤ 2

q(1−α)

  ∫ t 1 1−q+α ds M ‖x0 − y0 ‖α exp (1 − q + α q)B1 (t ) ‖u(s)‖Y

(4)

0

where B1 (t ) =



[

2B 1,

2q(1 − α) − 1

]

q(1−α) 1−q(1−α)

Lf Mα qΓ (2 − α)

[

q(1 − α)

]

1 1−q(1−α)

t

Γ (1 + q(1 − α))

2q(1−α)−1 1−q(1−α)

,

and t

∫

1 1−q+α

‖u(s)‖Y

ds ≤ T

1 m

[ ] 1−q1+α n 1−q+α ‖ u‖ Y .

0

Case 2: if

 1 1  1 − < q(1 − α) < , m

 0 < q(1 − α) <

1 2

2

,

for some 1 ≤ m < 2,

for some 2 ≤ m < ∞;

then q(1 − α) ∈ (0, 12 ) and 1 ≥

1−2[q(1−α)]2 . 1−[q(1−α)]2

By Lemma 2.12(2), one can obtain 1+3q(1−α)

‖x(t ) − y(t )‖α ≤ 2 1+4q(1−α) M ‖x0 − y0 ‖α exp



q(1 − α) 1 + 4q(1 − α)

B2 (t )

1+4q(1−α) q(1−α)

t

∫

‖u(s)‖Y



ds ,

(5)

0

where B2 (t ) =



[

2B 1,

4[q(1 − α)]2 1 + 3q(1 − α)

(1−α) [ ] 1+q3q (1−α)

Lf Mα qΓ (2 − α)

Γ (1 + q(1 − α))

] 1+4q(1−α) q(1−α)

×t

q(1−α)(1+4q(1−α))(q(1−α)−1)+(1+4q(1−α))−q(1−α) q(1−α)

,

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

5939

and 1+4q(1−α) q(1−α)

t

∫

‖u(s)‖Y

1



ds ≤ T m ‖u‖Ln¯



1 +4 q(1−α)

Y

0 n q(1−α)

where n¯ = + 4n. Thus, there exists a constant M ∗ > 0 such that

‖x(t ) − y(t )‖α ≤ 2MM ∗ ‖x0 − y0 ‖α , which implies the uniqueness of the solution.



3.2. Existence of optimal controls We first recall the following definition of approximately lower semicontinuous. Definition 3.2. The mapping g : J × Xα × Y → R¯ is called approximately lower semicontinuous if, for all ε > 0, there exists Jε ⊂ J such that λ(J \ Jε ) < ε and g |Jε ×Xα ×Y is lower semicontinuous. For the cost integrand g (·, ·, ·), we will make the following generally mild conditions. [Hg]: g : J × Xα × Y → R¯ is a mapping such that: (i) g : J × Xα × Y → R¯ + is approximately lower semicontinuous; (ii) for all t ∈ J , (x, u) → g (t , x, u) is lower semicontinuous from Xα × Yw into R¯ + , where Yw denotes Y with the weak topology; (iii) for all (t , x) ∈ J × Xα , u → g (t , x, u) is convex. We consider the optimal control problem (P): Find (x0 , u0 ) ∈ Cα × Uad = V such that

J ( x 0 , u0 ) =

inf

(x,u)∈V

J (x, u),

where x(·) denotes the mild solution of system (1) corresponding to the control u ∈ Uad . In order to obtain the existence of optimal controls we need the following important lemma. w

Lemma 3.3. Assume that uϑ (·) −→ u(·) in LnY and, for ϑ ≥ 1, xϑ (·) is the solution of system (1) corresponding to uϑ (·). Then, there exists a subsequence {xϑk (·)}k≥1 of {xϑ (·)}ϑ≥1 which converges pointwise to x(·), and x(·) is the solution of the system (1) corresponding to the control u(·). Proof. For every ϑ ≥ 1 and very t ∈ J, we have

∫ t    q −1  ‖xϑ (t ) − x(t )‖α =  ( t − s ) S ( t − s )[ f s , x ( s )) u ( s ) − f ( s , x ( s )) u ( s )] ds ( ϑ ϑ   0 α ∫ t    q −1  ≤  (t − s) S (t − s)[f (s, xϑ (s)) − f (s, x(s))]uϑ (s)ds 0 α ∫ t    q −1  +  (t − s) S (t − s)f (s, x(s))[uϑ (s) − u(s)]ds . α

