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Article

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

1
Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City 11837, Egypt
2
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria 21544, Egypt
*
Author to whom correspondence should be addressed.
Fractal Fract. 2022, 6(12), 721; https://doi.org/10.3390/fractalfract6120721
Submission received: 30 October 2022 / Revised: 25 November 2022 / Accepted: 2 December 2022 / Published: 5 December 2022
(This article belongs to the Section General Mathematics, Analysis)

Abstract

:
This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example.

1. Introduction

The controllability of various deterministic and stochastic control systems has been investigated in many works (see [1,2,3,4,5,6,7,8,9]). It should be emphasized that there are many different notions of controllability for fractiona-evolution systems—for example, approximate controllability, complete controllability, null controllability, and so on. However, most work on controllability has focused on deterministic models rather than stochastic models. However, deterministic models often fluctuate due to the presence of environmental noise. It is reasonable and practical to import stochastic effects into investigations with deterministic models. In recent years, fractional stochastic differential equations and fractional stochastic inclusions have attracted the attention of many researchers and have become increasingly popular due to their practical applications in various fields of science and engineering (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]). Moreover, Hilfer proposed a generalized Riemann–Liouville fractional derivative—for brevity, this is called the Hilfer fractional derivative—which includes the Riemann–Liouville fractional derivative and Caputo fractional derivative (see [25,26]). Subsequently, a few authors have studied the controllability of fractional stochastic differential inclusions involving Hilfer fractional derivatives—for example, Yang and Wang [27] studied the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Dineshkumar et al. [28] discussed the approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions. However, no work has been reported in the literature regarding the null controllability of nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. In order to complete this part, in the present work, we analyze the following system:
D 0 + , ϱ U ( ς ) U ( ς ) + γ ( ς , U ( ς ) ) + A X ( ς ) + ϑ ( ς , U ( ς ) ) d ϖ ( ς ) d ς + Ξ ( ς , U ( ς ) ) , ς T = ( 0 , q ] , I 0 + ( 1 ϱ ) ( 1 ) U ( 0 ) + μ ( U ) = U 0 ,
where D 0 + , ϱ is the Hilfer fractional derivative, 0 1 , 1 2 < ϱ < 1 , and ℧ is the infinitesimal generator of a compact semigroup { ( ς ) , ς 0 } in Hilbert space Λ , where sup ς T ( ς ) Π , Π > 1 .
In this paper, there exists a separable Hilbert space with norm · and inner product · , · . Assume that { ϖ ( ς ) } ς 0 is an -valued Wiener process with a finite trace nuclear covariance operator Φ 0 , L ( Ψ , Λ ) is the space of all bounded linear operators from Ψ into Λ , and Clarke’s subdifferential of Ξ ( ς , U ( ς ) ) is Ξ ( ς , U ( ς ) ) (see [21]). U ( · ) takes values in Λ , the function of the control is X ( · ) L 2 ( T , F ) for the Hilbert space of admissible control functions, where F is a Hilbert space, and A is a bounded linear operator from F into Λ . γ : T × Λ 2 Λ is a non-empty, bounded, closed, convex, and multivalued map, ϑ : T × Λ L Φ ( , Λ ) , and μ : C ( T , Λ ) Λ . In the current paper, the space of all Φ -Hilbert–Schmidt operators from Ψ to Λ is L Φ ( Ψ , Λ ) .

