Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

Yongqiang Fu; Lixu Yan

doi:10.1515/math-2020-0060

Open Access Published by De Gruyter Open Access November 5, 2020

Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

Yongqiang Fu and Lixu Yan

From the journal Open Mathematics

https://doi.org/10.1515/math-2020-0060

Abstract

The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.

Keywords: stochastic system; fractional Laplacian; weak solution; optimal control

MSC 2010: 60H15; 35A01; 47H06

1 Introduction

In this paper, we discuss a class of stochastic fractional reaction-diffusion systems in Ω :

(1) d u + ( − Δ ) α u d t = f ( u ) d t + g d t + σ d W , t ∈ ( 0 , T ] , x ∈ Ω , u ( t ) = 0 , t ∈ [ 0 , T ] , x ∈ ℝ d \ Ω , u ( 0 ) = u 0 , x ∈ Ω,

where Ω is a smooth bounded domain contained in ℝ d , α ∈ ( 0 , 1 ) , T ∈ ( 0 , + ∞ ) , u is a vector-valued function, σ is an operator-valued function and { W ( t ) } t ∈ [ 0 , T ] is the space-time noise.

Different from Laplacian, the fractional Laplacian is a nonlocal linear operator. A natural question arises: Whether we can extend the results of Laplacian problems to the fractional Laplacian ones or not? Unfortunately, these extensions are not always true, see [1,2,3]. In particular, Devillanova and Carlo Marano have studied the following fractional differential equation in [3]:

u ( 2 ) ( t ) + 2 ξ ω 2 − β u ( β ) ( t ) + ω 2 u ( 0 ) ( t ) = 0 .

Devillanova and Carlo Marano have delicately compared the nonfractional cases ( β = 0 , 1 ) and the fractional case ( 0 < β < 1 ) through the mathematical analysis and experimental data. Furthermore, the authors have dealt with the fractional derivative by using the Laplace transform and inversion procedure and some rich results have been obtained. In the last decade, there have been many significant investigations on fractional Laplacian problems in deterministic cases, see [4,5,6] and references therein. But the corresponding stochastic cases need to be further studied. It is well known that ( − Δ ) α → ( − Δ ) , as α → 1 . When α = 1 , system (1) becomes the standard stochastic reaction-diffusion system. In [7], Ahmed studied on the situation of α = 1 and g ≡ 0 , and the well-posedness of weak solution is obtained for p ∈ [ 1 , + ∞ ) . We extend this result to the fractional Laplacian case and obtain the existence and uniqueness of weak solutions when p ∈ [ 1 , 2 α ∗ / 2 ) .

Recently, there have been several papers on stochastic fractional system. In [8], Durga and Muthukumar have considered a type of stochastic time-fractional system as follows:

D t α u = Δ u + f + σ d W d t , t ∈ ( 0 , T ] , u ( 0 ) = u 0 ,

where D t α is the α th Caputo type of fractional derivative. Based on fractional calculus, the existence and regularity of mild solutions are obtained, whereas the existence of optimal controls for the corresponding optimal problem is proved by Balder’s method. In [9], Bezdek has considered the stochastic fractional reaction-diffusion system on a circle

d u − ℒ u d t = σ ( u ) d W , t ∈ ( 0 , + ∞ ) , u ( 0 ) = u 0 ,

where ℒ is a general fractional derivative operator and σ is a locally Lipschitz continuous mapping. Based on the assumption of the growth condition

| σ ( x ) | ≤ a | x | γ + b ,

where γ ≥ 1 , Bezdek proved the existence of mild solutions. In [10], Wang investigated the following stochastic fractional reaction-diffusion system:

(2) d φ + ( − Δ ) α φ d t = f ( φ ) d t + g d t + c φ d W , t > τ , x ∈ Ω , φ ( t ) = 0 , t > τ , x ∈ ℝ d \ Ω , φ ( τ ) = φ τ , x ∈ Ω,

where φ is the real valued function and c ∈ ( 0 , + ∞ ) . Under suitable assumptions on f, the existence and uniqueness of solutions are achieved. Wang et al. [11] keep discussing system (2) on the whole ℝ d . But only the existence of solutions is obtained.

Motivated by the aforementioned results, we focus our attention on the properties of weak solutions of system (1). These solutions satisfy (1) in the weak sense with probability 1. Additionally, we wish to extend the optimal control theory of deterministic linear control problems in [12,13] to stochastic control problems. We consider the stochastic control problem of (1) as follows:

d u + ( − Δ ) α u d t = f ( u ) d t + g d t + A v d t + σ d W , t ∈ ( 0 , T ] , u ( 0 ) = u 0

with the cost function

J ( v ) = E ∫ 0 T ∥ N u ( v ) − z d ∥ 2 d t + ( K v , v ) U ˜ ,

where v = v ( t , x ) is a stochastic control and A is a given operator which is called a controller.

The framework of this paper is organized as follows. In Section 2, the proper functional spaces and basic concepts are presented. In Section 3, the assumptions on the nonlinear term and noise are stated, then the existence and uniqueness of weak solutions are obtained by using Galërkin’s method along with a uniform estimate. In Section 4, the existence of optimal controls is proved. In Section 5, some sufficient conditions and examples are shown to illustrate our results.

2 Preliminaries

Stochastic fractional reaction-diffusion system (1) involves both stochastic term and fractional Laplacian. It is necessary to employ the theories of stochastic partial differential equations (see [14,15]) and fractional calculus (see [16,17]) to study this type of system.

Let S be the Schwartz space, the fractional Laplacian for φ ∈ S is defined by

( − Δ ) α φ ( x ) = C ( d , α ) P .V . ∫ ℝ d φ ( x ) − φ ( y ) | x − y | d + 2 α d y , x ∈ ℝ d ,

where α ∈ ( 0 , 1 ) and

1 C ( d , α ) = ∫ ℝ d 1 − cos ξ 1 | ξ | d + 2 α d ξ .

In this paper, we denote ( − Δ ) α u = ( ( − Δ ) α u 1 , ( − Δ ) α u 2 , … , ( − Δ ) α u N ) .

