Control Systems Described by a Class of Fractional Semilinear Evolution Equations and Their Relaxation Property

and Applied Analysis 3 Sukavanam 21 , Sakthivel et al. 22 . Wang and Zhou in 23 were concerned with the optimal control settings. 2. Preliminaries and Assumptions Let J 0, b be a closed interval of the real line with the Lebesgue measure μ and the σalgebra Σ of μ measurable sets. The norm of the space X or Y will be denoted by ‖ · ‖X or ‖·‖Y . For any Banach space V the symbolω−V stands for V equippedwith the weak σ V, V ∗ topology. The same notation will be used for subsets of V . In all other cases we assume that V and its subsets are equipped with the strong normed topology. We first recall the following known definitions from fractional differential theory. For more details, please see 11, 12 . Definition 2.1. The fractional integral of order α with the lower limit zero for a function f is defined as


Introduction
Let J 0, b and 0 < α < 1.We consider the following control system described by a class of fractional semilinear evolution equations of the form: Here C D α t is the Caputo fractional derivative of order α, b > 0 is a finite real number, A is the infinitesimal generator of a strongly continuous semigroup {T t , t ≥ 0} in a separable reflexive Banach space X, g : J × X → L Y, X L Y, X is the space of continuous linear operators from Y into X , h : J × X → X is a nonlinear function, and U : J × X → 2 Y \ {∅} is a multivalued mapping with closed values that is not necessarily convex.The space Y is a separable, reflexive Banach space modeling the control space.
We denote by C J, X the space of all continuous functions from J into X with the supremum norm given by x C sup t∈J x t X for x ∈ C J, X .Let B X ⊆ X be the open unit ball centered at zero.Consider the multivalued map here co stands for the closed convex hull of a set.The map 1.4 is usually called the convex upper semicontinuous regularization of U t, x .Along with the constraint 1.2 on the control we also consider the constraint u t ∈ V t, x t a.e. on J 1.5 on the control.Note that usually we have coU t, x ⊆ V t, x .
Definition 1.1.A solution of the control system 1.1 , 1.2 is defined to be a pair x • , u • consisting of a trajectory x ∈ C J, X and a control u ∈ L 1 J, Y satisfying 1.1 and the inclusion 1.2 a.e.
A solution of the control system 1.1 , 1.5 is defined similarly.We denote by R U , Tr U R V , Tr V the sets of all solutions, all admissible trajectories of the control system 1.1 and 1.2 the control system 1.1 and 1.5 .
Relaxation property 1 has important ramifications in control theory.There are many papers dealing with the verification of the relaxation property for various classes of control systems.For example, we refer to 2-5 for nonlinear evolution inclusions or equations, 6, 7 for control problems of subdifferential type and the references therein.In this paper, we investigate this property for control systems described by fractional semilinear evolution equations.We will prove that Tr V is a compact set in C J, X and

Preliminaries and Assumptions
Let J 0, b be a closed interval of the real line with the Lebesgue measure μ and the σalgebra Σ of μ measurable sets.The norm of the space X or Y will be denoted by • X or • Y .For any Banach space V the symbol ω−V stands for V equipped with the weak σ V, V * topology.The same notation will be used for subsets of V .In all other cases we assume that V and its subsets are equipped with the strong normed topology.
We first recall the following known definitions from fractional differential theory.For more details, please see 11, 12 .
Definition 2.1.The fractional integral of order α with the lower limit zero for a function f is defined as provided the right hand side is point-wise defined on 0, ∞ , where Γ • is the gamma function.
Definition 2.2.The Riemann-Liouville derivative of order α with the lower limit zero for a function f is defined as