0

For the moment, let us concentrate on the second summand. From the Hahn–Banach theorem, we know that we can find a x∗ϑ ∈ B∗1 (B∗1 is the dual unit ball) such that

∫ t   ∫ t     q −1 ∗    (t − s)q−1 S (t − s)f (s, x(s))[uϑ (s) − u(s)]ds , ( t − s ) S ( t − s ) f ( s , x ( s ))[ u ( s ) − u ( s )] ds , x = ϑ ϑ     0

α

0

which implies that

∫ t  ∫        t q −1 ∗ ∗ ∗ q −1     . ( t − s ) [ u ( s ) − u ( s )], f ( s , x ( s )) S ( t − s ) x ds = ( t − s ) S ( t − s ) f ( s , x ( s ))[ u ( s ) − u ( s )] ds ϑ ϑ ϑ     0

0

α

By Lemma 2.11(9), S ∗ (t − s) is compact for t > s, f ∗ (s, x(s))S ∗ (t − s) is compact for t > s. Also, by Alaoglu’s theorem, we know that B∗1 is weakly compact. So by passing to a subsequence if necessary, we may assume that w x∗ϑ −→ x∗ ∈ B∗1 .

Hence, s f ∗ (s, x(s))S ∗ (t − s)x∗ϑ −→ z ∗ (t , s) ∈ LnY ∗ ([0, t ]),

5940

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

which implies s f ∗ (·, x(·))S ∗ (t − ·)x∗ϑ −→ z ∗ (t , ·)

in LnY ∗ ([0, t ]).

Recalling that ∗ n [ Lm Y ] = LY ∗

and that L1Y ∗ ⊆ [LnY ]∗ ,

m = ∞, n = 1,

we arrive that

 ∫ t    ((t − s)q−1 [uϑ (s) − u(s)], f ∗ (s, x(s))S ∗ (t − s)x∗ )ds → 0, ϑ   0

which implies that

∫ t     (t − s)q−1 S (t − s)f (s, x(s))[uϑ (s) − u(s)]ds → 0,   α

0

as ϑ → ∞.

Then, by setting t

∫ 

Rϑ (t ) =  

0

  (t − s)q−1 S (t − s)f (s, x(s))[uϑ (s) − u(s)]ds  → 0, α

as ϑ → ∞,

we have

∫ t    q−1  ‖xϑ (t ) − x(t )‖α ≤ Rϑ (t ) +  (t − s) S (t − s)[f (s, xϑ (s)) − f (s, x(s))]uϑ (s)ds  0 α ∫ t Lf Mα qΓ (2 − α) ≤ Rϑ (t ) + (t − s)q−1−αq ‖uϑ (s)‖Y ‖xϑ (s) − x(s)‖α ds. Γ (1 + q(1 − α)) 0 Invoking Lemma 2.12 again, there exists a constant M ∗ > 0 such that

‖x(t ) − y(t )‖α ≤ 2M ∗ Rϑ (t ), which is yield to s

xϑ (t ) −→ x(t ),

as ϑ → ∞, for all t ∈ J . 

Using Theorem 3.1 and Lemma 3.3, we can prove the existence of optimal controls for problem (P). Theorem 3.4. In addition to the assumptions of Theorem 3.1 and Lemma 3.3, suppose the condition [Hg] holds. Then the problem (P) admits at least one optimal pair (x0 , u0 ) ∈ V such that

J (x0 , u0 ) =

inf

(x,u)∈V

J (x, u).

Proof. By Theorem 3.1, we know that system (1) has an unique mild solutions x given by x(t ) = T (t )x0 +

t

∫

(t − s)q−1 S (t − s)f (s, x(s))u(s)ds,

t ∈ J.

0

Let {(xϑ , uϑ )}ϑ≥1 be a minimizing sequence in V , i.e.,

J (xϑ , uϑ ) → ϵ =

inf

(x,u)∈V

J (x, u).

Note that

{uϑ (·)}ϑ≥1 ⊆ SUm = {w(·) ∈ Lm Y | w(t ) ∈ U (t ), a.e.}. From Theorem 4.2 of [40], we know that SUm is w -compact in Lm Y and, by Lemma 2.15, is w -sequentially compact. So passing to a subsequence if necessary, we may assume that w

uϑ (·) −→ u0 (·) ∈ SUm .

