Abstract
The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classical -transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in the -transform. We also prove forms of images in the -transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.
1. Introduction
Fractional operators are generalizations of the corresponding operators of integer order, as well in continuous as in discrete case. The particular interpretation of the fractional derivative was presented, for example, in [1] or [2]. Fractional differences used in models of control systems could be understand as an approximation of continuous operators (see [3]) and a possibility of involving some memory to difference systems, that is, systems in which the current state depends on the full history of systems’ states. Systems with fractional derivatives and differences are widely discussed in many papers. However, different authors could use various definitions of operators. Some comparisons between three basic types of fractional difference operators were studied in [4] and in [5] for multistep case. There are many papers, in which the authors usually involve in discrete case the Grünwald-Letnikov-type fractional operator, see, for example, [2, 6–14] for case and also for general case . However, the exact formulas for solutions are not used to considered linear initial value problems. The authors mainly use the recurrence method for stating their results. Using the particular type of definitions of fractional difference operators it was possible to state the exact formulas for solutions to initial value problems with the Riemann-Liouville- and the Caputo-type fractional difference operators. The clue point is that solutions to initial value problems with the Grünwald-Letnikov-type operator have the same values as those for systems with the Riemann-Liouville-type operator.
Basic properties of fractional sums and operators were developed firstly in [15] and continued by Atici and Eloe [16, 17], Baleanu and Abdeljawad [18, 19]. Another concept of the fractional sum/difference was introduced in [20–22].
Here we attempt to review methods of solutions for fractional difference systems. By comparing both types of solutions we produce the formula for the -transform of particular type of the discrete Mittag-Leffler matrix function. There are few papers where various kinds of discrete Mittag-Leffler functions are discussed. The important paper is [23], where discrete type and -discrete analogues of Mittag-Leffler function are presented. Their relations to fractional differences of the particular case are investigated. The author considers also applications of these functions to numerical analysis and integrable systems. However, the formulas of the considered functions mainly depend on the class of the type of calculus. For example in [19] the authors consider discrete Mittag-Leffler functions connected with nabla Caputo fractional linear difference systems and use there the nabla discrete Laplace transform. In [24] the authors consider a linear nabla -difference equations of noninteger order. In [25] the sequential discrete fractional equations with some particular convention of the fractional difference are solved using the kind of fractional nabla discrete Mittag-Leffler functions.
There is the possible use of the generalized Laplace transformation on time scales, for that see, for example, [26]. Such method was used in [16] and [27] under the method called the -transform and the Laplace transform, respectively, for the situation where on the right side of the system there are functions that depend only on time. Additionally, in the dissertation [28] the possible use of the Laplace transform for semilinear systems is mentioned. More classical solution of the problem is also presented in [29], where the authors use the -transform to give the conditions for stability of systems with the Caputo-type operator. The conclusions were derived based on the image in the complex domain. The main advantage of the use of the -transform is to introduce the natural language for discrete systems; it means to work with sequences instead of discrete functions defined on various domains.
The stability property is the main issue in dynamical systems. As the most of descriptions of behaviour of processes are digital due to computing tools used at computers, it is important to know some conditions for stability of discrete-time systems. The results are similar for different operators. In [30, 31] the study of the stability problem for discrete fractional systems with the nabla Riemann-Liouville-type difference operator is presented. The authors stress that the -transform can be used as a very effective method for stability analysis of systems. In engineering problems more often systems are used with the Grünwald-Letnikov-type fractional difference. One of the attempts to the stability of fractional difference systems is the notion of practical stability, see [6, 7]. Since the linear systems are defined by some matrices, the conditions connected with eigenvalues of these matrices are presented, for instance, in [11, 13, 14].
Our paper is rather motivated by the idea of presenting images via -transform of fractional difference operators and consequences into stability property. We receive results for stability of linear systems with three operators and with steps . In the case of the Grünwald-Letnikov-type fractional operator our results are similar to those in [11, 13, 14]. However, steps in the proof of stability condition are based on [31].
