Finite-approximate controllability of fractional stochastic evolution equations with nonlocal conditions

This paper deals with the finite-approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We establish sufficient conditions for the finite-approximate controllability of the control system when the compactness conditions or Lipschitz conditions for the nonlocal term and uniform boundedness conditions for the nonlinear term are not required. The discussion is based on the fixed point theorem, approximation techniques and diagonal argument. In the end, an example is presented to illustrate the abstract theory. Our result improves and extends some relevant results in this area.


Introduction
In this paper, we shall be concerned with the finite-approximate controllability (see Definition 2.8 in Sect. 2) for following fractional evolution equations with nonlocal initial conditions of the form x(t)) + σ (t, x(t)) dW (t) dt + Bu(t), t ∈ [0, b], x(0) = g(x), (1.1) where c D α t is the Caputo fractional derivative of order 1 2 < α ≤ 1, and confusion. We are also employing the same notation · for the norm of L(K, H), which denotes the space of all bounded linear operators from K into H. Suppose that {W (t) : t ≥ 0} is a K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q ≥ 0 defined on a filtered complete probability space (Ω, F, {F t } t≥0 , P). The control function u(·) belongs to the space L 2 F (J, U), a Banach space of admissible control functions, for a separable Hilbert space U, B : U → H is a bounded linear operator, while f , σ and h are appropriate functions to be given later.
It is well known that nonlocal problems have better properties in applications than the classical ones, so differential equations with nonlocal initial conditions have been studied by many authors, see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein. However, in order to establish the main results, the assumption of compactness or Lipschitz condition on the nonlocal term plays an important role in these articles. But this restriction is too strong and is not usually satisfied in practical applications.
Recently, Liang, Liu, and Xiao [14] investigated the existence of mild solutions for a class of nonlocal Cauchy problem under the hypothesis that the nonlocal term g satisfies the condition (H) For any u, w ∈ C([0, b]; E), there exists a constant δ ∈ (0, b) such that u(t) = w(t) (t ∈ [δ, b]) implies g(u) = g(w). Note that assumption (H) is for the case when the values of the solution x(t) for t near zero do not affect g (x). With the help of assumption (H), the authors relaxed the compactness and Lipschitz continuity on the nonlocal item g.
On the other hand, controllability for various linear and nonlinear dynamical systems have been considered in many publications by using different approaches due to its applications in many fields of science and engineering, see [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. It should be emphasized that there are many different notions of controllability for dynamical systems, for example, approximate controllability, exact controllability, null controllability, and so on. There have been many papers on the approximate controllability for semilinear evolution systems in abstract spaces, see [16][17][18][19][20][21]32] and the references therein. Several authors have studied exact controllability for differential control systems, see [22][23][24][25] and the references therein. It is worth mentioning that [15] studied simultaneous approximate and finite-dimensional exact controllability (finite-approximate controllability) of the following control system: (1.3) where f : [0, T] × X → X, g : C([0, T], X) → X, and X is a Hilbert space. In this paper, finiteapproximate controllability means that system (1.3) is approximately controllable in X as well as exactly controllable in a finite dimensional subspace E ⊂ X. The author obtained sufficient conditions for the finite-approximate controllability of system (1.3) when the nonlocal term g satisfies Lipschitz-type conditions and the nonlinear term f satisfies a growth condition.
In recent years, stochastic differential equations have attracted great interest due to their successful applications to problems in mechanics, electricity, economics, physics, and several fields in engineering. For details, see [33][34][35][36][37][38][39] and the references therein. In particular, some researchers investigated controllability of stochastic dynamical control systems in infinite-dimensional spaces, see [26][27][28][29][30][31]. However, there are few works that have reported about the study of controllability problems for stochastic evolution equations with nonlocal conditions, see [30,31], and the authors suppose that the nonlocal item g is a completely continuous map in these papers.
Inspired by the above discussions, especially [15], in this work, we will study the finite-approximate controllability for (1.1). The first novelty of this article is the finiteapproximate controllability which is a stronger version of the controllability concept. Up to now, no one has studied the finite-approximate controllability for a stochastic system, this paper fills this gap in the literature. The second novelty of this article is that the nonlocal term g(x) defined by (1.2) depends on all values of x on the whole interval [0, b], so the methods used in [14] are not valid for the present paper. By using stochastic analysis, approximation techniques, diagonal argument, and Schauder fixed-point theorem, the finite-approximate controllability results are established under weaker conditions in which g(x) is not necessarily Lipschitz continuous or has some compactness property. More precisely, the nonlocal term g(x) depends on all the values of x in the whole interval [0, b], is only continuous and satisfies some weak growth condition. The third novelty of this article is that in almost all the articles on the topic of approximate controllability, for example, see [16][17][18][19][20][21][28][29][30][31], the authors always require the nonlinear term f be uniformly bounded. In the present work we delete this restriction, and only need the nonlinear term which satisfies some natural growth conditions. So the theorems obtained here extend and complement those obtained in [16][17][18][19][20][21][28][29][30][31]. In addition, as a special case, the methods used in the present paper can be applied to study the finite-approximate controllability of deterministic systems with nonlocal conditions by suitably introducing the abstract space and norm. The corresponding results that appear are also new.
We organize the paper in the following way: In Sect. 2, we introduce some useful definitions and preliminary results to be used in this paper. In Sect. 3, we state and prove finite-approximate controllability results for fractional stochastic evolution equation with nonlocal conditions. Finally, in Sect. 4, an example is provided to illustrate the applications of the obtained results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this article.
Let (Ω, F, {F t } t≥0 , P) be a filtered complete probability space satisfying the usual conditions, which means that the filtration is a right-continuous increasing family and F 0 contains all P-null sets. Let {e k , k ∈ N} be a complete orthonormal basis of K. We denote by {W (t) : t ≥ 0} a cylindrical K-valued Brownian motion or Wiener process defined on the probability space (Ω, F, {F t } t≥0 , P) with a finite trace nuclear covariance operator Q ≥ 0, and we let Tr(Q) = ∞ k=1 λ k = λ < ∞, which implies that Qe k = λ k e k , k ∈ N. Let {W k (t), k ∈ N} be a sequence of one-dimensional standard Wiener processes mutually independent on (Ω, F, {F t } t≥0 , P) such that Furthermore, we assume that F t = σ {W (s), 0 ≤ s ≤ t} is the σ -algebra generated by W and Let L 0 2 = L 2 (Q 1 2 K, H) denote the space of all Hilbert-Schmidt operators from Q 1 2 K into H with the inner product φ, ϕ = Tr(φQϕ * ). It also turns out to be a separable Hilbert space. The collection of all F b -measurable, square-integrable H-valued random variables, denoted L 2 (Ω, H), is a Banach space equipped with the norm x L 2 = (E x(ω) 2 ) 1 2 , where E denotes the expectation with respect to the measure P. Let C([0, b], L 2 (Ω, H)) be the Banach space of all continuous mappings from The theory of stochastic integrals in Hilbert space can be found in [37,39].
In the rest of the manuscript, we suppose that A generates a compact C 0 -semigroup T(t) (t ≥ 0) of uniformly bounded linear operators in H. That is, there exists a positive constant M ≥ 1 such that T(t) ≤ M for all t ≥ 0. For any constant r > 0, let By [40, Proposition 2.8], we have the following result which will be used throughout this paper.