2. Preliminaries

Definition 1 
([25,26]). The Hilfer fractional derivative of order 0 1 and 0 < ϱ < 1 is defined as
D 0 + , ϱ U ( ς ) = I 0 + ( 1 ϱ ) d d t I 0 + ( 1 ) ( 1 ϱ ) U ( ς ) ,
where
I ϱ U = 1 Γ ( ϱ ) 0 ς U ( κ ) ( ς κ ) 1 ϱ d κ , ς > 0
and Γ ( · ) is the Gamma function.
Let ( Ω , Y , { Y ς } ς 0 , K ) be a complete probability space and let D : = C ( T , L 2 ( Y , Λ ) ) be a Banach space with norm U D = sup ς T E ς ( 1 ) ( 1 ϱ ) U ( ς ) 2 ) 1 / 2 , where L 2 ( Y , Λ ) = L 2 ( Ω , Y , K , Λ ) . Let L Y 2 ( T , Λ ) be the Hilbert space of all random processes that are Y ς -adapted measurable as defined on T with values in Λ and the norm U L Y 2 ( T , Λ ) = ( 0 q E U ( ς ) Λ 2 ) 1 / 2 < .
Definition 2 
([28]). Let J be a Banach space with the dual spaces J * and G : J R , which is a locally Lipschitz functional on J . Clarke’s generalized directional derivative of G at the point β J in the direction x J is defined by
G 0 ( β ; x ) = lim sup ν 0 + τ β G ( τ + ν x ) G ( τ ) ν .
Clarke’s generalized gradient of G at β J is given by
G ( β ) = { β * G * : G 0 ( β ; x ) β * , x , x J } .
Definition 3 
([26,29]). A Y ς stochastic process U D is a mild solution of (1) if U ( 0 ) = U 0 μ ( U ) Λ and B ( ς ) L Y 2 ( T , Λ ) such that B ( ς ) Ξ ( ς , U ( ς ) ) for ς T and
U ( ς ) = , ϱ ( ς ) [ U 0 μ ( U ) ] + 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + 0 ς P ϱ ( ς κ ) B ( κ ) d κ + 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) , ς T
where
, ϱ ( ς ) = I 0 + ( 1 ϱ ) P ϱ ( ς ) , P ϱ ( ς ) = ς ϱ 1 T ϱ ( ς ) , T ϱ ( ς ) = 0 ϱ θ Ψ ϱ ( θ ) ( ς ϱ θ ) d θ ,
where
Ψ ϱ ( θ ) = n = 1 ( θ ) n 1 ( n 1 ) ! Γ ( 1 n ϱ ) , θ ( 0 , ) .
Lemma 1 
([26]). The operators , ϱ and P ϱ have the following properties:
(i) 
{ P ϱ ( ς ) : ς > 0 } is continuous in the uniform operator topology.
(ii) 
For any fixed ς > 0 , , ϱ ( ς ) and P ϱ ( ς ) are linear and bounded operators, and
P ϱ ( ς ) U Π ς ϱ 1 Γ ( ϱ ) U , , ϱ ( ς ) U Π ς ( 1 ) ( 1 ϱ ) Γ ( ( 1 ϱ ) + ϱ ) U .
(iii) 
{ P ϱ ( ς ) : ς > 0 } and { , ϱ ( ς ) : ς > 0 } are strongly continuous.
Some assumptions are considered to establish the null controllability criteria for the nonlinear system (1):
(A1)
Let E A X ( ς ) 2 Π A E X ( ς ) 2 for all X ( ς ) F on T where Π A > 0 .
(A2)
γ : T × Λ 2 Λ is locally Lipschitz continuous for all ς T , z , U 1 , U 2 Λ , C 2 > 0 such that
E γ ( ς , U 1 ) γ ( ς , U 2 ) 2 C 2 ( E U 1 U 2 2 , E γ ( ς , U ) 2 C 2 ( 1 + E U 2 ) .
(A3)
ϑ : × Λ L Φ ( Ψ , Λ ) is locally Lipschitz continuous for all ς T , U , U 1 , U 2 Λ , and there exist constants C 3 > 0 such that
E ϑ ( ς , U 1 ) ϑ ( ς , U 2 ) Φ 2 C 3 ( E U 1 U 2 2 , E ϑ ( ς , U ) Φ 2 C 3 ( 1 + E U 2 ) .
(A4)
Ξ : T × Λ R such that:
()
Ξ ( · , U ) : T R is measurable U Λ ,
()
Ξ ( ς , · ) : Λ R is locally Lipschitz continuous for ς T ,
()
∃ a function ζ L 1 ( T , R + ) , C 4 > 0 , which satisfies
E Ξ ( ς , U ) 2 = sup { E B ( ς ) 2 : B ( ς ) Ξ ( ς , U ) } ζ ( ς ) + C 4 E U 2 ,
U Λ a.e. ς T and U Λ .
(A5)
μ : C ( T , Λ ) Λ is continuous, for any U , U 1 , U 2 C ( I , Λ ) C 5 > 0 , such that
E μ ( U 1 ) μ ( U 2 ) 2 C 5 E U 1 U 2 2 , E μ ( U ) 2 C 5 ( 1 + E U 2 ) .
Now, we define an operator Θ : L Y 2 ( T , Λ ) 2 L Y 2 ( T , Λ ) as follows: Θ ( U ) = { B L Y 2 ( T , Λ ) : B ( ς ) Ξ ( ς , U ( ς ) ) a.e. ς T for U L Y 2 ( T , Λ ) } .
Lemma 2 
([28]). The set Θ ( U ) has nonempty, convex, and weakly compact values for U L Y 2 ( T , Λ ) and if ( A 4 ) holds.
Lemma 3 
([30]). If ( A 4 ) holds, the operator Y satisfies the following: If U n U in L Y 2 ( T , Λ ) , ξ n ξ is weakly in L Y 2 ( T , Λ ) and ξ n Θ ( U n ) , then ξ Θ ( U ) .