J. Wang, Y. Zhou / Nonlinear Analysis 74 (2011) 5929–5942

5941

Let x0 (·) be the solution of system (1) corresponding to u0 (·). Then, from Lemma 3.3, we know that, by passing to a further subsequence if necessary, we may assume that, for all t ∈ J, s

xϑ (·) −→ x0 (·). So, we can apply Theorem 4 of [34] and get that limϑ→∞ J (xϑ , uϑ ) ≥ J (x0 , u0 ), which deduce that

ϵ ≥ J (x0 , u0 ) ≥ ϵ. Thus, we have

J (x0 , u0 ) = ϵ. This implies that (x0 , u0 ) is the desired optimal pair.



4. An example We are going to apply these results to a control system governed by a parabolic, fractional nonlinear integrodifferential equation. The system under consideration is the following:

 q ∫ z ∂ x ( t , y ) − 1 x ( t , y ) = h(t , x(t , y)) · u(y)dy, q 0 x∂(t0, y) = χ (z ),

y ∈ Ω , t ∈ [0, T ], q =

9 13

,

(6)

where Ω = [0, 1], h : J × R × R → R is a Carathéodory function such that |h(t , x)| ≤ Nh and |h(t , x)− h(t , z )| ≤ kˆ |x − z |, χ ∈ Ln ([0, 1]), 1 < n ≤ ∞. Our aim is to find the control u(t , y) that minimizes the performance index J (x, u) =

T

∫

∫ [0,1]

0

g (t , x(t , y), u(t , y))dydt

subject to the system (6). theorem, we can choose α =

z 0

11 24

then X 11 24

1 ,n

W0 ([0, 1]), and Ax = − ∂∂ y2x for x ∈ D(A). By Sobolev embedded ↩→ C 1 ([0, 1]). Define x(t )(y) = x(t , y), f : J × Xα → Lb (X ) by f (t , x)u(z ) =

Define X = Y = Ln ([0, 1]), D(A) = W 2,n ([0, 1])



2

h(t , x(y)) · u(y)dy. Repeating the same process in the example of [32], after some calculation, one can verify that

  = q 1 , 0 < q1 ≤ min{1 − 131 , 1 − 24×113 , 1 − 5m f (t , x) ∈ Lb (X ), i.e., ‖f (t , x)‖Lb (X ) ≤ Mf ∈ Lm }, for some + where m 24 1 1 1 ≤ m, m1 < ∞, 1 < n, n1 ≤ ∞ and m + 1n = 1, m11 + n11 = 1, and for all x ∈ Xα , t → f (t , x) is measurable from J into Lb (X ). Also, for all x, v ∈ Xα , then ‖f (t , x) − f (t , v)‖Lb (X ) ≤ Lf ‖x − v‖α . The controls are functions u : (U x)[0, 1] → R, such that u(·, ·) ∈ Ln (U x[0, 1]). Then t → u(t , ·) going from J into Y is measurable. We restrict the admissible controls to be all u(·, ·) ∈ Ln (U x)[0, 1] such that ‖u(t , ·)‖Y ≤ φ(t ), a.e., where φ(·) ∈ Ln+ (J ). Set U (t ) = {u ∈ Y | ‖u‖Y ≤ φ(t )}, we can check easily that U (·) satisfies the hypotheses in Section 3. Now, we write system (6) in the form of an evolution equation. So we have mn n

Dq x(t ) = −Ax(t ) + f (t , x(t ))u(t ), x(0) = χ ,

C

4m n

99m n

t ∈ J , q ∈ (0, 1),

(7)

with the cost functional

J ( x , u) =

T

∫

g (t , x(t ), u(t ))dt , 0

where g : J × Xα × Y → R¯ + is defined as before. It is easy to see that we can choose some 1 ≤ m < 2 such that 1 1 9 11 1− < q(1 − α) < for q = ,α= . m 2 13 24 Then it satisfies all the assumptions given in Theorem 3.4. Therefore, the system (6) has at least one optimal pair. References [1] A.A. Kilbas, Hari M. Srivastava, J. Juan Trujillo, Theory and applications of fractional differential equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [2] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009. [3] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

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