Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations
We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.
The purpose of this paper is to investigate the solvability and optimal controls for the following semilinear control systems with Riemann-Liouville fractional derivatives: where , denotes the Riemann-Liouville fractional derivative of order with the lower limit zero. is the infinitesimal generator of a -semigroup on a separable Banach space . is a given function to be specified later. The control function is given in a suitable admissible control set . is a linear operator from a separable reflexive Banach space into . The cost functional over the family of admissible state control pair is given by
In recent years, fractional calculus has been paid more and more attention that lies on the fact that it allows us to consider integration and differentiation of any order, not necessarily integer, and fractional-order models are more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of term which insures the history and its impact on the present and future in a model. Therefore, it has drawn great applications of the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, viscoelasticity, heat conduction, electricity mechanics, control theory, and so forth. For more details on these topics one can see, for instance, [1–17] and the reference therein.
The definitions of Riemann-Liouville fractional derivatives or integrals initial conditions play an important role in some practical problems. Heymans and Podlubny  have demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions.
Since the pioneering work on the fundamental solutions of Caputo fractional evolution equations associated with some probability densities has been reported by El-Borai [19, 20], the study of the existence, controllability, and optimal controls of the fractional semilinear functional differential equations and inclusions with the infinitesimal generator of a -semigroup have been extensively studied (one can see [21–23]). However, to the best of our knowledge, the solvability and optimal controls for fractional semilinear differential equations with Riemann-Liouville fractional derivatives are still untreated topics in the literature and this fact is the motivation of the present work. Our aim in this paper is to provide some suitable sufficient conditions for the existence and uniqueness of solutions and optimal control results corresponding to the admissible control sets of fractional abstract Cauchy problems with the Riemann-Liouville fractional derivatives.
The rest of this paper is organized as follows. In Section 2, we will present some basic definitions and preliminary facts, such as definitions, lemmas, and theorems, which will be used throughout the following sections. In Section 3, by applying the well-known fixed point theorem, some sufficient conditions are established for the existence and uniqueness of mild solutions of the system (1). In Section 4, we will study the optimal controls for semilinear differential equations with Riemann-Liouville fractional derivatives. Finally, we present an example to demonstrate our main results in Section 5.
In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach space will be denoted by . denotes the space of bounded linear operators from to . For the uniformly bounded -semigroup , we set . Let denote the Banach space of all -value continuous functions from into with the norm . Let be the space of functions which are absolutely continuous on and and . To define the mild solutions of (1), we consider the Banach space with the norm . Obviously, the space is a Banach space.
Definition 1. The integral is called Riemann-Liouville fractional integral of order , where is the gamma function.
Definition 2. For a function given in the interval , the expression where , denotes the integer part of number and is called the Riemann-Liouville fractional derivative of order .
Lemma 3 (see ). Let , let , and let be the fractional integral of order . If and , then one has the following equality:
The Laplace transform formula for the Riemann-Liouville fractional integral is defined by where is the Laplace of defined by
Lemma 4. Let and , , if , , and is a solution of the following problem: then, satisfies the following equation: where and is a probability density function defined on , that is,
Proof. Apply Riemann-Liouville fractional integral operator on both sides of (8); then, by Lemma 3, we get
Let ; taking the Laplace transformations
to (13), we obtain
Consider the one-sided stable probability density whose Laplace transform is given by
Hence, it follows from (15) and (17) that
According to the above work, we get
Now, we can invert the Laplace transform to (19) and obtain
Then, we get
This completes the proof of the lemma.
According to Lemma 4, we give the following definition.
Definition 5. A function is called a mild solution of (1) if it satisfies the following fractional integral equation:
Remark 6. A mild solution of the system (1) is referred to as a state trajectory of the fractional semilinear differential equation corresponding to the initial state and the control .
Due to the paper , we can obtain the following.
Lemma 7. The operator has the following properties.(i)For any fixed , is linear and bounded operators; that is, for any , (ii) is strongly continuous.(iii)For any , is also a compact operator if is compact.
Let us recall the following generalized Gronwall inequality which can be found in .
Lemma 8. Suppose , is a nonnegative function locally integrable on , and is a nonnegative, nondecreasing continuous function defined on and (constant), and suppose is nonnegative and locally integrable on with Then,
Remark 9. Under the hypotheses of Lemma 8, let be a nondecreasing function on . Then, where is the Mittag-Leffler function defined by
3. Existence of Mild Solutions
This section is devoted to the study of the existence and uniqueness results for a class of semilinear differential equations with Riemann-Liouville fractional derivatives.
In what follows, we will make the following hypotheses on the data of our problems::the function satisfies the following:(i) is measurable for all and is continuous for a.e. ;(ii)there exists a function , , and a constant such that (iii)there exists a function , , such that :the operator ;:the multivalued map (where is a class of nonempty closed and convex subsets of ) is measurable and there exists a function , , such that
Set the admissible control set
Then, (see Proposition 2.1.7 and Lemma 2.3.2 of ). And it is not difficult to check that is a closed and convex subset of .