3. Main Result

To prove the null controllability criteria for the nonlinear system (1), we present the linear Hilfer fractional stochastic differential equation as follows:
D 0 + , ϱ w ( ς ) = w ( ς ) + γ ( ς ) + A X ( ς ) + ϑ ( ς ) d ϖ ( ς ) d ς , ς T = ( 0 , q ] , I 0 + ( 1 ϱ ) ( 1 ) w ( 0 ) = w 0 ,
which is associated with the system (1)
Consider
L 0 q X = 0 q P ϱ ( q κ ) A X ( κ ) d κ : L 2 ( T , F ) Λ ,
where L 0 q X has a bounded inverse operator ( L 0 ) 1 with values in L 2 ( T , F ) / k e r ( L 0 q ) , and
N 0 q ( w , γ , ϑ ) = , ϱ ( ς ) w + 0 q P ϱ ( q κ ) γ ( κ ) d κ + 0 q P ϱ ( q κ ) ϑ ( κ ) d ϖ ( κ ) : Λ × L 2 ( T , F ) Λ .
Definition 4 
([31]). The linear system (3) is exactly null controllable on T if α > 0 such that ( L 0 q ) * w 2 α ( N 0 q ) * w 2 w Λ .
Lemma 4 
([32]). Let (3) be exactly null controllable on T ; then, the linear operator ( L 0 ) 1 N 0 q : Λ × L 2 ( T , Λ ) L 2 ( T , F ) is bounded and the control
X ( ς ) = ( L 0 ) 1 , ϱ ( ς ) w 0 + 0 q P ϱ ( q κ ) γ ( κ ) d κ + 0 b P ϱ ( q κ ) ϑ ( κ ) d ϖ ( κ ) ( ς )
transfers the system (3) from w 0 to 0.
Definition 5 
([30]). The system (1) is exact null controllable on T if ∃ a stochastic control X L 2 ( T , F ) such that the solution U ( ς ) of (1) satisfies U ( q ) = 0 .
To establish the null controllability, we add the following assumption:
(A6)
The fractional linear system (3) is exactly null controllable on T .
Definition 6. 
If the assumptions ( A 1 ) ( A 6 ) hold, then the system (1) is exactly null controllable on T provided that
2 = 25 C 5 Π 2 q 2 ( 1 ) ( 1 ϱ ) Γ 2 ( ( 1 ϱ ) + ϱ ) + 25 Π 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) [ ( C 2 + T r ( Φ ) C 3 ) + C 4 q ] × 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) < 1 .
Proof. 
We define W : D 2 D as follows:
W ( U ) = Z D : Z ( ς ) = , ϱ ( ς ) [ U 0 μ ( U ) ] + 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + 0 ς P ϱ ( ς κ ) B ( κ ) d κ + 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) , B Θ ( U )
where
X ( ς ) = ( L 0 ) 1 [ , ϱ ( q ) [ U 0 μ ( U ) ] + 0 q P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 q P ϱ ( ς κ ) B ( κ ) d κ + 0 q P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) ] ( ς ) .
In the following steps, we show that W has a fixed point:
  • Step 1: For each U D , W ( U ) has nonempty, convex, and weakly compact values.
    According to Lemma 2.2, it is easy to see that W ( U ) has nonempty and weakly compact values. Moreover, as Θ ( U ) has convex values, if B 1 , B 2 Θ ( U ) , then δ B 1 + ( 1 δ ) B 2 Θ ( U ) δ ( 0 , 1 ) , which clearly implies that W ( U ) is convex.
  • Step 2: The operator W is bounded on a bounded subset of D .
    Let us consider Δ = { U D : U D 2 } , > 0 . It is obvious to conclude that Δ is a bounded, closed, and convex set of D . We claim that there exists a constant ρ > 0 such that for each χ W ( U ) , U Δ , χ D 2 ρ .
    If χ W ( U ) , then there exists a B Θ ( U ) such that
    χ ( ς ) = , ϱ ( ς ) [ U 0 μ ( U ) ] + 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + 0 ς P ϱ ( ς κ ) B ( κ ) d κ + 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) , ς T .
    From ( A 1 ) ( A 5 ) and Lemma 1, we get
    χ ( ς ) D 2 25 sup ς T ς 2 ( 1 ) ( 1 ϱ ) { E , ϱ ( ς ) [ U 0 μ ( U ) ] 2 + E 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ 2 + E 0 ς P ϱ ( ς κ ) A X ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) B ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) 2 } 25 sup ς T ς 2 ( 1 ) ( 1 ϱ ) { E , ϱ ( ς ) [ U 0 μ ( U ) ] 2 + E 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ 2 + Π A ( L 0 ) 1 2 E 0 ς P ϱ ( ς κ ) [ , ϱ ( q ) [ U 0 μ ( U ) ] + 0 q P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 q P ϱ ( ς κ ) B ( κ ) d κ + 0 q P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) ] d κ 2 + E 0 ς P ϱ ( ς κ ) B ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) 2 } { 25 Π 2 Γ 2 ( ( 1 ϱ ) + ϱ ) E U 0 2 + C 5 ( 1 + ι ) + 25 Π 2 q 1 2 ( 1 ϱ ) ( 2 ϱ 1 ) Γ 2 ( ϱ ) ( C 2 + T r ( Φ ) C 3 ) ( 1 + ι ) + B L 1 ( I , R + ) + C 4 q } × 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) : = ρ .