The paper is organized as follows. In Section 2, we present the preliminary material needed for further reading. Solutions to the considered linear initial value problems with the Caputo-type fractional difference and the image of the unknown vector function are constructed in Section 3. As the simple consequence of considering two types of solution: one from the classical approach and the second from the -transform we can write the images of the one-parameter Mittag-Leffler functions. Sections 4 and 5 deal with images of the unknown vector functions for linear initial value problems with the Riemann-Liouville- and the Grünwald-Letnikov-type fractional differences, respectively. Some comparisons for the -transforms of Caputo-type and Riemann-Liouville-type operators are presented in Section 6. In Section 7 we study the stability of the considered systems. Section 8 provides the brief conclusions. Additionally, Table 1 presents basic -transform formulas used in the text.
2. Preliminaries
Now, we recall the necessary definitions and technical propositions that are used in the sequel therein the paper. Let , and define and for any . Hence for . For a function the forward -difference operator is defined as , where , see [21, 22]. For an arbitrary and such that , the -factorial function is defined by , where is the Euler gamma function, see [21, 22]. We use the convention that division at a pole yields zero. For we write shortly instead of and instead of .
Definition 1 (see [22]). For a function the fractional -sum of order is given by , where , and additionally, .
For we write shortly instead of . Note that . From Definition 1 it follows that for where . Observe that formula (1) has the form of the convolution of sequences. Let us call for the moment the family of binomial functions defined on , parameterized by , and given by values: for and for . Then using we can write formula (1) in the form of the convolution of and . In fact where “” denotes a convolution operator; that is, .
We recall here that the -transform of a sequence is a complex function given by , where is a complex number for which this series converges absolutely.
Note that . In the stability analysis it is important to prove to which classes of sequences belongs. Let be a sequence and let and be the norms defined in the space of sequences. Moreover, let , , be the spaces of all sequences satisfying .
Lemma 2. For and such that the sequence belongs to ; that is, .
Proof. Let , , and be the integer part of ; that is, and . Then for such that we have and for we get . Therefore Hence .
Proposition 3. For one has .
Proof. Note that and for . Hence and consequently, ; that is, .
Proposition 4. For let one denote and . Then where .
Proof. In fact as we have from (3) that , then . Moreover, , then we easily see equality (6).
For , (6) can be shortly written as , where is treated as a sequence.
Two fractional -sums can be composed as follows.
Proposition 5 (see [32]). Let be a real-valued function defined on , where . For the following equalities hold: , where .
In [22] the authors prove the following lemma that gives transition between fractional summation operators for any and .
Lemma 6. Let and . Then, , where and .
In our consideration the crucial role is played by the power rule formula, see [21]: , where , , . Then for we have . Nextly, taking , , we get . Particularly, we have .
Now, we adapt the above formulas to multivariable fractional-order linear case and define Mittag-Leffler matrix functions. In [19] the discrete Mittag-Leffler function is introduced in the following way.
Definition 7. Let with . The discrete Mittag-Leffler two-parameter function is defined as
We introduce the two-parameter Mittag-Leffler function defined in different manner and show that both definitions give the same values. Moreover, we use the function for the matrix case and prove by two ways its image in the -transform. Let us define the following function: In fact, in the paper we use two of them, namely, and . For we have that .
In our consideration an important tool is the image of with respect to the -transform.
Proposition 8. Let be defined by (8). Then where and .
Proof. By basic calculations we have where the summation exists for and .
Proposition 9. Let and . Let be the set of all roots of the following equation: If all elements from are strictly inside the unit circle, then .
Proof. If all roots of (11) are strictly inside the unit circle, then using theorem of final value for the -transform, we easily get the assertion.
By Proposition 9 we get that for some order there is a “good” such that all elements from are inside the unit circle.
Corollary 10. Let . All elements from (Proposition 9) are inside the unit circle if and only if .
