Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups

Lemma 2.1

If h : J × ℍ → L ( K ,   ℍ ) is continuous and x ∈ C ( J , L 2 ( Ω , ℍ ) ) , then

Definition 2.1

[10] The Riemann-Liouville fractional integral of order α > 0 of a function y : ( 0 , + ∞ ) → ℝ is given by

provided the right side is pointwise defined on ( 0 , + ∞ ) .

Definition 2.2

[10] The Riemann-Liouville fractional derivative of order α > 0 of a function y : [ 0 , + ∞ ) → ℝ is given by

where n = [ α ] + 1 , provided that the right side is pointwise defined on ( 0 , + ∞ ) .

Definition 2.3

[10] The Caputo fractional derivative of order α > 0 of a function y : [ 0 , + ∞ ) → ℝ is given by

where n = [ α ] + 1 , provided that the right side is pointwise defined on ( 0 , + ∞ ) .

Remark 2.1

1. In the case y ( t ) ∈ C n [ 0 , + ∞ ) , then

1. If y ( t ) is an abstract function with values in E, then the integrals which appear in Definitions 2.1, 2.2 and 2.3 are taken in Bochner’s sense.

1. The Caputo derivative of a constant is equal to zero.

Lemma 2.2

The operators T α ( t ) ( t ≥ 0 ) and S α ( t ) ( t ≥ 0 ) satisfy the following properties:

1. For any fixed t ≥ 0 , T α ( t ) and S α ( t ) are linear and bounded operators in ℍ , i.e., for any x ∈ ℍ ,

   (2.3) ∥ T α ( t ) x ∥ ≤ M ∥ x ∥ ,    ∥ S α ( t ) x ∥ ≤ M Γ ( α ) ∥ x ∥ .

2. For every x ∈ ℍ , t → T α ( t ) x and t → S α ( t ) x are continuous functions from [ 0 , ∞ ) into ℍ .

3. The operators T α ( t ) and S α ( t ) are strongly continuous.

4. If semigroup T ( t ) is an equicontinuous semigroup, T α ( t ) and S α ( t ) are also equicontinuous in ℍ for t > 0 .

Now, we recall some properties of the measure of noncompactness, which will be used later.

Let μ ( ⋅ ) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness [39]. Let X be a Banach space, for any D ⊂ C ( J , X ) and t ∈ J , let D ( t ) = { x ( t ) : x ∈ D } ⊂ X . If D is bounded in C ( J , X ) , then D ( t ) is bounded in X, and μ ( D ( t ) ) ≤ μ ( D ) .

Lemma 2.3

[40] Let X be a Banach space and D ⊂ C ( J , X ) be bounded and equicontinuous, where J is a finite closed interval in ℝ . Then, μ ( D ( t ) ) is continuous on J, and

Lemma 2.4

[41] Let D = { u n } ⊂ C ( J , X ) be a bounded and countable set. Then, μ ( D ( t ) ) is the Lebesgue integral on J, and

In this article, we adopt the following definition of the mild solution of (1.1) based on [27,36].

Definition 2.4

For any given u ∈ L ℱ 2 ( J ,   U ) , a stochastic process x ∈ ℋ is said to be a mild solution of (1.1) on J if x satisfies

1. x ( t ) is measurable and adapted to ℱ t ;

2. x ( t ) satisfies the following integral equation:

Let x ( b ; u ) be the state value of system (1.1) at terminal time b corresponding to control u. The set ℛ ( b ) = { x ( b ; u ) :   u ∈ L ℱ 2 ( J ,   U ) } is called the reachable set of (1.1) at the terminal time b.

Definition 2.5

[28] (Exact controllability) The fractional stochastic control system (1.1) is called exact controllable on the interval J if ℛ ( b ) = L 2 ( Ω , ℍ ) .

To prove the controllability result, the following hypotheses are necessary throughout the article: (H1) The linear operator Φ : L ℱ 2 ( J ,   U ) → ℍ defined by

satisfies:

1. Φ has an inverse operator Φ − 1 which takes values in L ℱ 2 ( J ,   U ) \ ker Φ , and there exist two constants L B and L Φ such that ∥ B ∥ ≤ L B , ∥ Φ − 1 ∥ ≤ L Φ ;

2. There exist a constant q 0 ∈ ( 0 ,   q ) and a function k Φ ∈ L 1 / q 0 ( J ,   ℝ + ) such that

for any countable subset D ⊂ ℍ .