2.2
Definition 2.3.The Caputo derivative of order α with the lower limit zero for a function f is defined as If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner's sense.
We now proceed to some basic definitions and results from multivalued analysis.For more details on multivalued analysis, see the books 24, 25 .
We use the following notations: P f Y is the set of all nonempty closed subsets of Y , P fb Y is the set of all nonempty, closed and bounded subsets of Y , and P fc Y is the set of all nonempty, closed, and convex subsets of Y .
On P bf Y , we have a metric known as the "Hausdorff metric" and defined by We say that a multivalued map F : and B X is the σ-algebra of the Borel sets in X.
Suppose V, Z are two Hausdorff topological spaces and F : V → 2 Z \ {∅}.We say that F is lower semicontinuous in the sense of Vietoris l.s.c. for short at a point F is said to be upper semicontinuous in the sense of Vietoris u.s.c. for short at a point x 0 ∈ V if for any open set W ⊆ Z, F x 0 ⊆ W, there is a neighborhood O x 0 of x 0 such that F x ⊆ W for all x ∈ O x 0 .For the properties of l.s.c and u.s.c, see the book 24 .
Besides the standard norm on L q J, Y here Y is a separable, reflexive Banach space , 1 < q < ∞, we also consider the so called weak norm: The space L q J, Y furnished with this norm will be denoted by L q ω J, Y .The following result establishes a relation between convergence in ω − L q J, Y and convergence in L q ω J, Y .Lemma 2.4 see 5 .If a sequence {u n } n≥1 ⊆ L q J, Y is bounded and converges to u in L q ω J, Y , then it converges to u in ω − L q J, Y .
We assume the following assumptions on the data of our problems in the whole paper.

H A :
The operator A generates a strongly continuous semigroup T t , t ≥ 0 in X, and there exists a constant M A ≥ 1 such that sup t∈ 0,∞ T t ≤ M A .For any t > 0, T t is compact.
Remark 2.5.Let us take X L 2 0, π and define the operator A by Aω ω with the domain D A {ω ∈ X: ω, ω are absolutely continuous, ω ∈ X, and ω 0 ω π 0}.Then Aω − ∞ n 1 n 2 ω, e n e n , ω ∈ D A , where e n z 2/π 1/2 sin nz, 0 ≤ z ≤ π, n 1, 2, . ... Clearly A generates a compact semigroup {T t : t > 0} in X and it is given by T t ω ∞ n 1 e −n 2 t ω, e n e n , ω ∈ X.In such a case, it is easy to see that H A holds 22 .H g : The operator g : J × X → L Y, X is such that 1 the map t → g t, x u is measurable for all x ∈ X and u ∈ Y ; 2 for a.e.t ∈ J, the map x → g * t, x h is continuous for all h ∈ X * , where g * t, x is the adjoint operator to g t, x ; 3 for a.e.t ∈ J and x ∈ X g t, x L Y,X ≤ d, with d > 0.

2.6
H h : The function h : J × X → X of Carathéodory type satisfies: there exists a constant 0 < β < α such that for a.e.t ∈ J and all x ∈ X, h t, x X ≤ a 1 t c 1 x X , where a 1 ∈ L 1/β J, R and c 1 > 0.
H U : The multivalued map U : J × X → P f Y is such that: 1 t, x → U t, x is Σ ⊗ B X measurable; 2 for a.e.t ∈ J, the map x → U t, x is l.s.c.; 3 for a.e.t ∈ J and all x ∈ X, U t, x Y sup{ v Y : v ∈ U t, x } ≤ a 2 t c 2 x X , where a 2 ∈ L 1/β J, R and c 2 > 0.
H M : For any M > 0, there exists a function l M ∈ L ∞ J, R such that for a.e.t ∈ J and for any We note that the condition similar to H M was also assumed in 6, 7 .
From the Definitions 2.1 and 2.2 and the results obtained in 18, 19 , Definition 1.1 can be rewritten in the following form.
x 0 and there exists u ∈ L 1 J, Y such that u t ∈ U t, x t a.e. on t ∈ J and A similar definition can be introduced for the system 1.1 and 1.5 .Here and ξ α is a probability density function defined on 0, ∞ 26 , that is Lemma 2.7 see 18, 19 .Let H(A) hold, then the operators P α and Q α have the following properties.
1 For any fixed t ≥ 0, P α t and Q α t are linear and bounded operators, that is, for any x ∈ X, 2 {P α t , t ≥ 0} and {Q α t , t ≥ 0} are strongly continuous.
3 For every t > 0, P α t and Q α t are compact operators.
Lemma 2.8 see 27, Theorem 3.1 .Let x t be continuous and nonnegative on 0, b .If where 0 ≤ γ < 1, ψ t is a non-negative, monotonic increasing continuous function on 0, b and M is a positive constant, then where E 1−γ z is the Mittag-Leffler function defined for all γ < 1 by 2.15