    Thus, W ( Δ ) is bounded in D .
  • Step 3: The set { W ( U ) : U Δ } is equicontinuous.
    For any U Δ , χ W ( U ) , there exists a B Θ ( U ) such that (4) holds for each ς T .
    For 0 < ς 1 < ς 2 < q , we get
    χ ( ς 2 ) χ ( ς 1 ) D 2 25 , ϱ ( ς 2 ) , ϱ ( ς 1 ) [ U 0 μ ( U ) ] D 2 + 25 0 ς 2 P ϱ ( ς 2 κ ) γ ( κ , U ( κ ) ) d κ 0 ς 1 P ϱ ( ς 1 κ ) γ ( κ , U ( κ ) ) d κ D 2 + 25 0 ς 2 P ϱ ( ς 2 κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) 0 ς 1 P ϱ ( ς 1 κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) D 2 + 25 0 ς 2 P ϱ ( ς 2 κ ) B ( κ ) d κ 0 ς 1 P ϱ ( ς 1 κ ) B ( κ ) d κ D 2 + 25 0 ς 2 P ϱ ( ς 2 κ ) A X ( κ ) d κ 0 ς 1 P ϱ ( ς 1 κ ) A X ( κ ) d κ D 2 = 25 , ϱ ( ς 2 ) , ϱ ( ς 1 ) [ U 0 μ ( U ) ] D 2 + 25 ς 1 ς 2 P ϱ ( ς 2 κ ) γ ( κ , U ( κ ) ) d κ D 2 + 25 0 ς 1 P ϱ ( ς 2 κ ) P ϱ ( ς 1 κ ) γ ( κ , U ( κ ) ) d κ D 2 + 25 ς 1 ς 2 P ϱ ( ς 2 κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) D 2 + 25 0 ς 1 P ϱ ( ς 2 κ ) P ϱ ( ς 1 κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) D 2 + 25 ς 1 ς 2 P ϱ ( ς 2 κ ) B ( κ ) d κ D 2 + 25 0 ς 1 P ϱ ( ς 2 κ ) P ϱ ( ς 1 κ ) B ( κ ) d κ D 2 + 25 ς 1 ς 2 P ϱ ( ς 2 κ ) A X ( κ ) d κ D 2 + 25 0 ς 1 P ϱ ( ς 2 κ ) P ϱ ( ς 1 κ ) A X ( κ ) d κ D 2 .
    Since ( ς ) ( ς > 0 ) is compact, then, the right-hand side of (5) tends to zero as ς 2 ς 1 . Thus, W ( U ) ( ς ) is continuous in ( 0 , q ] . In addition, for ς 1 = 0 and 0 < ς 2 q , E χ ( ς 2 ) χ ( 0 ) D 2 0 with respect to U Δ as ς 2 0 .
    Hence, we conclude that { W ( U ) ( ς ) : U Δ } is equicontinuous in D .
  • Step 4: W is completely continuous.
    We show that ∀ ς T , > 0 , the set H ( ς ) = { χ ( ς ) : χ W ( Δ ) } is relatively compact in Λ . Clearly, H ( 0 ) is compact. Let 0 < ς q be fixed, 0 < ϵ < ς , for U Δ , and we define
    χ ϵ , j ( ς ) = ϱ Γ ( ( 1 ϱ ) ) 0 ς ϵ j θ ( ς κ ) ( 1 ϱ ) 1 κ ϱ 1 Ψ ϱ ( θ ) ( κ ϱ θ ) [ U 0 μ ( U ) ] d θ d κ + ϱ 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) γ ( κ , U ( κ ) ) d θ d κ + ϱ 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) A X ( κ ) d θ d κ + ϱ 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) B ( κ ) d θ d κ + ϱ 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) ϑ ( κ , U ( κ ) ) d θ d κ = ϱ ( ϵ ϱ j ) Γ ( ( 1 ϱ ) ) 0 ς ϵ j θ ( ς κ ) ( 1 ϱ ) 1 κ ϱ 1 Ψ ϱ ( θ ) ( κ ϱ θ ϵ ϱ j ) [ U 0 μ ( U ) ] d θ d κ + ϱ ( ϵ ϱ j ) 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ϵ ϱ j ) γ ( κ , U ( κ ) ) d θ d κ + ϱ ( ϵ ϱ j ) 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ϵ ϱ j ) A X ( κ ) d θ d κ + ϱ ( ϵ ϱ j ) 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ϵ ϱ j ) B ( κ ) d θ d κ + ϱ ( ϵ ϱ j ) 0 ς ϵ j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ϵ ϱ j ) ϑ ( κ , U ( κ ) ) d θ d κ .
    Since ( ϵ ϱ j ) , ϵ ϱ j > 0 is a compact operator, the set H ϵ , j ( ς ) = { χ ϵ , j ( ς ) : χ ϵ , j W ( Δ ) } is relatively compact in Λ . In addition, we have
    E χ ( ς ) χ ϵ , j ( ς ) D 2 = sup ς T ς 2 ( 1 ) ( 1 ϱ ) E χ ( ς ) χ ϵ , j ( ς ) 2 25 ϱ 2 Γ 2 ( ( 1 ϱ ) ) sup ς T ς 2 ( 1 ) ( 1 ϱ ) × E 0 ς 0 j θ ( ς κ ) ( 1 ϱ ) 1 κ ϱ 1 Ψ ϱ ( θ ) ( κ ϱ θ ) [ U 0 μ ( U ) ] d θ d κ 2 + 25 ϱ 2 Γ 2 ( ( 1 ϱ ) ) sup ς T ς 2 ( 1 ) ( 1 ϱ ) × E ς ϵ ς j θ ( ς κ ) ( 1 ϱ ) 1 κ ϱ 1 Ψ ϱ ( θ ) ( κ ϱ θ ) [ U 0 μ ( U ) ] d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E 0 ς 0 j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) γ ( κ , U ( κ ) ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E ς ϵ ς j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) γ ( κ , U ( κ ) ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E 0 ς 0 j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) A X ( κ ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E ς ϵ ς j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) A X ( κ ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E 0 ς 0 j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) B ( κ ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E ς ϵ ς j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) B ( κ ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E 0 ς 0 j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) ϑ ( κ , U ( κ ) ) d θ d κ 2 + 25 ϱ 2 sup ς T ς 2 ( 1 ) ( 1 ϱ ) E ς ϵ ς j θ ( ς κ ) ϱ 1 Ψ ϱ ( θ ) ( ( ς κ ) ϱ θ ) ϑ ( κ , U ( κ ) ) d θ d κ 2 .