Let L p ( ℝ d , ℝ N ) be the space of all pth integrable functions. For any u ∈ L p ( ℝ d , ℝ N ) and 1 ≤ p < + ∞ , the norm is defined by

∥ u ∥ L p ( ℝ d , ℝ N ) = ∑ i = 1 N ∫ ℝ d | u i ( x ) | p d x ) 1 p .

Since fractional Laplacian is a nonlocal operator, let

L p ≜ { u ∈ L p ( ℝ d , ℝ N ) : u i = 0 a .e . on ℝ d \ Ω , i = 1 , 2 , … , N } .

Noting that, for a given function f ∈ L p ( Ω , ℝ N ) , it is only defined on Ω , we say f ∈ L p means that we extend the domain of f to ℝ d by setting

f ( x ) = 0 , x ∈ ℝ d \ Ω .

The function space H α ( ℝ d , ℝ N ) is given by

H α ( ℝ d , ℝ N ) = u ∈ L 2 ( ℝ d , ℝ N ) : ∑ i = 1 N ∬ ℝ d × ℝ d | u i ( x ) − u i ( y ) | 2 | x − y | d + 2 α d x d y < + ∞ .

Let

X 0 ≜ { u ∈ H α ( ℝ d , ℝ N ) : u i = 0 a .e . in ℝ d \ Ω , i = 1 , 2 , … , N }

be endowed with the norm

∥ u ∥ X 0 = ∑ i = 1 N ∬ ℝ d × ℝ d | u i ( x ) − u i ( y ) | 2 | x − y | d + 2 α d x d y 1 2 .

It is easily seen that ( X 0 , ∥ ⋅ ∥ X 0 ) is a Hilbert space under the inner product

( u , v ) X 0 = ∑ i = 1 N ∬ ℝ d × ℝ d ( u i ( x ) − u i ( y ) ) ( v i ( x ) − v i ( y ) ) | x − y | d + 2 α d x d y .

Based on [16, Proposition 1.18] and [17, Corollary 7.2], we get the following.

Lemma 2.1

Let α ∈ ( 0 , 1 ) , then for any u ∈ X 0 ,

(3) 〈 ( − Δ ) α u , u 〉 X 0 ∗ , X 0 = 1 2 C ( d , α ) ∥ u ∥ X 0 2 .

Lemma 2.1 extends Proposition 1.18 of [16] directly, and they have similar proofs, so we omit the proof of Lemma 2.1. Setting p α ∗ = d p d − α p for p ∈ ( 0 , + ∞ ) , we state the following.

Lemma 2.2

Let α ∈ ( 0 , 1 ) and 2 α < d . If p ∈ [ 1 , 2 α ∗ / 2 ] , then

X 0 ↪ L 2 p ↪ L 2 .

Thus, there exists c 0 > 0 such that

(4) ∥ u ∥ L 2 ≤ c 0 ∥ u ∥ X 0

Moreover, the embedding X 0 ↪ ↪ L 2 p is compact when p ∈ [ 1 , 2 α ∗ / 2 ) .

We work in a complete probability space ( Σ , ℱ , P ) equipped with a filtration { ℱ t } t ∈ [ 0 , T ] . Let Y be a separable Hilbert space, σ ( t ) be a bounded linear operator from Y to L 2 for any t ∈ [ 0 , T ] and W be a Y-valued Q-Wiener process defined on Σ , ℱ , P , ℱ t ∈ [ 0 , T ] . We recall a result of [15] as follows.

Lemma 2.3

[15, Proposition 4.1] Let Q ∈ L ( Y ) be a symmetric nonnegative operator with Tr Q < + ∞ , then for ∀ t ∈ [ 0 , T ] and ∀ y ∈ Y :

W is a Gaussian process on Y and
E ( W ( t ) , y ) Y = 0 , E ( W ( t ) , y ) Y 2 = t ( Q y , y ) .
W has the following expression:

W ( t ) = ∑ j = 1 + ∞ λ j β j ( t ) e j ,

where { e j } is an orthonormal basis of Y, { λ j } is the sequence of eigenvalues of Q and { β j ( t ) } is a sequence of Brownian motions which are independent from each other on probability space Σ , ℱ , P , ℱ t ∈ [ 0 , T ] .

Next, we introduce the space of random processes. For any p ∈ [ 1 , + ∞ ) , let

L p ℱ ( [ 0 , T ] , X ) = u is an ℱ t ∈ [ 0 , T ] adapted random process: E ∫ 0 T ∥ u ( t ) ∥ X p d t < ∞

and define the norm

∥ u ∥ L p ℱ ( [ 0 , T ] , X ) = E ∫ 0 T ∥ u ( t ) ∥ X p d t 1 p .

While p = + ∞ , let

L ∞ ℱ ( [ 0 , T ] , X ) = u is an ℱ t ∈ [ 0 , T ] adapted random process : ess sup t ∈ [ 0 , T ] E ∥ u ( t ) ∥ X 2 < ∞

and endow u with the following norm:

∥ u ∥ L ∞ ℱ ( [ 0 , T ] , X ) = ess sup t ∈ [ 0 , T ] ( E ∥ u ( t ) ∥ X 2 ) 1 2 .

( L p ℱ ( [ 0 , T ] , X ) , ∥ ⋅ ∥ L p ℱ ( [ 0 , T ] , X ) ) is a Banach space for any p ∈ [ 1 , + ∞ ] . Moreover, it is a reflexive Banach space for any p ∈ [ 1 , + ∞ ) . In this paper, we confine X to be L 2 , X 0 or L 2 p .

We denote L 2 ℱ 0 ( Σ , X ) by the space of all ℱ 0 measurable random variables with bounded second moments. Let

C T 1 ( [ 0 , T ] ) = { ϕ ∈ C 1 ( [ 0 , T ] ) : ϕ vanished at T } ,

then C T 1 ( [ 0 , T ] ) is a sub-space of C 1 ( [ 0 , T ] ) . We give the concept of weak solutions for system (1) as follows.