In order to discuss the solvability and optimal control of system (1), we need to consider the following.
Lemma 10. Assume that – hold. Then, there exists a constant such that
Now, we are in the position to present our first result.
Theorem 11. Assume that the hypotheses – are satisfied. Then, the problem (1) has a unique mild solution on provided that
Proof. Consider the operator defined by
Choose and let . It is obvious that is a bounded, closed, and convex subset of .
Firstly, we show that maps into itself. In fact, for any and , like the proof of Lemma 10, we can easily obtain which means that .
Next, we show that is a contraction operator on .
Indeed, let , and ; then we obtain
Since , so is a contradiction operator. According to Banach’s fixed point theorem, we obtain the problem (1) that has a unique mild solution on . The proof is completed.
Remark 12. In Theorem 11, we investigate the existence of a local mild solution of (1). Next, if we assume that the semigroup is compact, then we can get the global version of a mild solution for system (1).
Firstly, for a compact semigroup, we have the following.
Lemma 13 (see ). Let be a -semigroup. If is a compact semigroup, then is continuous in the uniform operator topology for .
The key tool in the existence result is the following Schaefer fixed point theorem.
Theorem 14. Let be a Banach space and let be a completely continuous operator. If the set is bounded, then has at least a fixed point.
Now, we are ready to state the existence result which is based on Theorem 14.
Theorem 15. Assume that – hold; if is the infinitesimal generator of a compact semigroup , then the problem (1) has at least one mild solution on .
Proof. We consider the operator defined by
For the sake of convenience, we subdivide the proof into several steps.
Step 1. maps bounded sets into bounded sets in .
In fact, it is enough to show that, for any , there exists a such that, for each , we have .
For each , we obtain which implies that
Thus, we know that is a bounded set in .
Step 2. We prove that is continuous.
Let be a sequence such that in as . Then, for each , we obtain
Hence, we get
Step 3. We prove that is an equicontinuous set in .
Firstly, for any , by Lemma 7(ii), is strongly continuous; then, there exists a , such that
Thus, for the above , there exists , such that, for any , , , one can obtain
Hence, by the definition of equicontinuity, we get that is equicontinuous on .
Next, for any and , we obtain
By the assumptions and Holder’s inequality, we have
Similarly, we obtain and, for , , there exists a which is small enough, such that
Since the compactness of and Lemma 13 imply the continuity of in in the uniform operator topology, it can be easily seen that and tend to zero independently of as , . It is also clear that tend to zero as does not depend on particular choice of . Thus, we get that tends to zero independently of as which implies that is equicontinuous on .
Therefore, by all of the above work, we can get that is an equicontinuous set in .
Step 4. We show that is compact.
Let be fixed; we show that the set is relatively compact in .
Clearly, is compact, so it is only necessary to consider . For each , , and and any , we define where
From the boundedness of and the compactness of , we obtain that the set is relatively compact set in for each and . Moreover, we have
Since , the last inequality tends to zero when and . Therefore, there are relatively compact sets arbitrarily close to the set . Hence, the set is also relatively compact in .
Step 5. A priori bounds.
Now, it remains to show that the set is bounded.
Let ; then for some . Similarly to the proof of Lemma 10, we obtain
This shows that the set is bounded.
As a result, by the conclusion of Theorem 14, we obtain that has a fixed point . Therefore, system (1) has at least one mild solution on . The proof is completed.
4. Optimal Control Results
In this section, we are concerned with the following Lagrange problem (P).
Minimize a cost function of the form among all the admissible state control pairs of the semilinear differential equations with Riemann-Liouville fractional derivatives (1); that is, find an admissible state control pair such that where denotes the mild solution of system (1) corresponding to the control .
For the existence of solution for problem (P), we will introduce the following assumptions::the function satisfies the following:(i)the function is Borel measurable;(ii) is sequentially lower semicontinuous on for almost all ;(iii) is convex on for each and almost all ;(iv)there exist constants , , is nonnegative, and such that
Next, we can give the following result on existence of optimal controls for problem (P).
Theorem 16. Let the assumptions of Theorem 15 and hold. Then, Lagrange problem (P) admits an optimal pair; that is, there exists an admissible control pair such that
Proof. If , then it is clear that the Lagrange problem (P) has an optimal pair.
Without loss of generality, we assume that . Using (iv), we have . By definition of infimum, there exists a minimizing sequence feasible pair that is a mild solution of system (1) corresponding to , such that as . Since , is a bounded subset of the separable reflexive Banach space , there exists a subsequence, relabeled as , and such that
Since is closed and convex, by Mazur’s lemma, .
Let denote the sequence of solutions of the system (1) corresponding to ; that is, Now, we prove that is relatively compact on .
Firstly, it follows the boundedness of