    We see that when ϵ 0 + and j 0 + , the inequality (6) tends to zero. Thus, the set H ( ς ) is relatively compact in Λ . Hence, from Step 3 and the Arzela–Ascoli theorem, W is completely continuous.
  • Step 5: W has a closed graph.
    Let us consider U n U * in D , χ n W ( U n ) , and χ n χ * in D . We will prove that χ * W ( U * ) .
    Actually, χ n W ( U n ) shows that there exists a B n Θ ( U n ) such that
    χ n ( ς ) = , ϱ ( ς ) [ U 0 μ ( U n ) ] + 0 ς P ϱ ( ς κ ) γ ( κ , U n ( κ ) ) d κ + 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + 0 ς P ϱ ( ς κ ) B n ( κ ) d κ + 0 ς P ϱ ( ς κ ) ϑ ( κ , U n ( κ ) ) d ϖ ( κ ) , ς T
    From ( A 2 ) ( A 5 ) , it is easy to verify that { μ ( U n ) , γ ( · , U n ) , B n , ϑ ( · , U n ) } n 1 Λ × Λ × L Υ 2 ( I , Λ ) × L Φ is bounded. Then, we get
    ( μ ( U n ) , γ ( · , U n ) , B n , ϑ ( · , U n ) ) ( μ ( U * ) , γ ( · , U * ) , B * , ϑ ( · , U * ) ) w e a k l y i n Λ × Λ × L Υ 2 ( I , Λ ) × L Φ .
    According to (7), (8), and the compactness of the operator ( ς ) , we have that
    χ n ( ς ) , ϱ ( ς ) [ U 0 μ ( U * ) ] + 0 ς P ϱ ( ς κ ) γ ( κ , U * ( κ ) ) d κ + 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + 0 ς P ϱ ( ς κ ) B * ( κ ) d κ + 0 ς P ϱ ( ς κ ) ϑ ( κ , U * ( κ ) ) d ϖ ( κ ) .
    Concentrating on χ n χ * in D and B n Θ ( U n ) , from (9) and Lemma 3, we can get B * Θ ( U * ) . Therefore, we can show that χ * W ( U * ) , which shows that W has a closed graph. By using Proposition 3.3.12(2) of [30], we find the conclusion that W is upper semicontinuous.
  • Step 6: An a priori estimate.
    From previous steps, we found that W is compact convex valued and upper semicontinuous and that W ( Δ ) is relatively compact. By Theorem 2.10 from [29], we can prove that the set = { U D : η U W , η > 1 } is bounded.
    Let us consider U and assume that there occurs a B Θ ( U ) such that
    U ( ς ) = η 1 , ϱ ( ς ) [ U 0 μ ( U ) ] + η 1 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + η 1 0 ς P ϱ ( ς κ ) A X ( κ ) d κ + η 1 0 ς P ϱ ( ς κ ) B ( κ ) d κ + η 1 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) .
    From ( A 1 ) ( A 5 ) and Lemma 1, we can get
    U ( ς ) 2 25 | η 1 | 2 { E , ϱ ( ς ) [ U 0 μ ( U ) ] 2 + E 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ 2 + E 0 ς P ϱ ( ς κ ) A X ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) B ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) 2 } 25 { E , ϱ ( ς ) [ U 0 μ ( U ) ] 2 + E 0 ς P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ 2 + Π A ( L 0 ) 1 2 E 0 ς P ϱ ( ς κ ) [ , ϱ ( q ) [ U 0 μ ( U ) ] + 0 q P ϱ ( ς κ ) γ ( κ , U ( κ ) ) d κ + 0 q P ϱ ( ς κ ) B ( κ ) d κ + 0 q P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) ] d κ 2 + E 0 ς P ϱ ( ς κ ) B ( κ ) d κ 2 + E 0 ς P ϱ ( ς κ ) ϑ ( κ , U ( κ ) ) d ϖ ( κ ) 2 } { 25 Π 2 q 2 ( 1 ) ( 1 ϱ ) Γ 2 ( ( 1 ϱ ) + ϱ ) ( E U 0 2 + C 5 ( 1 + E U ( ς ) 2 ) + 25 Π 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) ( C 2 + T r ( Φ ) C 3 ) ( 1 + E U ( ς ) 2 ) + B L 1 ( T , R + ) + C 4 q E U ( ς ) 2 } × 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) 1 + 2 E U ( ς ) 2
    where
    1 = 25 Π 2 q 2 ( 1 ) ( 1 ϱ ) ( E U 0 2 + C 5 ) Γ 2 ( ( 1 ϱ ) + ϱ ) + 25 Π 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) ( C 2 + T r ( Φ ) C 3 ) + B L 1 ( I , R + ) × 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) ,
    and
    2 = 25 C 5 Π 2 q 2 ( 1 ) ( 1 ϱ ) Γ 2 ( ( 1 ϱ ) + ϱ ) + 25 Π 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) [ ( C 2 + T r ( Φ ) C 3 ) + C 4 q ] { 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) } .
    Since 2 < 1 , from ( 3.9 ) , we obtain
    U D 2 = sup ς T E ς ( 1 ) ( 1 ϱ ) U ( ς ) 2 1 + 2 U D 2 .
    Then, U D 2 1 1 2 implies that the set is bounded. According to Theorem 2.10 from [29], W has a fixed point. Any fixed point of W is a mild solution of the system (1) on T . Hence, the system (1) is exact null controllable on T .