Definition 2.4

An ℝ N -valued stochastic process u ∈ L ∞ ℱ ( [ 0 , T ] , L 2 ) ∩ L 2 ℱ ( [ 0 , T ] , X 0 ) ∩ L 2 p ℱ ( [ 0 , T ] , L 2 p ) is called a weak solution of system (1), if u satisfies

(5) − ( u 0 , ϕ ( 0 ) ν ) L 2 − ∫ 0 T ( u ( t ) , ϕ ̇ ν ) L 2 d t + ∫ 0 T 〈 ( − Δ ) α u ( t ) , ν 〉 X 0 ∗ , X 0 ϕ ( t ) d t = ∫ 0 T 〈 f ( u ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t + ∫ 0 T ( g ( t ) , ϕ ( t ) ν ) L 2 d t + ∫ 0 T ( ϕ ( t ) ν , σ ( t ) d W ( t ) ) , a .s . ,

for any ν ∈ X 0 and ϕ ∈ C T 1 ( [ 0 , T ] ) .

3 The existence and uniqueness of weak solutions

From [18, proposition 9] and [19, proposition 4], we know that there exists a sequence of eigenfunctions of fractional Laplacian which constructs an orthonormal basis of L 2 and an orthogonal basis of X 0 . This result ensures that we can use Galërkin’s method to obtain the existence of weak solutions of (1).

First, due to the theory of semigroups of nonlinear operators (see [20, Chapter 4]), we assume that f : L 2 p → ( L 2 p ) ∗ is m-dissipative. We also impose the following conditions on f and the operator-valued function σ ( t ) .

(H1): There exist c 1 ∈ [ 0 , + ∞ ) and c 2 ∈ ( 0 , + ∞ ) , such that
〈 f ( u ) , u 〉 ( L 2 p ) ∗ , L 2 p ≤ c 1 ∥ u ∥ L 2 2 − c 2 ∥ u ∥ L 2 p 2 p .
(H2): There exist c 3 ∈ ( 0 , + ∞ ) and c 4 ∈ ( 0 , + ∞ ) , such that
∥ f ( u ) ∥ ( L 2 p ) ∗ ≤ c 3 + c 4 ∥ u ∥ L 2 p 2 p − 1 .
(H3): f ( u ) is lower semi-continuous and
〈 f ( u ) − f ( v ) , u − v 〉 ( L 2 p ) ∗ , L 2 p ≤ 0 .
(H4): There exists an M ∈ ( 0 , + ∞ ) , such that

∥ σ ( t ) ∥ ≤ M , ∀ t ∈ [ 0 , T ] , uniformly in x ∈ Ω .

Next, we give an important result on the nonlinear term f.

Lemma 3.1

Let p ∈ [ 1 , + ∞ ) and { u n } be bounded in L 2 p ℱ ( [ 0 , T ] , L 2 p ) . If (H2)–(H3) hold, then for any ν ∈ X 0 and ϕ ∈ C T 1 ( [ 0 , T ] ) , there exists u ˜ ∈ L 2 p ℱ ( [ 0 , T ] , L 2 p ) such that

∫ 0 T 〈 f ( u n ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t → ∫ 0 T 〈 f ( u ˜ ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t , a . s .

Proof

Since { u n } is bounded in L 2 p ℱ ( [ 0 , T ] , L 2 p ) , there exists u ˜ ∈ L 2 p ℱ ( [ 0 , T ] , L 2 p ) such that

(6) u n → u ˜ weakly in L 2 p ℱ ( [ 0 , T ] , L 2 p ) .

We need to show that for any ξ ∈ L ∞ ( Σ ) ,

E ξ ∫ 0 T 〈 f ( u n ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t → E ξ ∫ 0 T 〈 f ( u ˜ ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t .

In virtue of (H2) and basic inequality ( a + b ) p ≤ 2 p − 1 ( a p + b p ) , p > 0 , we obtain

(7) ∥ f ( u n ) ∥ L 2 p ℱ [ 0 , T ] , L 2 p ∗ ≤ E ∫ 0 T c 3 + c 4 ∥ u n ∥ L 2 p 2 p − 1 2 p / ( 2 p − 1 ) d t ( 2 p − 1 ) / 2 p ≤ a 1 + a 2 ∥ u n ∥ L 2 p ℱ ( [ 0 , T ] , L 2 p ) 2 p − 1 ,

where a 1 and a 2 are constants dependent on { c 3 , c 4 , p } . This inequality implies that { f ( u n ) } is bounded in ( L 2 p ℱ ( [ 0 , T ] , L 2 p ) ) ∗ = L 2 p / ( 2 p − 1 ) ℱ ( [ 0 , T ] , L 2 p / ( 2 p − 1 ) ) . We also use { f ( u n ) } as the subsequence if necessary, thus there exists h ∈ ( L 2 p ℱ ( [ 0 , T ] , L 2 p ) ) ∗ such that

f ( u n ) → h weakly ∗ in ( L 2 p ℱ ( [ 0 , T ] , L 2 p ) ) ∗ .

In addition, by Hölder inequality and (7), we get

E ξ ∫ 0 T 〈 f ( u n ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t ≤ a 3 + a 4 ∥ u n ∥ L 2 p ℱ ( [ 0 , T ] , L 2 p ) 2 p − 1 ,

where a 3 and a 4 are constants dependent on { a 1 , a 2 , ∥ ξ ∥ L ∞ ( Σ ) , ∥ ϕ ∥ L 1 / ( 2 p − 1 ) ( [ 0 , T ] ) , ∥ ν ∥ X 0 } . So

E ξ ∫ 0 T 〈 f ( u n ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t → E ξ ∫ 0 T 〈 h ( t ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t

holds by the Dominated Convergence Theorem.

Now let us verify

h = f ( u ˜ ) , a .s .

From (6), the sequence { u n } is weakly convergence to u ˜ in L 2 p ℱ ( [ 0 , T ] , L 2 p ) , further by the Mazur theorem, there exists a sequence { ζ n } which is a suitable convex combination of { u n } and { ζ n } that converges strongly to u ˜ , that is,

ζ n → u ˜ in L 2 p ℱ ( [ 0 , T ] , L 2 p ) .

