Research Article | Open Access
The Solvability and Optimal Controls for Some Fractional Impulsive Equation
This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations. Firstly, we introduce the fractional calculus, Gronwall inequality, and Leray-Schauder’s fixed point theorem. Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.
In this paper, we study some fraction evolution with finite impulsive: where is the standard Caputo fractional derivative of order , , , is a generator of a semigroup defined on a complex Banach space , let be a given function and satisfying some assumptions that will be specified later, the function is continous, and , , and denote the right and the left limits of at , is a given control function in another Banach space , and is a linear operator from to .
The fractional calculus and fractional difference equations have attracted lots of authors during the past years, and they gave some outstanding work [1–4], because they described many phenomena in engineering, physics, science, and controllability. Delay evolution equation allows someone to think after-effect, so it is a relative important equation. There are some significant development; for example, Wang et al. [5, 6] consider the following fractional delay nonlinear integrodiffrential controlled system: and they used laplace transform and probability density functions to prove some sufficient conditions of some fractional nonlinear finite time delay evolution equations. Shu et al.  used the solution operator of semigroup to investigate the system given by Benchohra et al.  deal with existence of mild solutions of some fractional functional evolution equations with infinite delay. Balachandran et al.  concerned the relative controllability of fractional dynamical with delays in control. Specially, Cuevas and Lizama  studied some sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the following semilinear fractional differential equation fractional differential equations:
Motivated by the abovementioned works, we study (1). The rest of this paper is organized as follows. In Section 2, some notation and preparation are given. In Section 3, some mainly results of (1) are obtained. At last, an example is given to demonstrate our results.
In this section, we will give some definitions and preliminar which will be used in the paper. The norm of the space will be defined by . Let denote the Banach space of all value continuous functions from into , the norm . Let the another banach space , , there exist , , , , . We can use to denote the Banach space of all Lebesgue measurable functions from to with , and denote the Banach space of functions which are Bochner integrable normed by , .
Let that , , and is a suitable function.
Definition 1 (Riemann-Liouville fractional integral and derivative operators). The integral operator is defined on by The derivative operators are defined as , where and
Definition 2. Caputo fractional derivative of of order is defined as
If , we can write the Caputo derivative of the function , via the above Riemann-Liouville fractional derivative as
Let us recollect the generalized Gronwall inequality which can be found in  and will be used in our main result.
Lemma 3. Suppose , is a nonnegative function locally integrable on , and is a nonnegative, nondecreasing continuous function defined on , (constant), and is nonnegative and locally integrable on with Then
Remark 4. Under the hypothesis of Lemma 3, let be a nondecreasing function on . Then where is the Mittag-Leffler function defined by
Lemma 5 (see ). Let satisfy the following inequality: where are constants. Then where .
Specially, if ,
We also introduce the following theorem that will be used in our mainly result.
Theorem 6 (Hölder’s inequality). Assume that , , and ; if and then and .
Theorem 7 (Arzela-Ascoli theorem). If a sequence in is bounded and equicontinuous, then it has a uniformly convergent subsequence.
Theorem 8 (Leray-Schauder’s fixed point theorem). If is a closed bounded and convex subset of Banach space and is completely continuous, then the has a fixed point in .
3. Existence and Uniqueness of Mild Solution
In this section, we will investigate the existence and uniqueness for impulsive fractional differential equations with the help of the Leray-Schauder’s fixed point theorem and someone else. Without loss of generality, let , .
Firstly, we will make the following assumptions be satisfied on the data of our problem. is a compact semigroup, and there exists a constant , such that .The function satisfies the following:(i) is measurable for all ;(ii) there exists a constant such that , for all ;(iii) there exists a real function , , and a constant , such that , for a.e. and all . () satisfies the following:(i) maps a bounded set to a bounded set;(ii) there exist constants () such that (iii).Let be a separable reflexive Banach space. Operator , , stands for the norm of operator on Banach space .The multivalued maps (where is a class of nonempty closed and convex subsets of ) are measurable and where are a bounded set of .
Set the admissible control set: Then, (see Proposition 2.1.7 and Lemma 2.3.2 of ). And it is obvious that for all .
Definition 9. A function is said to be a solution (mild solution) of the problem (1) if such that where where is a probability density function defined on , that is
Lemma 10 (see ). The operators and have the following properties and there exists as described in .(i)For any fixed , and are linear and bounded operators; that is, for any , (ii) and are strongly continuous.(iii)For any , and are also compact operators if is compact.
Lemma 11. If the assumptions – are satisfied and (1) is mildly solvable on , then there exists a constant such that .
Proof. If (1) can be solvable on , we may suppose is the mild solution of it, so must satisfy (19) as follows: For , , through calculating, we can get that Let , then so it follows from Lemma 5, where The proof is completed.