4. Example

The following control system is described by Hilfer fractional stochastic partial differential inclusions with the Clarke subdifferential and nonlocal conditions:
D 0 + 1 5 , 3 4 U ( ς , z ) 2 z 2 ( U ( ς , z ) + 1 20 s i n ( U ( ς , z ) ) + γ ( ς , z ) + 1 5 c o s ( U ( ς , z ) ) d ϖ ( ς ) d ς + Ξ ( ς , U ( ς , z ) ) , ς T = ( 0 , 1 ] , U ( ς , 0 ) = U ( ς , 2 ) = 0 , ς T , I 0 + 3 5 U ( 0 , z ) + i = 1 p a i U ( ς i , z ) = U 0 ( z ) , 0 z 3 ,
where D 0 + 1 5 , 3 4 is the Hilfer fractional derivative of order = 1 5 , ϱ = 3 4 , 0 < ς 0 < ς 1 < . . . < ς p < 1 , U 0 ( z ) Λ = L 2 ( [ 0 , 3 ] ) , and ϖ is a Wiener process. The functions are defined as U ( ς ) ( z ) = U ( ς , z ) , 1 8 s i n ( U ( ς , z ) ) = γ ( ς , U ( ς , z ) ) , 1 20 c o s ( U ( ς , z ) ) = ϑ ( ς , U ( ς , z ) ) , Ξ ( ς , U ( ς ) ) ( z ) = Ξ ( ς , U ( ς , z ) ) , and X ( ς ) ( z ) = γ ( ς , z ) . The bounded linear operator A is defined by A y = ϱ ( ς , z ) , ς T , 0 z 2 , y F .
Let Λ = Ψ = F = L 2 ( [ 0 , 3 ] ) , and let the operator : D ( ) Λ Λ be given by = 2 z 2 , with D ( ) = { U Λ , z , z z being absolutely continuous, 2 U z 2 Λ , U ( 0 ) = U ( 2 ) = 0 } .
Then, ℧ can be written as
U = n = 1 ( n 2 ) ( U , U n ) U n , z D ( ) ,
where U n ( s ) = 2 π sin n s , n = 1 , 2 , . . . is the orthogonal set of eigenvectors of .
Furthermore, for U Λ , we have
( ς ) z = n = 1 e n 2 ς 1 + n 2 ( z , U n ) U n .
The operator ℧ is the infinitesimal generator of a compact semigroup { ( ς ) } ς 0 in Λ . From the above choice, the system (12) can be written in the abstract form of (1), all assumptions of Theorem 4 are satisfied, and
25 C 5 Π 2 q 2 ( 1 ) ( 1 ϱ ) Γ 2 ( ( 1 ϱ ) + ϱ ) + 25 Π 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) [ ( C 2 + T r ( Φ ) C 3 ) + C 4 q ] × 1 + 25 Π 2 Π A ( L 0 ) 1 2 q 2 ϱ 1 ( 2 ϱ 1 ) Γ 2 ( ϱ ) < 1 .
Thus, the system (12) is null controllable on ( 0 , 1 ] .