On the other hand, from assumption (H3), we can deduce that for ∀ ψ ∈ L 2 p ,

∫ 0 T 〈 f ( ψ ) − h , ψ − u ˜ 〉 ( L 2 p ) ∗ , L 2 p d t = ∫ 0 T 〈 f ( ψ ) − f ( ζ n ) + f ( ζ n ) − h , ψ − ζ n + ζ n − u ˜ 〉 ( L 2 p ) ∗ , L 2 p d t ≤ 0 , a .s .

Furthermore, since f is lower semi-continuous, choosing ψ = u ˜ + ϵ μ for any ϵ ∈ ( 0 , + ∞ ) and μ ∈ L 2 p ℱ ( [ 0 , T ] , L 2 p ) , we derive that

(8) ∫ 0 T 〈 f ( u ˜ ) − h , μ 〉 ( L 2 p ) ∗ , L 2 p d t ≤ lim ̲ ϵ → 0 ∫ 0 T 〈 f ( u ˜ + ϵ μ ) − h , μ 〉 ( L 2 p ) ∗ , L 2 p d t ≤ 0 , a .s .

In view of the arbitrariness of μ , inequality (8) is available if and only if h = f ( u ˜ ) , a .s . □

The main result of this section is given as follows.

Theorem 3.2

Let α ∈ ( 0 , 1 ) , 2 α < d and p ∈ [ 1 , 2 α ∗ / 2 ) . If (H1)–(H4) hold, then system (1) has unique weak solution u, restrict to any g ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) and u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) .

Proof

We start with the uniqueness of weak solutions and then we prove the existence of weak solutions by using the Galërkin method. This proof is divided into four steps.

Step 1. The uniqueness of weak solutions.

Assume that u ˆ , u ˇ ∈ L ∞ ℱ ( [ 0 , T ] , L 2 ) ∩ L 2 ℱ ( [ 0 , T ] , X 0 ) ∩ L 2 p ℱ ( [ 0 , T ] , L 2 p ) are the weak solutions with initial states u ˆ 0 , u ˇ 0 and g ˆ , g ˇ , respectively. Since u ˆ and u ˇ satisfy system (1) in the weak sense, from integrating by parts, we deduce that

1 2 ∥ u ˆ − u ˇ ∥ L 2 2 + ∫ 0 t ( ( − Δ ) α ( u ˆ − u ˇ ) , u ˆ − u ˇ ) X 0 ∗ , X 0 , d s = 1 2 ∥ u ˆ 0 − u ˇ 0 ∥ L 2 2 + ∫ 0 t ( f ( u ˆ ) − f ( u ˇ ) , u ˆ − u ˇ ) ( L 2 p ) ∗ , L 2 p d s + ∫ 0 t ( g ˆ − g ˇ , u ˆ − u ˇ ) L 2 d s + ∫ 0 t ( σ ∗ u ˆ − σ ∗ u ˇ , d W ) .

Due to assumption (H3), (3), (4) and Young’s inequality, we get that for arbitrarily ε > 0 ,

1 2 ∥ u ˆ − u ˇ ∥ L 2 2 + 1 2 C ( d , α ) ∫ 0 t ∥ u ˆ − u ˇ ∥ X 0 2 d s ≤ 1 2 ∥ u ˆ 0 − u ˇ 0 ∥ L 2 2 + c 0 ε ∫ 0 t ∥ u ˆ − u ˇ ∥ X 0 2 d s + c ε ∫ 0 t ∥ g ˆ − g ˇ ∥ L 2 2 d s + ∫ 0 t ( σ ∗ u ˆ − σ ∗ u ˇ , d W ) ,

where c ε = ε − 1 . Choosing ε such that c 0 ε = C ( d , α ) / 4 , then taking the expectation we obtain

E ∥ u ˆ − u ˇ ∥ L 2 2 + 1 2 C ( d , α ) E ∫ 0 t ∥ u ˆ − u ˇ ∥ X 0 2 d s ≤ E ∥ u ˆ 0 − u ˇ 0 ∥ L 2 2 + 2 c ε E ∫ 0 t ∥ g ˆ − g ˇ ∥ L 2 2 d s ,

where the expectation of ∫ 0 t ( σ ∗ u ˆ − σ ∗ u ˇ , d W ) equals zero. Setting u ˆ 0 = u ˇ 0 , g ˆ = g ˇ , it is easy to see that u ˆ = u ˇ .

Step 2. The existence of solutions for the finite dimensional truncated system.

In view of [18], we let { e ¯ i } be an orthonormal sequence of eigenfunctions of fractional Laplacian and { e ¯ i } is also an orthonormal basis of L 2 and an orthogonal basis of X 0 . As in Lemma 2.3, { e i } is an orthonormal basis of Y and B n ≜ ( β 1 , β 2 , … , β n ) ′ is an n dimensional Brownian motion. We consider the truncation of system (1):

(9) d u n + ( − Δ ) α u n d t = f ( u n ) d t + g n d t + σ d W n , t ∈ ( 0 , T ] , x ∈ Ω , u n ( t ) = 0 , t ∈ [ 0 , T ] , x ∈ ℝ d \ Ω , u n ( 0 ) = ∑ j = 1 n ( u 0 , e ¯ j ) L 2 e ¯ j ≜ ∑ j = 1 n ϑ n j ( 0 ) e ¯ j , x ∈ Ω,

where u n ( t ) ≜ ∑ j = 1 n ϑ n j ( t ) e ¯ , g n ( t ) ≜ ∑ j = 1 n ( g ( t ) , e ¯ j ) L 2 e ¯ j and W n ( t ) ≜ ∑ j = 1 n ( W ( t ) , e j ) Y e j . Consequently, let ℒ be an n × n matrix, F and G be n-vectors and Θ be an n × n matrix, whose elements are given by:

L i j = 〈 ( − Δ ) α e ¯ i , e ¯ j 〉 X 0 ∗ , X 0 , F i ( ϑ n ) = f ∑ j = 1 n ϑ n j , e ¯ i ( L 2 p ) ∗ , L 2 p , G i ( t ) = ( g ( t ) , e ¯ i ) L 2 , θ i j ( t ) = λ j ( e ¯ i , σ ( t ) e j ) L 2 ,

where i , j = 1 , 2 , … , n . Next, we study the following n dimensional stochastic system:

(10) d ϑ n = ℒ ϑ n d t + F ( ϑ n ) d t + G d t + Θd B n , t ∈ [ 0 , T ] .