Lemma 2.6
The operators T α (t) (t ≥ 0) and S α (t) (t ≥ 0) satisfy the following properties: (i) 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,

also compact operators in
H for t > 0, and hence they are norm-continuous.
In this paper, we adopt the following definition of the mild solution of (1.1).

Definition 2.7 For any given
is measurable and adapted to F t ; (ii) x(t) satisfies the following integral equation: Let E be a finite-dimensional subspace of L 2 (Ω, H) and denote by π E the orthogonal projection from This means that the control u ε can be chosen such that x(b; u ε ) satisfies (2.4) and simultaneously a finite number of constraints, that is, condition (2.5).
To prove the main result, we need the following restrictions: and a nondecreasing continuous function (H4) The linear fractional differential system It is known that system (2.6) approximately controllable on [0, b] if and only if the condition

Lemma 2.9
Suppose that Assumptions (H1)-(H3) are satisfied. Then the following conditions hold: Moreover, for any t ∈ [0, b], using Hölder inequality and Lebesgue dominated convergence theorem, we can get On the other hand, from Lemma 2.1, Hölder inequality and Lebesgue dominated convergence theorem, we obtain Meanwhile, by (H3), we see that According to the inequality obtained above, we obtain the following relation: Therefore, p is continuous in B r . Next, we prove (ii). For all ∈ (0, b) and all ν > 0, define an operator F ,ν on B r by the formula for any x ∈ B r . Applying (H1)-(H3), Lemmas 2.1 and 2.6, and Hölder inequality, we have Therefore, there are relatively compact sets arbitrarily close to the set { (F 1 x) This completes the proof of Lemma 2.9.