5. Conclusions

In this paper, a control system described by Hilfer fractional stochastic differential inclusions with the Clarke subdifferential and nonlocal conditions was presented. By using the fixed-point technique, fractional calculus, stochastic analysis, properties of the Clarke subdifferential, and non-smooth analysis, the null controllability of the considered system was investigated. Moreover, we provided an example in order to illustrate the applicability of the results.
In future work, we can present the boundary null controllability of non-instantaneous impulsive fractional stochastic evolution inclusions.

Author Contributions

H.M.A. and M.M.E.-B.: methodology, validation, and supervision; W.E.-S.: writing—original draft preparation and writing—review and editing; A.E.: formal analysis, investigation, and resources. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Dineshkumar, C.; Udhayakumar, R. Results on approximate controllability of fractional stochastic Sobolev-type Volterra–Fredholm integro-differential equation of order 1<r<2. Math. Methods Appl. Sci. 2022, 45, 6691–6704. [Google Scholar]
  2. Priyadharsini, J.; Balasubramaniam, P. Controllability of fractional noninstantaneous impulsive integrodifferential stochastic delay system. IMA J. Math. Control Inf. 2021, 2, 654–683. [Google Scholar] [CrossRef]
  3. Sathiyaraj, T.; Wang, J.; Balasubramaniam, P. Controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systems. Appl. Math. Optim. 2021, 3, 2527–2554. [Google Scholar] [CrossRef]
  4. Dhayal, R.; Malik, M. Existence and controllability of impulsive fractional stochastic differential equations driven by Rosenblatt process with Poisson jumps. J. Eng. Math. 2021, 130, 11. [Google Scholar] [CrossRef]
  5. Ahmed, H.M. Controllability of fractional stochastic delay equations. Lobachevskii J. Math. 2009, 30, 195–202. [Google Scholar] [CrossRef]
  6. Wang, J.; Ahmed, H.M. Null controllability of nonlocal Hilfer fractional stochastic differential equations. Miskolc Math. Notes 2017, 18, 1073–1083. [Google Scholar]
  7. Alnafisah, Y.; Ahmed, H.M. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump. Int. J. Nonlinear Sci. Numer. Simul. 2021. [Google Scholar] [CrossRef]
  8. Yan, Z.; Zhou, Y.H. Optimization of exact controllability for fractional impulsive partial stochastic differential systems via analytic sectorial operators. Int. J. Nonlinear Sci. Numer. Simul. 2021, 22, 559–579. [Google Scholar] [CrossRef]
  9. Wang, J.; Sathiyaraj, T.; O’Regan, D. Relative controllability of a stochastic system using fractional delayed sine and cosine matrices. Nonlinear Anal. Model. Control 2021, 26, 1031–1051. [Google Scholar] [CrossRef]
  10. Luo, D.; Tian, M.; Zhu, Q. Some results on finite-time stability of stochastic fractional-order delay differential equations. Chaos Solitons Fractals 2022, 158, 111996. [Google Scholar] [CrossRef]
  11. Luo, D.; Zhu, Q.; Luo, Z. A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients. Appl. Math. Lett. 2021, 122, 107549. [Google Scholar] [CrossRef]
  12. Luo, D.; Zhu, Q.; Luo, Z. An averaging principle for stochastic fractional differential equations with time-delays. Appl. Math. Lett. 2020, 105, 106290. [Google Scholar] [CrossRef]
  13. Moghaddam, B.P.; Lopes, A.M.; Machado, J.A.T.; Mostaghim, Z.S. Computational scheme for solving nonlinear fractional stochastic differential equations with delay. Stoch. Anal. Appl. 2019, 37, 893–908. [Google Scholar] [CrossRef]
  14. Sakthivel, R.; Revathi, P.; Ren, Y. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. Theory, Methods Appl. 2013, 81, 70–86. [Google Scholar] [CrossRef]
  15. Ahmed, H.M. Conformable fractional stochastic differential equations with control function. Syst. Control Lett. 2021, 158, 105062. [Google Scholar] [CrossRef]