Because f is m-dissipative, F is m-dissipative. Hence, we can use the linear interpolation methods and Crandall-Liggett’s theory to show the existence of solutions of system (10). Let Π k = 0 = t k 0 < t k 1 < , … , < t k k = T be the kth uniform partition of [ 0 , T ] with | Π k | → 0 , as k → + ∞ . Denote δ k ≜ | Π k | , then the sequence of approximate solutions { ϑ k ( t ) } is given by

ϑ k ( t k i ) = ( I − δ k F ) − 1 [ ϑ k ( t k i − 1 ) + δ k ℒ ϑ k ( t k i − 1 ) + δ k G ( t k i − 1 ) + Θ ( t k i − 1 ) ( B n ( t k i ) − B n ( t k i − 1 ) ) ] ,

where i = 1 , 2 , … , k . According to [20, Theorem 4.7], there exists ϑ ∈ C ℱ ( [ 0 , T ] , L 2 ) such that

ϑ k → ϑ , uniformly on [ 0 , T ] , as k → ∞ ,

that is, the process

ϑ = ( ϑ 1 , ϑ 2 , … , ϑ n ) ′

is a solution of (10). Thus, u n = ∑ j = 1 n ϑ j e ¯ j is a solution of (9).

Step 3. A uniform estimate.

Integrating by parts on both sides of (9), we obtain

1 2 ( u n ( t ) , u n ( t ) ) L 2 + ∫ 0 t 〈 ( − Δ ) α u n ( s ) , u n ( s ) 〉 X 0 ∗ , X 0 d s = 1 2 ( u n ( 0 ) , u n ( 0 ) ) L 2 + ∫ 0 t 〈 f ( u n ( s ) , u n ( s ) 〉 ( L 2 p ) ∗ , L 2 p d s + ∫ 0 t ( g n ( s ) , u n ( s ) ) L 2 d s + ∫ 0 t ( u n ( s ) , σ ( s ) d W n ( s ) ) .

Consequently, similar to the calculation of Step 1, we have

(11) E ∥ u n ( t ) ∥ L 2 2 + 1 2 C ( d , α ) ∫ 0 t E ∥ u n ( s ) ∥ X 0 2 d s + 2 c 2 ∫ 0 t E ∥ u n ( s ) ∥ L 2 p 2 p d s ≤ E ∥ u n ( 0 ) ∥ L 2 2 + 2 c 1 ∫ 0 t E ∥ u n ( s ) ∥ L 2 2 d s + 2 c ε ∫ 0 t E ∥ g n ( s ) ∥ L 2 2 d s

by using (H1), (3), (4) and Young’s inequality. Since u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) and g ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) , using (11) and Gronwall inequality, we get that

E ∥ u n ( t ) ∥ L 2 2 ≤ c e 2 c 1 T ,

where c = c ∥ u 0 ∥ L 2 ℱ 0 ( Σ , L 2 ) , ∥ g ∥ L 2 ℱ ( [ 0 , T ] , L 2 ) , T . Hence using (11) once more, we deduce that { u n } is bounded in L ∞ ℱ ( [ 0 , T ] , L 2 ) ∩ L 2 ℱ ( [ 0 , T ] , X 0 ) ∩ L 2 p ℱ ( [ 0 , T ] , L 2 p ) . So there exists a subsequence of { u n } , which is also denoted by { u n } and an element u ˜ , such that

(12) u n → u ˜ weakly ∗ in L ∞ ℱ ( [ 0 , T ] , L 2 ) , u n → u ˜ weakly in L 2 ℱ ( [ 0 , T ] , X 0 ) , u n → u ˜ weakly in L 2 p ℱ ( [ 0 , T ] , L 2 p ) .

Step 4. The existence of weak solutions for (1).

We assert that u ˜ is a weak solution of (1), that is, we need to show u ˜ satisfies formula (5). Taking any ϕ ∈ C T 1 ( [ 0 , T ] ) and ξ ∈ L ∞ ( Σ ) , by (9), we can compute that

0 = E ξ ( u n ( 0 ) , ϕ ( 0 ) e ¯ i ) + E ξ ∫ 0 T ( u n ( t ) , ϕ ̇ e ¯ i ) L 2 d t − E ξ ∫ 0 T 〈 ( − Δ ) α u n ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t + E ξ ∫ 0 T 〈 f ( u n ( t ) ) , ϕ ( t ) e ¯ i 〉 ( L 2 p ) ∗ , L 2 p d t + E ξ ∫ 0 T ( g n ( t ) , ϕ ( t ) e ¯ i ) L 2 d t + E ξ ∫ 0 T ( ϕ ( t ) e ¯ i , σ ( t ) d W n ( t ) ) ≜ I 1 + I 2 − I 3 + I 4 + I 5 + I 6 ,

where i = 1 , 2 , … , n . Following the fact that ϕ e ¯ i is the deterministic function, we will use (12) and the Dominated Convergence Theorem to prove the assertion.

Start with I 1 = E { ξ ( u n ( 0 ) , ϕ ( 0 ) e ¯ i ) } . Noting that u n ( 0 ) is the n-dimensional truncation of u 0 , thus we derive

u n ( 0 ) → u 0 in L 2 ℱ 0 ( Σ , L 2 ) .

Moreover, since there exists b 1 ∈ ( 0 , + ∞ ) such that

E{ ξ ( u n ( 0 ) , ϕ ( 0 ) e ¯ i ) } ≤ b 1 ∥ u 0 ∥ L 2 ℱ 0 ( Σ , L 2 ) 2 ,

where b 1 = b 1 ( ∥ ξ ∥ L ∞ ( Σ ) , ϕ ( 0 ) ) , then we have

E{ ξ ( u n ( 0 ) , ϕ ( 0 ) e ¯ i ) } → E { ξ ( u 0 , ϕ ( 0 ) e ¯ i ) } , n → ∞ .