(H2) The function f : J × ℍ → ℍ satisfies:

1. For any t ∈ J , f ( t ,   ⋅ ) is continuous, and f ( ⋅ ,   x ) is strongly measurable for all x ∈ ℍ ;

2. For some r > 0 , there exists a function ξ f ∈ L 1 ( J ,   ℝ + ) such that

   sup ∥ x ∥ 2 ≤ r E ∥ f ( t ,   x ) ∥ 2 ≤ ξ f ( t ) , t ∈ J ,

   where ξ f ( t ) satisfies

   lim inf r → + ∞ ∥ ξ f ∥ L 1 r ≜ Λ 1 < ∞ ;

3. For any bounded and countable set D ⊂ ℍ , there exist a constant q 1 ∈ ( 0 ,   α ) and a function k f ∈ L 1 / q 1 ( J ,   ℝ + ) such that

(H3) The function σ : J × ℍ → L 2 0 satisfies the following conditions:

1. For t ∈ J , σ ( t ,   ⋅ ) is continuous, and σ ( ⋅ ,   x ) is strongly measurable for all x ∈ ℍ ;

2. For some r > 0 , there exist a constant q 2 ∈ ( 0 ,  2 α − 1 ) and a function ξ σ ∈ L 1 / q 2 ( J ,   ℝ + ) such that

   sup ∥ x ∥ 2 ≤ r E ∥ σ ( t ,   x ) ∥ L 2 0 2 ≤ ξ σ ( t ) , t ∈ J ,

   where ξ σ ( t ) satisfies

   lim inf r → + ∞ ∥ ξ σ ∥ L 1 / q 2 r ≜ Λ 2 < ∞ ;

3. For any bounded and countable set D ⊂ ℍ , there exist a constant q 3 ∈ ( 0 , 2 α − 1 2 ) and a function k σ ∈ L 1 / q 3 ( J ,   ℝ + ) such that

(H4) g : ℋ → ℍ is continuous and there exists a constant K > 0 such that for any r > 0

For any x 1 ∈ L 2 ( Ω , ℍ ) , we introduce a control u ( t ) ≔ u ( t ; x ) by

Next, we estimate some properties of control u ( t ) defined above.

Lemma 2.5

Suppose that assumptions (H1)–(H4) are satisfied, then for any x ∈ B r , the following conclusions hold:

1. E ∥ u ( t ; x ) ∥ 2 ≤ L u ;

2. u ( t ; x ) is continuous in B r ,

Proof

For t ∈ J and x ∈ B r , by the Hölder inequality, conditions (H1)(i), (H2)(i)(ii), (H3)(i)(ii), (H4), (2.3) and Lemma 2.1, we get that

Next, we prove (ii). Let x n → x in B r , then we have

Moreover, for all t ∈ J , using the Hölder inequality and Lebesgue dominated convergence theorem, we can get

On the other hand, from Lemma 2.1, the Hölder inequality and Lebesgue dominated convergence theorem, we obtain

According to the inequality obtained above, we obtain the following relation:

Therefore, u ( t ; x ) is continuous in B r . This completes the proof of Lemma 2.5.□

Lemma 2.6

[42] Let Ω be a closed convex subset of a Banach space X and θ ∈ Ω . Assume that P :   Ω → Ω is a continuous map, which satisfies Mönch’s condition, i.e., for D ⊂ Ω is countable and D ⊂ c o ¯ ( { θ } ∪ P ( D ) ) ⇒ D is compact. Then, P has at least one fixed point in X.

Theorem 3.1

Assume that − A generates an equicontinuous C 0 -semigroup T ( t ) ( t ≥ 0 ) of uniformly bounded operators in Hilbert space ℍ . If assumptions (H1)–(H4) are satisfied, then fractional stochastic control systems with nonlocal conditions ( 1.1 ) are exact controllability on J provided that

and

where

M ¯ = M K 1 / 2 + 2 M b 1 − q 1 ∥ k f ∥ L 1 / q 1 Γ ( α ) ( 1 − q 1 α − q 1 ) 1 − q 1 + M ∥ k σ ∥ L 1 / q 3 ( 2 T r ( Q ) b 2 α − 1 − 2 q 3 ) 1 / 2 Γ ( α ) ( 1 − 2 q 3 2 α − 1 − 2 q 3 ) 1 − 2 q 3 / 2 .

Proof

To begin with, we define an operator F :   ℋ → ℋ as follows:

It is easy to see that the mild solution of control system (1.1) is equivalent to the fixed point of the operator F defined by (3.3). Next, we complete the proof by four steps.

Step 1. We show that there exists a positive constant r such that F ( B r ) ⊂ B r . If this is not true, then for any r > 0 , there would exist x r ∈ B r and t r ∈ J such that E ∥ ( F x r ) ( t r ) ∥ 2 > r . From Lemma 2.5, we obtain

Dividing both sides by r and taking the lower limit as r → + ∞ , we obtain

which contradicts (3.1). Hence, for some positive integer r > 0 , F ( B r ) ⊂ B r .

Step 2. we prove that F : B r → B r is a continuous operator. Let x n ,   x ∈ B r and x n → x ( n → ∞ ) , then for any t ∈ J , we have

For t ∈ J , (2.4), (2.5) and the Lebesgue dominated convergence theorem then give

Therefore, ∥ ( F x n ) ( t ) − ( F x ) ( t ) ∥ ℋ → 0 ( n → ∞ ) , which means that F is continuous in B r .

Step 3. We prove that F ( B r ) is an equicontinuous family of functions on [ 0 , b ] . For any x ∈ B r and 0 ≤ t 1 < t 2 ≤ b , we get that

In order to prove that E ∥ ( F x ) ( t 2 ) − ( F x ) ( t 1 ) ∥ 2 → 0 ( t 2 − t 1 → 0 ) , we only need to check I i → 0 independently of x ∈ B r when t 2 − t 1 → 0 for i = 1 , 2 , … , 7 .