  16. Balasubramaniam, P.; Kumaresan, N.; Ratnavelu, K.; Tamilalagan, P. Local and Global Existence of Mild Solution for Impulsive Fractional Stochastic Differential Equations. Bull. Malays. Math. Sci. Soc. 2014, 38, 867–884. [Google Scholar] [CrossRef]
  17. Ahmed, H.M. Sobolev-type fractional stochastic integrodifferential equations with nonlocal conditions in Hilbert space. J. Theoret. Probab. 2017, 30, 771–783. [Google Scholar] [CrossRef]
  18. Li, L.; Liu, J.G.; Lu, J. Fractional stochastic differential equations satisfying fluctuation-dissipation theorem. J. Stat. Phys. 2017, 169, 316–339. [Google Scholar] [CrossRef] [Green Version]
  19. Lv, J.; Yang, X. A class of Hilfer fractional stochastic differential equations and optimal controls. Adv. Differ. Equ. 2019, 2019, 17. [Google Scholar] [CrossRef] [Green Version]
  20. dos Santos Lima, L. Fractional Stochastic Differential Equation Approach for Spreading of Diseases. Entropy 2022, 24, 719. [Google Scholar] [CrossRef]
  21. Omar, O.A.; Elbarkouky, R.A.; Ahmed, H.M. Fractional stochastic models for COVID-19: Case study of Egypt. Results Phys. 2021, 23, 104018. [Google Scholar] [CrossRef] [PubMed]
  22. Atangana, A.; Bonyah, E. Fractional stochastic modeling: New approach to capture more heterogeneity. Chaos: Interdiscip. J. Nonlinear Sci. 2019, 29, 013118. [Google Scholar]
  23. Omar, O.A.; Elbarkouky, R.A.; Ahmed, H.M. Fractional stochastic modelling of COVID-19 under wide spread of vaccinations: Egyptian case study. Alex. Eng. J. 2022, 61, 8595–8609. [Google Scholar] [CrossRef]
  24. Li, Y.X.; Lu, L. Existence and controllability for stochastic evolution inclusions of Clarke’s subdifferential type. Electron. J. Qual. Theory Differ. Equ. 2015, 2015, 1–16. [Google Scholar] [CrossRef]
  25. Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
  26. Gu, H.; Trujillo, J.J. Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 2015, 257, 344–354. [Google Scholar] [CrossRef]
  27. Yang, M.; Wang, Q. Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math. Methods Appl. Sci. 2016, 40, 1126–1138. [Google Scholar] [CrossRef]
  28. Lu, L.; Liu, Z.; Bin, M. Approximate controllability for stochastic evolution inclusions of Clarke’s subdifferential type. Appl. Math. Comput. 2016, 286, 201–212. [Google Scholar] [CrossRef]
  29. Dineshkumar, C.; Nisar, K.S.; Udhayakumar, R.; Vijayakumar, V. A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions. Asian J. Control 2021, 24, 2378–2394. [Google Scholar] [CrossRef]
  30. Migórski, S.; Ochal, A.; Sofonea, M. Nonlinear Inclusions and Hemivariational Inequalities. In Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics; Springer: New York, NY, USA, 2013; Volume 26. [Google Scholar]
  31. Fu, X.L.; Zhang, Y. Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions. Acta Math. Sci. Ser. B 2013, 33, 747–757. [Google Scholar] [CrossRef]
  32. Park, J.Y.; Balasubramaniam, P. Exact null controllabiliyt of abstract semilinear functional integrodifferential stochastic evolution equations in Hilbert space. Taiwan J. Math. 2009, 13, 2093–2103. [Google Scholar] [CrossRef]
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Ahmed, H.M.; El-Borai, M.M.; El-Sayed, W.; Elbadrawi, A. Null Controllability of Hilfer Fractional Stochastic Differential Inclusions. Fractal Fract. 2022, 6, 721. https://doi.org/10.3390/fractalfract6120721

AMA Style

Ahmed HM, El-Borai MM, El-Sayed W, Elbadrawi A. Null Controllability of Hilfer Fractional Stochastic Differential Inclusions. Fractal and Fractional. 2022; 6(12):721. https://doi.org/10.3390/fractalfract6120721

Chicago/Turabian Style

Ahmed, Hamdy M., Mahmoud M. El-Borai, Wagdy El-Sayed, and Alaa Elbadrawi. 2022. "Null Controllability of Hilfer Fractional Stochastic Differential Inclusions" Fractal and Fractional 6, no. 12: 721. https://doi.org/10.3390/fractalfract6120721

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