Theorem 12. Assume that the hypotheses – are satisfied, and then the problem (1) has an unique mild solution on provided that
Proof. Transform the problem (1) into a fixed point theorem. Consider the operator defined by
Clearly, the problem of finding mild solutions of (1) is reduced to find the fixed points of the , the proof base on Theorem 8. Now we prove that the operator satisfies all the conditions of the Theorem 8.
Firstly, choose and consider the bounded set .
Next, for the sake of convenient, we divide the proof into several steps.
Step 1. We prove that .
In fact, for each , , , we have Hence, we can deduce that .
Step 2. We show that is continuous.
Let be a sequence such that in as . Then, for each , , we obtain as , and it is easy to see that that is, is continuous.
Step 3. is equicontinuous on .
Let ; then, for each , we obtain
Let By (ii) of Lemma 10, we have By the assumption , we obtain and we get
Combining the estimations for , , let and , and we know that , which implies that is equicontinuous.
Step 4. Now we show that is compact.
Let , be fixed, and we show that the set is relatively compact in .
Clearly, is compact, so it is only necessary to consider . For each , , and any , we define where From the compactness of , we obtain that the set is relatively compact set in for each and . Moreover, we have when and , we can easily find , , , . Therefore, there are relatively compact sets arbitrarily close to the set . Hence the set is also relatively compact in .
As a result, by the conclusion of Theorem 8, we obtain that has a fixed point on . So system (1) has a unique mild solution on . The proof is completed.
4. Optimal Control Results
In the following, we will consider the Lagrange problem (P).
Find a control pair such that where and denotes the mild solution of system (1) corresponding to the control .
For the existence of solution for problem (P), we shall introduce the following assumption. 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 13. Let the assumptions of Theorem 12 and hold. Suppose that is a strongly continuous operator. Then Lagrange problem () admits at least one optimal pair; that is, there exists an admissible control pair such that
Proof. If , there is nothing to prove.
Without loss of generality, we assume that . Using , we have . By definition of infimum, there exists a minimizing sequence feasible pair is a mild solution of system (31) 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, due to Marzur lemma, . Let denote the sequence of solutions of the system (1) corresponding to , is the mild solution of the system (1) corresponding to . and satisfy the following integral equation, respectively:
It follows the boundedness of , and Lemma 11, one can check that there exists a positive number such that , .
For , we obtain By (ii), we have
Using Lemma 10(i) and by (ii), one can obtain
Similarly, one has
Since is strongly continuous, we have which implies Thus by virtue of singular version Gronwall inequality (i.e., Lemma 5), we obtain This yields that
Note that implies all of the assumptions of Balder (see , Theorem 2.1) are satisfied. Hence, by Balders theorem, we can conclude that is sequentially lower semicontinuous in the strong topology of . Since , is weakly lower semicontinuous on , and since, by (iv), attains its infimum at ; that is, The proof is completed.
5. An Example
Consider the following initial-boundary value problem of fractional impulsive parabolic control system with the cost function where , is continuous, , .
Take and the operator is defined by where the domain is given by Then can be written as where is an orthonormal basis of . It is well known that is the infinitesimal generator of a compact semigroup in given by
It is easy to see that and then
The authors thank the referees for their careful reading of the paper and insightful comments, which help to improve the quality of the paper. They would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
- K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer, Berlin, Germany, 2010.
- J. Wang, Y. Zhou, and M. Medved, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31–50, 2012.
- J. Wang, Y. Zhou, and W. Wei, “A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4049–4059, 2011.
- X. B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis, vol. 74, no. 5, pp. 2003–2011, 2011.
- M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008.
- K. Balachandran, Y. Zhou, and J. Kokila, “Relative controllability of fractional dynamical systems with delays in control,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3508–3520, 2012.
- C. Cuevas and C. Lizama, “Almost automorphic solutions to a class of semilinear fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 12, pp. 1315–1319, 2008.
- E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, 2001.
- L. Xue and J. Xiong, “Existence and uniqueness of mild solutions for abstract delay fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1398–1404, 2011.
- H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
- J. Wang, M. Fečkan, and Y. Zhou, “Relaxed controls for nonlinear fractional impulsive evolution equations,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 13–32, 2013.
- S. Hu and N. S. Papageorgiou, Handbook of multivalued Analysis, Kluwer Academic Publishers, London, UK, 1997.
- J. Wang, M. Fečkan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345–361, 2011.
- Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
- E. J. Balder, “Necessary and sufficient conditions for -strong-weak lower semicontinuity of integral functionals,” Nonlinear Analysis, vol. 11, no. 12, pp. 1399–1404, 1987.
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