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

Lemma 2.1

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

Lemma 2.2

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

Thus, there exists c 0 > 0 such that

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

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 :

1. W is a Gaussian process on Y and

   E ( W ( t ) , y ) Y = 0 , E ( W ( t ) , y ) Y 2 = t ( Q y , y ) .

2. W has the following expression:

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 ] .

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

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

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

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

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

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

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

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

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

holds by the Dominated Convergence Theorem.

Now let us verify

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,

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

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

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

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

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

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

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):

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:

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

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

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

that is, the process

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

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

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

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

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