Auxiliary Results
In this section, we will give some auxiliary results needed in the proof of the main results.We begin with the a priori estimation of the trajectory of the control systems.
Lemma 3.1.For any admissible trajectory x of the control system 1.1 and 1.5 , that is, x ∈ Tr V , there is a constant L such that x C ≤ L.

3.1
Proof.Let any x ∈ Tr V , from Definition 2.6, we know that there exists a u with u t ∈ V t, x t a.e. and Then by Lemma 2.7, we get

3.3
From H h and H ölder inequality, we have

3.4
Similarly, by H g 3 and H U 3 , x s X ds.

3.6
From the above inequality, using the well-known singular version of the Gronwall inequality see Lemma 2.8 , we can deduce that there exists a constant This map is Lipschitz continuous.We define U 1 t, x U t, pr L x .Obviously, U 1 satisfies H U 1 and H U 2 .Moreover, by the properties of pr L , we have for a.e.t ∈ J, all x ∈ X and all u ∈ U 1 t, x the estimates Hence, Lemma 3.1 is still valid with U t, x substituted by U 1 t, x .Therefore, we assume without any loss of generality that for a.e.t ∈ J, and all x ∈ X Similarly, we can assume that for a.e.t ∈ J and all

3.12
It follows from assumption H g that for any h ∈ X * , the function h, g t, x u g * t, x h, u is measurable in t and continuous in x, u almost everywhere.Hence, for any measurable functions x : J → X and u : J → Y , the function t → g t, x t u t is scalarly measurable 28 .The separability of the space X implies that the function t → g t, x t u t is measurable.Therefore, according to H g and H h , for any x ∈ L 1/β J, X and u ∈ L 1/β J, Y , the function t → g t, x t u t h t, x t is an element of the space L 1/β J, X .Hence we can consider the operator A : L 1/β J, X × L 1/β J, Y → L 1/β J, X defined by the rule A x, u t g t, x t u t h t, x t .

3.13
Lemma 3.2.The operator x, u → A x, u is sequentially continuous as an operator from

3.15
Using the preceding four formulae and Lebesgue's theorem of dominated convergence, we obtain 3.17 Then it follows from 3.16 that x t h t , u t dt.

3.18
Since h t , g t, x t u t g * t, x t h t , u t and h ∈ L 1/ 1−β J, X * is arbitrary, by 3.17 and 3.18 , we deduce that

3.19
It follows from 3.9 , 3.10 , and 3.12 that {A x n , u n } n≥1 is a subset of X ϕ which is a metrizable compact set in ω − L 1/β J, X .If the sequence A x n , u n , n ≥ 1, does not converge to A x, u in ω − L 1/β J, X , then it has a subsequence A x n i , u n i , i ≥ 1, such that none of its subsequences converges to A x, u in ω − L 1/β J, X .Applying the above arguments to this very subsequence x n i , u n i , i ≥ 1, we obtain a contradiction.The lemma is proved.
Lemma 3.3.For a.e.t ∈ J, the multivalued map x → V t, x defined by 1.4 from X to 2 Y is u.s.c. with convex closed values.
Proof.From the definition of V t, x , it is clear that V t, x is closed convex valued.Since δ → U δ t, x is increasing in the sense of inclusion , and letting

3.21
Let x 0 ∈ X and W be an open set in Y such that V t, x 0 ⊆ W. By 3.21 , we can find an n 0 ≥ 1 such that U 1/n 0 t, x 0 ⊆ W.