We further consider I 2 = E ξ ∫ 0 T ( u n ( t ) , ϕ ̇ e ¯ i ) L 2 d t . By the Hölder inequality, there exists b 2 ∈ ( 0 , + ∞ ) such that

E ξ ∫ 0 T ( u n ( t ) , ϕ ̇ e ¯ i ) L 2 d t ≤ ∥ ξ ∥ L ∞ ( Σ ) ∫ 0 T ( E ∥ u n ( t ) ∥ L 2 ) 1 2 ( E ∥ ϕ ̇ ( t ) e ¯ i ∥ L 2 ) 1 2 d t ≤ b 2 ∥ u n ∥ L ∞ ℱ ( [ 0 , T ] , L 2 ) ,

where b 2 = b 2 ( ∥ ξ ∥ L ∞ ( Σ ) , ∥ ϕ ̇ ∥ L 1 ( [ 0 , T ] ) ) . Combining (12) with the Dominated Convergence Theorem, we get

E ξ ∫ 0 T ( u n ( t ) , ϕ ̇ e ¯ i ) L 2 d t → E ξ ∫ 0 T ( u ˜ ( t ) , ϕ ̇ e ¯ i ) L 2 d t , n → ∞ .

Consider I 3 = E{ ξ ∫ 0 T 〈 ( − Δ ) α u n ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t } . Recalling that p ∈ [ 1 , 2 α ∗ / 2 ) and ϕ e ¯ i ∈ L 2 ( [ 0 , T ] , X 0 ) , we use Lemma 2.2 and (12) to obtain

∫ 0 T 〈 ( − Δ ) α u n ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t = C ( d , α ) 2 ∫ 0 T ∬ ℝ d × ℝ d ( u n i ( t , x ) − u n i ( t , y ) ) ( ϕ ( t ) e ¯ i ( x ) − ϕ ( t ) e ¯ i ( y ) ) | x − y | d + 2 α d x d y d t → C ( d , α ) 2 ∫ 0 T ∬ ℝ d × ℝ d ( u ˜ i ( t , x ) − u ˜ i ( t , y ) ) ( ϕ ( t ) e ¯ i ( x ) − ϕ ( t ) e ¯ i ( y ) ) | x − y | d + 2 α d x d y d t .

Therefore, there exists b 3 ∈ ( 0 , + ∞ ) such that

E ξ ∫ 0 T 〈 ( − Δ ) α u n ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t ≤ b 3 ∥ ξ ∥ L ∞ ( Σ ) .

Thus, we conclude that

E ξ ∫ 0 T 〈 ( − Δ ) α u n ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t → E ξ ∫ 0 T 〈 ( − Δ ) α u ˜ ( t ) , e ¯ i 〉 X 0 ∗ , X 0 ϕ ( t ) d t .

We consider the fourth term I 4 . Since { u n } is bounded in L 2 p ℱ ( [ 0 , T ] , L 2 p ) , Lemma 3.1 implies that

E ξ ∫ 0 T 〈 f ( u n ( t ) ) , ϕ ( t ) e ¯ i 〉 ( L 2 p ) ∗ , L 2 p d t → E ξ ∫ 0 T 〈 f ( u ˜ ( t ) ) , ϕ ( t ) e ¯ i 〉 ( L 2 p ) ∗ , L 2 p d t ,

as n → ∞ .

We further consider I 5 = E ξ ∫ 0 T ( g n ( t ) , ϕ ( t ) e ¯ i ) L 2 d t . As g n is the n-dimensional truncation of g, it is clear that

g n → g in L 2 ℱ ( [ 0 , T ] , L 2 ) .

Using the Dominated Convergence Theorem, we get

E ξ ∫ 0 T ( g n ( t ) , ϕ ( t ) e ¯ i ) L 2 d t → E ξ ∫ 0 T ( g ( t ) , ϕ ( t ) e ¯ i ) L 2 d t .

For I 6 = E{ ξ ∫ 0 T ( ϕ ( t ) e ¯ i , σ ( t ) d W n ( t ) ) } , the fact that W n is the n-dimensional truncation of W leads to

W n → W in Y .

From assumption (H4), we deduce

E ξ ∫ 0 T ( ϕ ( t ) σ ∗ ( t ) e ¯ i , d W n ( t ) ) → E ξ ∫ 0 T ( ϕ ( t ) σ ∗ ( t ) e ¯ i , d W ( t ) ) .

To sum up, because { e ¯ i } is a basis of X 0 , for any ν ∈ X 0 , ϕ ∈ C T 1 ( [ 0 , T ] ) and ξ ∈ L ∞ ( Σ ) , we get

− E ξ ( u ˜ 0 , ϕ ( 0 ) ν ) − E ξ ∫ 0 T ( u ˜ ( t ) , ϕ ̇ ν ) L 2 d t + E ξ ∫ 0 T 〈 ( − Δ ) α u ˜ ( t ) , ν 〉 X 0 ∗ , X 0 ϕ ( t ) d t = E ξ ∫ 0 T 〈 f ( u ˜ ( t ) ) , ϕ ( t ) ν 〉 ( L 2 p ) ∗ , L 2 p d t + E ξ ∫ 0 T ( g ( t ) , ϕ ( t ) ν ) L 2 d t + E ξ ∫ 0 T ( ϕ ( t ) σ ∗ ( t ) ν , d W ( t ) ) .