Lemma 2.10
Suppose that Assumptions (H1)-(H4) are satisfied. Then the functional J ε satisfies the following properties: x) is continuous and strictly convex; (ii) For any r > 0, . Evidently, φ n L 2 = 1, so we can extract a subsequence (still denoted by φ n ), which weakly converges to an element φ in H. By the com- Observe that (2.12) and Fatou lemma implies By (H4), we get φ = 0, that is, φ n weakly converges to 0 in H. As E is finite-dimensional, the orthogonal projection π E is compact. Moreover, we have Therefore, which contradicts (2.12). So (ii) holds. This completes the proof of Lemma 2.10.

Lemma 2.11
If Assumptions (H1)-(H4) are satisfied, then the following conclusions hold: (i) For any x ∈ B r , there exists a constant L ε > 0 such that E Φ ε (x) 2 ≤ L ε ; (ii) For any x n , x ∈ B r satisfying x n → x in H([0, b], L 2 (Ω, H)), it holds that Proof (i) By Lemma 2.10, there exists L ε > 0 such that On the other hand, by the definition of Φ ε , (2.14) Therefore, by (2.13) and (2.14), we obtain that . By (i), we know that { φ ε,n } is bounded, thus we suppose that φ ε,n weakly converges to φ ε in H. According to the definition of J ε , Fatou lemma and the optimality of both φ ε,n = Φ ε (x n ) and φ ε = Φ ε (x), we have , that is, φ ε is also a minimum of J ε (·, x). By the uniqueness of the minimum of J ε (·, x), we get φ ε = φ ε . Moreover, we have From these relations, we easily see that Since H is a Hilbert space, by (2.16) and since φ ε,n φ ε weakly in H, we have This completes the proof of Lemma 2.11. Now, we introduce a control u ε (t, x) by From Lemma 2.11, we can get the following obvious result:

Lemma 2.12
If Assumptions (H1)-(H4) are satisfied, then for any x ∈ B r , the following conclusions hold: To discuss the finite-controllability for system (1.1), we need the following lemmas in this paper.

Lemma 2.13 Assume that -A generates a compact C 0 -semigroup T(t) (t ≥ 0) of uniformly bounded operators in a Hilbert space H. Let Assumptions (H1)-(H4) hold. Suppose, in addition, that the following condition is satisfied:
. Then the nonlocal problem (1.1) has at least one mild solution in B R provided that there exists a positive constant R such that Proof For any r > 0, define It is easily seen that for each x ∈ B r (δ), there exists a function y ∈ B r satisfying x(t) = y(t), t ∈ [δ, b]. Define the following mappings on B r (δ) by Then, by conditions (H1)-(H3) and (H5), it is easy to see that f * , σ * , g * is well defined on B r (δ) and continuous. In addition, (2.18) Define an operator F δ on B r (δ) as follows: Evidently, the results in Lemma 2.12 hold for u * ε (s, x). Next we prove that F δ has a fixed point by Schauder's fixed point theorem. For this purpose, we first check that there is a positive number R such that F δ maps B R (δ) into itself. For any x ∈ B R (δ) and t ∈ [δ, b], it follows from (2.17), (2.18), Lemmas 2.1 and 2.6, and Hölder inequality that It then follows that F δ maps B R (δ) to B R (δ).
Secondly, with a method similar to that in the proof of Lemma 2.9, we can also prove that F δ : B R (δ) → B R (δ) is a continuous operator and the set In what follows, we will show that F δ (B R (δ)) is an equicontinuous family of functions on [δ, b]. For any x ∈ B R (δ) and δ ≤ t 1 < t 2 ≤ b, we get that = I 0 + I 1 + I 2 + I 3 + I 4 + I 5 + I 6 .
For each δ ∈ (0, b) and arbitrary x ∈ H([0, b], L 2 (Ω, H)), write and It is easy to see that f δ , σ δ , and h δ defined above satisfy condition (H5), thus we obtain  (2.20) has at least one mild solution in B R provided that there exists a positive constant R such that (2.17) is satisfied.