3.22
For an arbitrary y ∈ x 0 1/n 0 B X , we can find a δ > 0 such that y δB X ⊆ x 0 1/n 0 B X .Therefore we obtain Then it is clear that V t, y ⊆ W, for all y ∈ x 0 1/n 0 B X .This means that x → V t, x is u.s.c.
Let C X {z k } k≥1 be a dense countable subset of the ball B X .We put

3.24
Lemma 3.4.For a.e.t ∈ J, let U 1/n t, x be defined by 3.20 , then we have where the closure is taken in Y .
Proof.We recall that coA coA for any subset A ⊆ Y .Hence it is sufficient to prove that for a.e.t ∈ J,

3.26
That the left hand side of 3.26 is contained in its right hand side is obvious.
x .Therefore 3.26 holds.The lemma is proved.Now we consider the following auxiliary problem: x 0 x 0 .

Abstract and Applied Analysis 11
It is clear that for every f ∈ L 1/β J, X , 3.27 has a unique mild solution S f ∈ C J, X which is given by The following lemma describes a property of the solution map S which is crucial in our investigation.
Lemma 3.5.The solution map S : Proof.Consider the operator H : L 1/β J, X → C J, X defined by

3.29
We know H is linear.From simple calculation, one has that is, the operator H is continuous from L 1/β J, X into C J, X , hence H is also continuous from ω − L 1/β J, X into ω − C J, X .
Let any B ∈ P b L 1/β J, X and suppose that for any f ∈ B, f L 1/β J,X ≤ K K > 0 is a constant .Next we will show that H is completely continuous.a From 3.30 , we know that H f t X is uniformly bounded for any t ∈ J and f ∈ B.
b H is equicontinuous on B. Let 0 ≤ t 1 < t 2 ≤ b and any f ∈ B, we get

3.31
By using analogous arguments as in Lemma 3.1, we find

3.32
For t 1 0, 0 < t 2 ≤ b, it is easy to see that I 3 0.For t 1 > 0 and > 0 be enough small, we have

3.33
Combining the estimations for I 1 , I 2 , and I 3 , and letting t 2 → t 1 and → 0 in I 3 , we obtain that H is equicontinuous.For more details, please see 19 .
c The set Π t {H f t : f ∈ B} is relatively compact in X.Clearly, Π 0 {0} is compact, and hence, it is only necessary to consider t > 0. For each h ∈ 0, t , t ∈ 0, b , f ∈ B and δ > 0 be arbitrary, we define where

3.35
From the compactness of T h α δ h α δ > 0 , we obtain that the set Π h,δ t is relatively compact in X for any h ∈ 0, t and δ > 0.Moreover, we have

3.36
In virtue of 2.11 , the last term of the preceding inequality tends to zero as h → 0 and δ → 0. Therefore, there exist relatively compact sets arbitrarily close to the set Π t , t > 0. Hence the set Π t , t > 0 is also relatively compact in X.Since X ϕ is a convex compact metrizable subset of ω − L 1/β J, X , it suffices to prove the sequential continuity of the map S. Now let {f n } n≥1 ⊆ X ϕ such that

3.37
By the property of the operator H, we have From the definitions of the operators S and H, we have that S f t P α t x 0 H f t .Then due to the arguments above, we have S f n → S f in C J, X .This completes the proof of the lemma.