In other words, u ˜ satisfies formula (5).□

4 The existence of optimal controls

Let U ˜ be a real Hilbert space, U = L ∞ ℱ ( [ 0 , T ] , U ˜ ) be a control functional space and A be a bounded linear operator from U ˜ to L 2 . We consider the following stochastic control problem:

(13) d u + ( − Δ ) α u d t = f ( u ) d t + g d t + A v d t + σ d W , t ∈ ( 0 , T ] , u ( 0 ) = u 0 ,

where u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) , g ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) , v ∈ U and A v ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) . We know that there exists unique weak solution of (13) by Theorem 3.2. So we are allowed to define the solution map as follows:

Ψ : U → L ∞ ℱ ( [ 0 , T ] ; L 2 ) ∩ L 2 ℱ ( [ 0 , T ] ; X 0 ) ∩ L 2 p ℱ ( [ 0 , T ] ; L 2 p ) ; v ↦ u ( v ) ,

where u ( v ) is the state of control problem (13). The observed state is denoted by z ( v ) = N u ( v ) and the desired state is denoted by z d ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) . We consider the cost function with the following form:

(14) J ( v ) = E ∫ 0 T ∥ N u ( v ) − z d ∥ L 2 2 d t + ( K v , v ) U ˜ ,

where N ∈ L ( L 2 ℱ ( [ 0 , T ] , X 0 ) ; L 2 ) and K ∈ L ( U ˜ , U ˜ ) , the operator K satisfies

( K v , v ) U ˜ = ( v , K v ) U ˜ ≥ k ∥ v ( t ) ∥ U ˜ 2 ,

where k ∈ [ 0 , + ∞ ) being a constant. Let U ad ⊂ U be an admissible set. We call v 0 ∈ U ad be the optimal control of J ( v ) if

J ( v 0 ) = min v ∈ U ad J ( v ) .

Thus, we have the following result.

Theorem 4.1

Suppose all assumptions in Theorem 3.2 hold and U ad is a compact subset of U , then stochastic control problem (13) with (14) has at least one optimal control v 0 ∈ U ad .

Proof

Since U ad is compact, it suffices to prove that Ψ is continuous and J is lower semicontinuous. Let { v k } ⊂ U ad and

v k → v ¯ in U ad .

Let { u k } , u ¯ denote the corresponding weak solutions of problem (13) with the same initial state u 0 and g. The proof is divided into two steps.

Step 1. Ψ is continuous.

Since { u k } and u ¯ are weak solutions of (13), u k − u ¯ satisfies

d u k − d u ¯ + ( − Δ ) α u k − ( − Δ ) α u ¯ d t = f ( u k ) d t − f ( u ¯ ) d t + A v k d t − A v ¯ d t

in the weak sense. From integrating by parts, (H3) and Hölder inequality, we get

(15) E ∥ u k − u ¯ ∥ L 2 2 + C ( d , α ) ∫ 0 t E ∥ u k − u ¯ ∥ X 0 2 d s ≤ 2 E ∫ 0 t 〈 f ( u k ) − f ( u ¯ ) , u k − u ¯ 〉 ( L 2 p ) ∗ , L 2 p d s + 2 E ∫ 0 t ( A v k − A v ¯ , u k − u ¯ ) L 2 d s ≤ 2 E ∫ 0 t ( A v k − A v ¯ , u k − u ¯ ) L 2 d s ≤ 2 E ∫ 0 t ∥ A ( v k − v ¯ ) ∥ L 2 2 p 2 p − 1 d s 2 p − 1 2 p E ∫ 0 t ∥ u k − u ¯ ∥ L 2 2 p d s 1 2 p .

On the other hand, since weak solutions satisfy inequality (11), { u k } is bounded in L ∞ ℱ ( [ 0 , T ] ; L 2 ) ∩ L 2 ℱ ( [ 0 , T ] ; X 0 ) ∩ L 2 p ℱ ( [ 0 , T ] ; L 2 p ) . Therefore, there exists a constant M > 0 such that

max { ∥ u ¯ ∥ L 2 p ℱ ( [ 0 , T ] ; L 2 p ) , ∥ u k ∥ L 2 p ℱ ( [ 0 , T ] ; L 2 p ) , k ∈ ℕ } ≤ M .

Thus, from inequality (15) and Lemma 2.2, there exist c > 0 such that

E ∥ u k − u ¯ ∥ L 2 2 + C ( d , α ) ∫ 0 t E ∥ u k − u ¯ ∥ X 0 2 d s ≤ 4 M c E ∫ 0 t ∥ A ( v k − v ¯ ) ∥ L 2 2 p 2 p − 1 d s 2 p − 1 2 p .

Taking to the limit, we obtain

lim k → ∞ sup t ∈ [ 0 , T ] E ∥ u k − u ¯ ∥ L 2 2 + C ( d , α ) ∫ 0 t E ∥ u k − u ¯ ∥ X 0 2 d s = 0 .

Furthermore, we deduce that

u k → u ¯ in L ∞ ℱ ( [ 0 , T ] ; L 2 ) ∩ L 2 ℱ ( [ 0 , T ] ; X 0 ) .

This proves that Ψ is continuous.

Step 2. J is lower semi-continuous.

First, write

J ( v ) = E ∫ 0 T ∥ N u ( t , v ) − z d ( t ) ∥ L 2 2 d t + E { ( K v ( t ) , v ( t ) ) U ˜ } ≜ J 1 ( v ) + J 2 ( v )

and

J 1 ( v ¯ ) = E ∫ 0 T ∥ N u ¯ ( t , v ) − z d ( t ) ∥ L 2 2 − ∥ N u k ( t , v ) − z d ( t ) ∥ L 2 2 d t + E ∫ 0 T ∥ N u k ( t , v ) − z d ( t ) ∥ L 2 2 d t .

Since N ∈ L ( L 2 ℱ ( [ 0 , T ] , X 0 ) , L 2 ) , for any ϵ > 0 , there exists N ϵ , when k > N ϵ ,

∥ N u ¯ ( t , v ) − z d ( t ) ∥ L 2 2 − ∥ N u k ( t , v ) − z d ( t ) ∥ L 2 2 < ϵ .

Thus, we get

J 1 ( v ¯ ) ≤ T ϵ + E ∫ 0 T ∥ N u k ( t , v ) − z d ( t ) ∥ L 2 2 d t = T ϵ + J 1 ( v k )

if k > N ϵ . So we arrive at

J 1 ( v ¯ ) ≤ lim ̲ k → ∞ J 1 ( v k )

by the arbitrariness of ϵ .

Finally, according to v k → v ¯ in U , we deduce v k ( t ) → v ¯ ( t ) , a.e., for all t ∈ [ 0 , T ] . Hence, { v k ( t ) } is a bounded subset of U ˜ . Since K ∈ L ( U ˜ , U ˜ ) ,

l i m ̲ k → ∞ ( K v k ( t ) , v k ( t ) ) U ˜ ≤ ( K v ¯ ( t ) , v ¯ ( t ) ) U ˜ , a .s .

