APPROXIMATION OF FRACTIONAL RESOLVENTS AND APPLICATIONS TO TIME OPTIMAL CONTROL PROBLEMS∗

We investigate the approximation of fractional resolvents, extending and improving some corresponding results on semigroups and resolvents. As applications, we utilize the approach of Meyer approximation to analyze the time optimal control problem of a Riemann-Liouville fractional system without Lipschitz continuity. A fractional diffusion model is also presented to confirm our theoretical findings.


Introduction
Since the notion of resolvent was firstly proposed and studied by Da Prato and Iannelli in [13,14], there has been considerable interest in introducing and analyzing the notions related to resolvent, such as solution operators, (a, k)-regularized families, α-order fractional resolvents and so on (refer to [6,8,15]). Recently, Li and Peng [6] investigated a Riemann-Liouville fractional evolution problem by introducing the notion of α-order fractional resolvent.
As we all know, the approximation of semigroups is of great importance in the study of optimal control problems (see [17,18]). Many researchers thereby show tremendous interest in analyzing the approximation of semigroups and resolvents. For example, the approximation of solution operators was analyzed in [1]. The approximation of (a, k)-regularized families was studied in [9]. However, limited work has been done in the approximation of α-order fractional resolvents. Considering that the resolvent technique is a convenient and efficient approach in studying fractional evolution systems [6,[22][23][24], we will investigate the approximation of fractional resolvents. The main difficulty in the study is that these resolvents possess singularity at zero. In this article, by introducing a new concept of exponential boundedness for s ≥ s 0 and constructing resolvents with parameters, we explore the approximation problem.
On the other hand, time optimal control problems for evolution systems have recent years drawn tremendous attention based upon their broad applications in many fields, such as control theory, industrial application and space technology, etc. Many researchers analyzed them by setting up time optimal sequence pairs (see [4,5,7,10,19]). Recently, with the aid of the Lipschitz assumption on the nonlinear term f and the approximation of semigroups, the problems were tackled by Meyer approximation (refer to [16][17][18]). That is, the authors formulated a sequence of Meyer problems to approximate the time optimal control problems. Naturally, one may ask how to analyze the problems for Riemann-Liouville evolution systems by Meyer approximation if we remove the Lipschitz continuity. Therefore, the present paper is intended to conduct some investigations on time optimal control problems for a Riemann-Liouville fractional evolution system by Meyer approximation, when the Lipschitz condition is not satisfied. In this paper, we first transform the original fractional evolution system into an approximate system and propose a Meyer problem. Then, we deal with the Meyer problem by constructing minimizing sequences twice. Finally, we analyze the time optimal problem by Meyer approximation.
The following enumerates two aspects of novelties of this paper: 1) Considering that the fractional resolvent has singularity at zero, we introduce a new concept of exponential boundedness for s ≥ s 0 . Furthermore, we analyze the approximation of fractional resolvents by setting up resolvents with parameters.
2) We combine the approximation of fractional resolvents and the time control problem organically. In addition, the new method of constructing minimizing sequences twice is employed to compensate the lack of Lipschitz assumption, when addressing the Meyer problem.
The outline of this article is as follows. We present some preliminaries in Section 2. Section 3 deals with the approximation of fractional resolvents. As applications, in Section 4, we treat the time optimal control problem of a Riemann-Liouville fractional system by Meyer approximation and present an example on a Riemann-Liouville fractional partial differential system to confirm our theoretical findings.

Preliminaries
We compile here some preliminaries, including the notions and facts about fractional resolvents. Let V be a Banach space and Y a reflexive separable Banach space. From now on, unless otherwise stated, we assume that 0 < β < 1. Set J = [0, T ], J ′ = (0, T ] and represent the collection of all bounded linear operators from Y to V and L (V ) denote L (V ; V ). Furthermore, we utilize the symbol P f c (Y ) to stand for a class of nonempty closed convex subset of Y and the notation * to mean the convolution, i.e., (f * g)(s) = s 0 f (s − τ )g(τ )dτ . Definition 2.1 ( [6]). By a β-order fractional resolvent, we understand a strongly continuous family where g β (τ ) = τ β−1 Γ(β) , τ > 0 and the notation J β τ stands for the β-order fractional integral operator, i.e., J β τ f (τ ) = (g β * f )(τ ) (see [12]). In addition, the generator A :
Proof. With the help of Lemma 2.2, we can easily verify the statement of this lemma by following the verification of Lemmas 3.1 and 4.2 in [23].