Existence Results for the Control Systems
In the present section, we are interested in the existence results for the control systems 1.1 , 1.2 and 1.1 , 1.5 .
Let Λ S X ϕ , from Lemma 3.5, we have that Λ is a compact subset of C J, X .It follows from formulae 3.9 , 3.10 , and 3.12 that Proof.By the hypothesis H U 1 , we have that for any measurable function x : J → X, the map t → U t, x t is measurable and has closed values.Therefore it has measurable selectors 29 .So the operator U is well defined and its values are closed decomposable subsets of L 1/β J, Y .We claim that x → U x is l.s.c.Let x * ∈ C J, X , h * ∈ U x * and let {x n } n≥1 ⊆ C J, X be a sequence converging to x * .It follows from Lemma 3.2 in 30 that there exists a sequence h n ∈ U x n such that

4.3
This together with 3.9 implies that h n → h * in L 1/β J, Y .Therefore the map x → U x is l.s.c.By Proposition 2.2 in 31 , there is a continuous function m : Abstract and Applied Analysis 15 Consider the map P : L 1/β J, X → L 1/β J, Y defined by P f m S f .Due to Lemma 3.5 and the continuity of m, the map P is continuous from ω − X ϕ into L 1/β J, Y .Then by Lemma 3.2, we deduce that the map f → A S f , P f is continuous from ω − X ϕ into ω − L 1/β J, X .It follows from 3.9 , 3.10 , 3.12 , and 3.13 that A S f , P f ∈ X ϕ for every f ∈ X ϕ .Therefore, the map  In virtue of 4.8 and 4.9 , and for a.e.t ∈ J, u n t ∈ V t, x n t , n ≥ 1, we obtain that u t ∈ V t, x t a.e.t ∈ J.This means that R V is compact in C J, X × ω − L 1/β J, Y .The proof is complete.

Main Results
In this section, we will prove the relaxation result.But first, we give a lemma which is important in the proof of our relaxation theorem.
Lemma 5.1.For any pair x * • , u * • ∈ R V , there exists a sequence of simple functions y n : J → X and a sequence v n ∈ L 1/β J, Y , n ≥ 1, such that From Lemma 3.4, we have that for a.e.t ∈ J, n ≥ 1 The map t → ∞ k 1 U k 1/n t, x * t is measurable see Propositions 2.3 and 2.6 in 29 and, by 3.9 , is integrally bounded.Therefore, from 5.4 and Theorem 2.2 in 32 , we have that there exists an We know that the map t → U k 1/n t, x * t is measurable and its value are closed, then following Theorem 5.6 in 29 also Proposition 2.2.3 in 24 , there exists a sequence of measurable selectors Therefore we have From 5.5 and 5.7 , according to Lemma 1.3 in 33 also see Proposition 2.3.6 24 , there is a finite measurable partition J 1 , J 2 , . . ., J l n of J such that where χ J i is the characteristic function of the set J i .Now let 5.9 Formula 5.9 implies that y n is a simple function, v n ∈ L

5.11
It is clear that the function t → r n t, x, u is measurable and the function x, u → r n t, x, u is continuous in view of H g and the fact that if x : J → X is a measurable function, x t X is a measurable real-valued function .According to the Theorem 2.4 in 34 , there exists a sequence of nested in the sense of inclusion closed sets For every k ≥ 1 the graph of the map

5.13
Hypothesis H M together with 5.12 implies that U n t, x is nonempty for a.e.t ∈ J and all x ∈ Q.Since the map U t, x is l.s.c. on J k × Q, k ≥ 1, and the graph of the map H n t, x is an open subset of J k × Q × Y , k ≥ 1, then according to Proposition 1.2.47 in 24 , we obtain that the map U n t, x is l.s.c. on J k × Q, k ≥ 1. Hence the map U n t, x U n t, x is l.s.c. on J k × Q, k ≥ 1.Therefore, for every continuous function x : J → Q the map t → U n t, x t is measurable and the map x → U n t, x is l.s.c. on Q for a.e.t ∈ J.
It is clear that U n t, x ⊆ U t, x .Consider the system 1.1 with the constraint U n t, x t on the control.The arguments used in the proof of the Theorem 4.1 enable us to obtain the existence of a solution x n • , u n • ∈ R U and g t, x * t y n t v n t h t, x * t y n t − g t, x n t u n t −h t, x n t X − l L 1 t x * t y n t − x n t X − 1 n ≤ 0. x n t P α t x 0 t 0 t − s α−1 Q α t − s g s, x n s u n s h s, x n s ds.