Due to the Fatou lemma, we only need to show

(16) lim ̲ k → ∞ E { ( K v k ( t ) , v k ( t ) ) U ˜ } < ∞ .

Following

( K v k ( t ) , v k ( t ) ) U ˜ ≤ ∥ K v k ( t ) ∥ U ˜ ∥ v k ( t ) ∥ U ˜ ≤ ∥ K ∥ ∥ v k ( t ) ∥ U ˜ 2 , a .s . ,

inequality (16) holds. Thus, we get that

J 2 ( v ¯ ) ≤ lim ̲ k → ∞ J 2 ( v k ) .

To sum up, J attains its minimum at v ¯ .□

5 Examples

In this section, we give some sufficient conditions and two examples to illustrate the results. Assume that

(A1): f ∈ C ( ℝ N × Ω ¯ × [ 0 , T ] , ℝ N ) ;
(A2): There exists an M ∈ ( 0 , + ∞ ) such that
| f ( ξ , x , t ) | ≤ M | ξ | 2 p − 1 , ∀ ( ξ , x , t ) ∈ ℝ N × Ω × [ 0 , T ] ;
(A3): There exists a θ ∈ ( 0 , + ∞ ) such that
f ( ξ , x , t ) ξ ≤ − θ | ξ | 2 p , ∀ ( ξ , x , t ) ∈ ℝ N × Ω × [ 0 ; T ] ;
(A4): For any ξ , η ∈ ℝ N , x ∈ Ω and t ∈ [ 0 , T ] ,

( f ( ξ , x , t ) − f ( η , x , t ) ) ( ξ − η ) ≤ 0 .

Thus from (A2) and (A3), we obtain

〈 f ( u ) , u 〉 ( L 2 p ) ∗ , L 2 p ≤ − θ ∥ u ∥ L 2 p 2 p , ∥ f ( u ) ∥ ( L 2 p ) ∗ ≤ M ∥ u ∥ L 2 p 2 p − 1 ,

then f satisfies (H1) and (H2). Moreover by (A1) and (A4), we get

〈 f ( u ) − f ( v ) , u − v 〉 ( L 2 p ) ∗ , L 2 p ≤ 0 ,

that is, f satisfies (H3). So we deduce the following result.

Corollary 5.1

Let α ∈ ( 0 , 1 ) , 2 α < d and p ∈ [ 1 , 2 α ∗ / 2 ) . If assumptions (A1)–(A4) and (H4) hold, then system (1) has unique weak solution u for any g ∈ L 2 ℱ ( [ 0 , T ] , L 2 ) and u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) .

Remark 5.2

Under the assumptions in Corollary 5.1, the result of Section 4 remains valid.

Next, two examples of optimal control problems are provided as follows. These examples are inspired by [12].

Example 5.3

Let Ω = B 0 be a unit ball in ℝ 3 and U ˜ = L 2 ( B 0 ) . We consider the control problem as follows:

(17) d u + ( − Δ ) α u d t = − ψ u 2 p − 1 d t + v d t + ( 1 + t ) k 0 d W , t ∈ ( 0 , T ] , x ∈ B 0 , u ( t ) = 0 , t ∈ [ 0 , T ] , x ∈ ℝ 3 \ B 0 , u ( 0 ) = a 0 e − b 0 | x | , x ∈ B 0 ,

where k 0 , a 0 and b 0 are strictly positive constants, ψ = ψ ( x , t ) ≥ 0 is a deterministic function and ψ ∈ L ∞ ( [ 0 , T ] × B 0 ) . The cost function is

J ( v ) = E ∫ 0 T ∥ u ( v ) − z d ( t ) ∥ L 2 2 d t .

We check that f = − ψ u 2 p − 1 satisfies (A1)–(A4), σ = ( 1 + t ) k 0 satisfies (H4) and u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) . A ∈ L ( U ˜ , L 2 ) is an identity, N is an injection from X 0 into L 2 and K is a null operator. Therefore, using Corollary 5.1 and Remark 5.2, we obtain that optimal control problem (17) has at least one optimal control.

Example 5.4

Let Ω = B 0 be a unit ball in ℝ 3 , C = ∂ B 0 and U ˜ = L 2 ( C ) . Consider the following control problem:

(18) d u + ( − Δ ) α u d t = − γ 0 | u | 2 p − 2 u d t + A v d t + 1 1 + t d W , t ∈ ( 0 , T ] , x ∈ B 0 , u ( t ) = 0 , t ∈ [ 0 , T ] , x ∈ ℝ 3 \ B 0 , u ( 0 ) = a 0 e − b 0 | x | , x ∈ B 0 ,

where γ 0 , a 0 and b 0 are strictly positive constants and A is given by

( A v , u ) = ∫ C u v d S .

The cost function J is chosen to be

J ( v ) = E ∫ 0 T ∂ ∂ n u ( v ) − z d ( t ) L 2 ( C ) 2 d t + λ ∫ C | v ( t ) | 2 d S ,

where λ is a positive parameter.

It is easy to see that f = − γ 0 | u | 2 p − 2 u satisfies (A1)–(A4) and σ = 1 1 + t satisfies (H4) and u 0 ∈ L 2 ℱ 0 ( Σ , L 2 ) . By simple calculations, we compute that the observed state ∂ ∂ n u ( v ) ∈ L 2 ℱ ( [ 0 , T ] , L 2 ( C ) ) . So we get the existence of optimal controls for optimal control problem (18) by Corollary 5.1 and Remark 5.2.

Acknowledgment

The authors are supported by the National Natural Science Foundation of China (Grant No. 11771107).

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Received: 2020-04-11

Revised: 2020-07-15

Accepted: 2020-07-15

Published Online: 2020-11-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

Abstract

1 Introduction

2 Preliminaries

Lemma 2.1

Lemma 2.2

Lemma 2.3

Definition 2.4

3 The existence and uniqueness of weak solutions

Lemma 3.1

Proof

Theorem 3.2

Proof

4 The existence of optimal controls

Theorem 4.1

Proof

5 Examples

Corollary 5.1

Remark 5.2

Example 5.3

Example 5.4

Acknowledgment

References

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