Main results
In this section, we shall discuss the finite-approximate controllability of the fractional stochastic dynamical control system (1.1) by using the approximation techniques and a diagonal argument. Proof To begin with, let {δ n : n ∈ N} be a decreasing sequence in (0, b) with lim n→∞ δ n = 0.
For every n, according to Lemma 2.14, we claim that the following system: has a mild solution x n ∈ B R if constant R satisfies (2.17), which is expressed by then v n ∈ B R . In view of the definitions of f δ n , σ δ n , and h δ n , we conclude that Next, we will show that the set {x n : n ∈ N} is precompact in H([0, b], L 2 (Ω, H)). For this purpose, we introduce the following definition: Therefore, we only need to show that the sets {ρ n : n ∈ N} and {ϕ n : n ∈ N} are precompact in H([0, b], L 2 (Ω, H)). From the expression of v n (t), we know that v n ∈ B R . This implies that (H1)-(H3) hold for f (s, v n (s)), σ (s, v n (s)) and h(s, v n (s)). Moreover, u ε (s, v n ) satisfies the estimates (i) and (ii) in Lemma 2.12. Hence, it is not difficult to prove that the set {ϕ n : n ∈ N} is precompact in H([0, b], L 2 (Ω, H)) by the arguments similar to those in the proof of Lemma 2.13. In the sequel, we will show that the set {ρ n : n ∈ N} is also precompact in H([0, b], L 2 (Ω, H)). In fact, we only need to prove that the set { b 0 h(s, v n (s)) ds : n ∈ N} is precompact in H([0, b], L 2 (Ω, H)).
Let {η n : n ∈ N} be a decreasing sequence in (0, b) such that lim n→∞ η n = 0. For every n ∈ N and t ∈ [η 1 , b], define function ω n : [η 1 , b] → H by ω n (t) = x n (t). Note that v n ∈ B R and hence, { b 0 h(s, v n (s)) ds : n ∈ N} is bounded. Meanwhile, T α (t) is compact and norm-continuous for t > 0, which implies that the set {T α (t) b 0 h(s, v n (s)) ds : n ∈ N} is precompact in H for any t ∈ [η 1 , b] and {T α (·) b 0 h(s, v n (s)) ds : n ∈ N} is equicontinuous. By Arzela-Ascoli theorem, we conclude that {T α (·) b 0 h(s, v n (s)) ds : n ∈ N} is precompact in H([η 1 , b], L 2 (Ω, H)). Combining this with the fact that {ϕ n : n ∈ N} is precompact in H([0, b], L 2 (Ω, H)), we claim that {ω n : n ∈ N} is precompact in H([η 1 , b], L 2 (Ω, H)). Hence, we can find a subsequence {x 1 n : n ∈ N} ⊂ {x n : n ∈ N} which is a Cauchy sequence in H ([η 1 , b], L 2 (Ω, H)). In the same way, we can select a subsequence {x 2 n : n ∈ N} ⊂ {x 1 n : n ∈ N} which is a Cauchy sequence in H([η 2 , b], L 2 (Ω, H)). Repeating the above reasoning and applying a diagonal argument, we know that there exist a subsequence {x * n : n ∈ N} ⊂ {x n : n ∈ N} which is a Cauchy sequence in H([η n , b], L 2 (Ω, H)). Moreover, for every t ∈ (0, b], {x * n (t) : n ∈ N} is a Cauchy sequence in H. So there exists a continuous function x * : (0, b] → L 2 (Ω, H) such that for each η k , We further show that {g(x * n ) : n ∈ N} is a Cauchy sequence in H. Let δ ∈ (0, b), then for any Therefore, for ∀ε > 0, there exists a positive constant δ 0 < b such that for any x 1 , Let y(t) be the function defined by From the definition of L δ , we can see easily that By the continuity of g, we can find a natural number N such that Therefore, for any m, n > N , we have By this inequality, for λ > 0, we have Setting λ → 0 + in the inequality above, we have Similarly to the process above with λ < 0, we get from the definition of p(x ε ) and (3.8)-(3.9), for any ψ ∈ H, it follows that By this and the properties of the orthogonal projection, we have Therefore, the fractional stochastic control system (1.1) is finitely-approximately controllable on [0, b]. This completes the proof of Theorem 3.3. The corresponding results that appear are also new.