5.16
Theorem 4.1 and R U ⊆ R V imply that we can assume, possibly up to a subsequence, that the sequence x n • , u n • → x • , u • ∈ R V in C J, X × ω − L 1/β J, Y .Subtracting 5.15 from 5.16 , we have

5.17
Here the linear operator H is defined by 3.29 .In virtue of 5.1 , we have x * y n → x * in L 1/β J, X .Since v n → u * in ω − L 1/β J, Y see 5.3 , from Lemma 3.2 we have that A x * y n , v n → A x * , u * in ω − L 1/β J, X .By the property of the operator H in Lemma 3.5, we obtain D 1 −→ 0 for every t ∈ J.

5.18
Due to 5.14 , we have t − s α−1 Q α t − s x * s y n s − x n s X ds αM A b α nαΓ 1 α .

5.19
Note that x * t X ≤ L, x n t X ≤ L for any n ≥ 1, t ∈ J. Combining 5.18 , 5.19 with 5.17 , let n → ∞, we get

5.20
This together with Lemma 2.8 implies that x * x.Hence we have that the sequence x n ∈ Tr U , n ≥ 1, converges to x * ∈ Tr V in C J, X .From Theorem 4.1, we know that Tr V is compact in C J, X .Now it follows from Tr U ⊆ Tr V and the proof above that the relation 5.10 holds.The proof is complete.

5 . 14 Nowt x 0 t 0 t
by x * • , u * • ∈ R V and x n • , u n • ∈ R U , n ≥ 1, we have x * t P α − s α−1 Q α t − s g s, x * s u * s h s, x * s ds, 5.15 t, x n t h t is a scalarly measurable function from J to Y * .Hence it is measurable.Consider a subsequence x n k , k ≥ 1, of the sequence x n , n ≥ 1, converging to x a.e. in t ∈ J.By H g , H h , and 3.10 , we have g * t, x n k t h t −→ g * t, x t h t a.e.t ∈ J in Y * , g * t, x n k t h t Y * ≤ d h t X * a.e.t ∈ J, Y .Take an arbitrary u ∈ Y and any h ∈ L 1/ 1−β J, X * .H g and the equality h t , g t, x n t u g * t, x n t h t , u 3.14 imply that t → g * Since the map y → U t, y is l.s.c., by the Proposition 1.2.26 in 24 , the function y → d Y h * t , U t, y is u.s.c. for a.e.t ∈ J.It follows from 4.2 that for a.e.t ∈ J X , Schauder's fixed point theorem implies that this map has a fixed point f * ∈ X ϕ , that is, f * A S f * , P f * .Let u * P f * and x * S f * , then we have u * m x * and f * A x * , u * .That means • is a solution of the control system 1.1 and 1.2 .Hence R U is nonempty.It is easy to see * • , u * Y .DenoteHence, to prove that x • , u • ∈ R V , we only need to verify that u t ∈ V t, x t a.e.t ∈ J.Since u n → u in ω − L 1/β J, Y , by Mazur's theorem we have 1/β J, Y and 5.1 , 5.2 hold.By Lemma 2.4, 5.5 and 5.8 , we obtain that 5.3 holds.The lemma is proved.Now we are ready to present our main result.
Theorem 5.2.The set Tr V is compact in C J, X and the following relation holdsTr V Tr U , 5.10where the bar stands for the closure in C J, X .Proof.Let x * • , u * • ∈ R V and {v n } n≥1 , {y n } n≥1 be given as in Lemma 5.1.Put Q {h ∈ X : h X ≤ L}, for fixed n ≥ 1, we consider the function defined by r n t, x, u g t, x * t y n t v n t h t, x * t y n t −g t, x u − h t, x X − l L 1 t x * t y n t − x X .