Approximation of fractional resolvents
This section is intended to display some approximation theorems of the fractional resolvent {R β (s)} s>0 .
As we all know, any C 0 -semigroup is (M, ω) type. However, the resolvent {R β (s)} s>0 is not (M, ω) type since it has singularity at zero. Now, we introduce the following new definition: For convenience, we employ the symbol A ∈ C β s0 (M , ω) to mean that A generates a resolvent {R β (s)} s>0 satisfying (3.1).
. Then by using (3.1) and Remark 2.2, it is easy to show that R β (s) is Laplace transformable for λ > ω. In fact, for t > s 0 , we have In such a case, {R β (s)} s>0 is a resolvent generated by A.
Proof. (Necessity) Let A ∈ C β s0 (M , ω). Then (a), (b) and (c) hold. Based upon Remark 3.1, R β (s) is Laplace transformable for λ > ω. Set Due to (a) and (b) of Lemma 2.1, we can employ Laplace transform to get that for any z ∈ D(A), Thus, according to (d) of Lemma 2.1, we can deduce that for λ > ω and z ∈ V , (Sufficiency) We suppose that (a), (b), (c) and (d) hold. In view of the resolvent identity, we get Hence, by utilizing inverse Laplace transform, we derive Proof. Due to Lemma 2.1, we have Proof. For clarity, we verify this lemma by considering the following two cases.
, we can easily derive the strong continuity of {R k β (s)} s>0 and the commutativity of R k β (s) and R k β (τ ). Moreover, we can see Thus, based on Lemmas 3.1 and 3.2, {R k β (s)} s>0 is a resolvent generated by kA. Case 2 k = 0. Firstly, we see at once that {R k β (s)} s>0 is strongly continuous, where the notation 0 stands for zero operator.
Here the notation s → means the strong operator topology.
Proof. We consider the following cases. Case 1 k n = 0. In view of Lemma 3.3, the proof is immediate. Case 2 k n > 0 and k ε = 0. Let z ∈ V . If t = 0, according to Lemmas 3.1 and 3.3, we derive If t > 0, we get On the other hand, due to k ε = 0 and Lemma 3.3, we have Firstly, the fact that k n → k ε , n → ∞ implies the boundedness of {k n }. In addition, if s 0 ∈ (0, tk 1 β ε ), by employing the strong continuity of {R β (t)} t>0 , the exponential boundedness for t ≥ s 0 and k n → k ε , one can easily see that (3.2) holds. If s 0 ≥ tk 1 β ε , by utilizing the boundedness of tk ε , s 0 and the strong continuity of {R β (t)} t>0 , we can easily conclude that (3.2) holds. As Hence, it is easily seen that where the notation τu → stands for the uniform operator topology.
Proof. It follows from lim n→∞ k n = k ε , k n , k ε ∈ (0, +∞) and Thus, due to the assumption on uniformly in t, which establishes the conclusion. Proof. Since k > 0 and t 1−β R k β (t) = tk , we can see that is compact and equicontinuous, by the compactness and equicontinuity of {t 1−β R β (t)} t>0 .

Remark 3.2.
With the aid of the resolvent properties, we have explored the approximation of resolvents by introducing the notion of exponential boundedness for s ≥ s 0 and constructing resolvents with parameters. Emphasis here is that our method differs from the approach in [11,17]. Moreover, our technique can also be applied to the case of C 0 -semigroups. However, the question whether the Trotter-Kato type approximation theorem holds for fractional resolvents is at present far from being solved, since these resolvents have singularity at zero.

Time optimal control problems
In this section, with the help of the approximation theory in Section 3, we deal with the time optimal control problem of a Riemann-Liouville fractional evolution system. We first propose the time optimal control problem (P ) of a Riemann-Liouville fractional control system and the Meyer problem (P ε ) of a transformation system. Then, we tackle the Meyer problem (P ε ) by constructing minimizing sequences twice. Finally, we deal with the problem (P ) by Meyer approximation.
Consider the following evolution control system with a β-order Riemann-Liouville fractional derivative: (HB) B ∈ L ∞ (J, L (Y, V )). Furthermore, we introduce an admissible control set is a measurable multi-valued mapping, G ⊆ Y is a bounded set and U (·) ⊆ G. Thanks to Proposition 2.1.7 in [3], we know that V ad ̸ = ∅. In addition, condition (HB) indicates that for all v ∈ V ad , Bv ∈ L p (J; V ).
Due to Lemma 2.3 and our previous work ( see Lemma 3.2 in [22]), we introduce the notion of mild solutions to system (4.1).

Definition 4.1.
For fixed v ∈ V ad , by a mild solution to system (4.1) associated with v, we mean the function z ∈ C 1−β (J; V ) satisfying For convenience, put S(v) = z ∈ C 1−β (J; V ) : z is a mild solution to (4.1) depending on v ∈ V ad . By the standard technique utilized for Riemann-Liouville fractional evolution systems (see Theorem 3.1 in [22]), we exhibit the existence result of (4.1).
For simplicity, set x is a mild solution of (4.2) related to w ∈ W .
By means of Theorem 4.1, we can establish the following existence result: Then, we analyze the following Meyer problem (P ε ) of system (4.2): Seek a state-control pair (x ε , w ε ) to guarantee that To treat the Meyer problem (P ε ), we need the following lemma: Since the uniqueness of the solutions cannot be acquired, the method of setting up a minimizing state-control pair sequence in [16][17][18] breaks down. We now utilize a new technique of establishing minimizing sequences twice to deal with the Meyer problem (P ε ). Proof. For clarity, we split the proof into the following procedures.
Step 1 For fixed w ∈ W , we will seek x ∈ S(w) to ensure that Below, we consider the following two cases.
is a constant, hence that the proof is obvious. Case 2 k > 0. Since it is trivial for the two cases when J ε (w) = +∞ or the solution set S(w) possesses only finite elements, we can suppose that J ε (w) < +∞. Thus, we can take {x n } n≥1 ∈ S(w) to guarantee that lim n→∞ J ε (x n , w) = J ε (w). By {x n } n≥1 ∈ S(w), we get that for s > 0, Since {s 1−β R k β (s)} s>0 is compact and equicontinuous, we can obtain the compactness of {x n } n≥1 in C k β ([0, 1]; V ). This follows by the same argument as in Step 3 of Theorem 3.1 in [22]. As such, we can choose x ∈ C k β ([0, 1]; V ) and a subsequence extracted from {x n } n≥1 , still written {x n } n≥1 , such that lim n→∞ x n = x. Therefore, by letting n → ∞ on both sides of (4.3), we conclude from the dominated convergence theorem that hence that x ∈ S(w), finally that This indicates that J ε (x, w) = J ε (w).
Step 2 We shall look for w ε satisfying J ε (w ε ) = inf w∈W J ε (w). For simplicity of notation, set m ε = inf w∈W J ε (w). We only need to consider the case m ε < +∞, since the case m ε = +∞ is trivial. Due to m ε < +∞, we can pick {w n } n≥1 ⊆ W to ensure that lim Based on the definitions of m ε , J ε (w), J ε (x, w) and W , {k n } is bounded. Accordingly, one can extract a subsequence from {k n }, relabeled by it again, to guarantee that k n → k ε , n → ∞, for some k ε ∈ [0, +∞).
Thanks to {u n } n≥1 ∈ V ad , a subsequence of {u n } n≥1 ⊆ V ad can be extracted, written {u n } n≥1 again, such that u n w → u ε , n → ∞, for some u ε ∈ V ad . Since V ad is close and convex, we infer from Mazur lemma that u ε ∈ V ad .
By virtue of Step 1, one can take x n ∈ S(w n ) to ensure that J ε (x n , w n ) = J ε (w n ). On account of x n ∈ S(w n ), it yields that for k n > 0 and s ∈ (0, 1], (4.4) and for k n = 0, Below, we consider the following three cases. Case 1 k n > 0 and k ε > 0. We see from Theorem 3.3 that {t 1−β R kn β (t)} t>0 is compact and equicontinuous, hence that {x n } n≥1 is compact by the same method in Step 1, finally that we can suppose, without loss of generality, that x n → x ε .
Firstly, according to k n → k ε and k ε > 0, we can derive the boundedness of . Thus, from sup [0, ∞) and Theorem 3.1, we can deduce that for fixed s ∈ (0, 1], Secondly, we deduce from Lemma 4.1 and u n w → u ε that for s ∈ (0, 1], In addition, according to the definition of V ad , we get {u n (s) : n ≥ 1, a.e. s ∈ [0, 1]} ⊆ G. Since G is bounded, we can suppose that ∥u n (s)∥ ≤ M 1 , uniformly for s ∈ [0, 1] and n ≥ 1. As such, which is due to the p-mean continuity (see problem 23.9 on page 445 in [20]). Furthermore, on account of sup ∥u n (s)∥ ≤ M 1 , uniformly in s ∈ [0, 1] and n ≥ 1, we derive that Additionally, we can infer from Theorem 3.2 and the dominated convergence theorem that Similarly, we have Therefore, letting n → ∞ on both sides of (4.4), we get Thus, x ε ∈ S(w ε ), where w ε = (u ε , k ε ). In addition, according to the boundedness of {k n }, x n → x ε and k n → k ε , we get In addition, Definition 4.2 shows that where x ε ∈ S(w ε ) and w ε = (u ε , 0). Hence, Case 3 k n = 0. From k n → k ε , n → ∞, we get k ε = 0. Moreover, due to Definition 4.2, we have where x ε ∈ S(w ε ) and w ε = (u ε , 0). As such, Thus, combining above cases, we can assert that which indicates that (x ε , w ε ) is an optimal trajectory-control pair.
We are now in a position to analyze the time optimal control problem (P ) by Meyer approximation. Proof. The proof will be divided into the following steps.

Remark 4.2.
In most of the existing results on the time optimal control problems of evolution systems, many researchers explored them by setting up time optimal sequences (see [4,5,10,19]). With the aid of the Lipschitz assumption on f , the authors in [16][17][18] coped with them by Meyer approximation. In the present paper, we have investigated the time control problem by Meyer approximation, when the Lipschitz condition is not satisfied. Thus, our results extend and generalize some recent results about time optimal controls of all evolution systems.
Finally, we address a fractional diffusion model by employing our theoretical findings.
Then by means of Theorem 4.4, the time optimal control problem of system (4.5) possesses optimal trajectory-control pairs.