3. Caputo-Type Operator
In this section we recall the definition of the Caputo-type operator and consider initial value problems of fractional-order systems with this operator. We discuss the problem of solvability of fractional-order systems defined by difference equations with the Caputo-type operator. In [33] versions of solutions of scalar fractional-order difference equation are given.
Nextly, we define the family of functions , parameterized by and by with the following values:
Proposition 11. Let be the function defined by (12). Then for such that .
Proof. This needs the following simple calculations: where to ensure the existence of the summation.
Note that, for any , , the following relations hold:(1), , and ;(2) and as the division by pole gives zero, the formula works also for ;(3);(4);(5) for ;(6)For we have and . According to item in the properties of functions for , we put some comment. Note that and . Therefore, , but taking so .
Taking particular value of and considering the family of functions the particular case of the Mittag-Lefler function from Definition 7 can be rewritten as , where . Note that in fact , so the right hand side is finite and consequently; values always exist. For there is the delta exponential function: .
The properties of two-indexed functions were given in [34, Proposition 2.5] and generalized for -indexed functions in [34] and in [35] with multiorder . In the paper therein, we use functions with step and one order. The next proposition is the particular case for the multi-indexed functions.
Proposition 12 (see [35]). Let and . Then for one has
The next step is to present Caputo-type -difference operator.
Definition 13 (see [32]). Let . The Caputo-type -difference operator of order for a function is defined by where .
Note that . Moreover, for , the Caputo-type -difference operator takes the form: , where . Observe that using function we can write the Caputo-type difference in the following way: where and .
For the Caputo-type fractional difference operator there exists the inverse operator that is the tool in recurrence and direct solving fractional difference equations.
Proposition 14 (see [32]). Let , , , and be a real-valued function defined on . The following formula holds , .
Similarly as in Lemma 6 there exists the transition formula for the Caputo-type operator between the cases for any and .
Lemma 15 (see [21]). Let and . Then, , where and .
From this moment for the case we write and similarly, number 1 is omitted for two other operators and fractional summations. Then the result from Proposition 14 can be stated as where , and .
Proposition 16. For , let one define , where and . Then where and .
Proof. The equality follows from . Then using the formula for after simple calculations we get (19).
Observe that for we have , which agrees with the transform of difference of . Using the -transform of the Caputo-type operator for , we can calculate that for the following relation holds: while for it becomes zero; that is, .
Firstly, let us consider systems with Caputo-type operator for , given by the following form: where , , , and is real matrix. Let us consider the following initial condition: for (21). System (21) with can be rewritten in the form where , , and . We also use the notation , what helps to avoid the problem with domains and allows to see the variable vector as a sequence.
Proposition 17 (see [4]). Let and . The linear initial value problem has the unique solution given by the formula where and .
The next proposition presents the proof of the -transform of the one-parameter Mittag-Leffler matrix function, where the idea of transforming a linear equation is used.
Proposition 18. Let and . Then the -transform of the one-parameter Mittag-Leffler function is given by
Proof. Let us consider problem (22a)-(22b) and take the -transform on both sides of (22a). Then for from (19), we get , where and . And after calculations we see that ; hence we get , that is, the solution of the considered initial value problem. And then taking initial conditions as vectors from standard basis from we get the fundamental matrix , whose transform form should agree with the given.
Let us consider the Caputo difference initial value problem for the linear system that contains only one equation with , that is, , . Then is the scalar one-parameter Mittag-Leffler function. Observe that the -transform of this scalar Mittag-Leffler function can be obtained from Proposition 18 by substituting and into (25).
4. Riemann-Liouville-Type Operator
In this section we recall the definition of the Riemann-Liouville-type operator and consider initial value problems of fractional-order systems with this operator. Similarly as in the case the Caputo-type operator we discuss the problem of solvability of fractional-order systems of difference equations. Family of functions is useful for solving systems with Caputo-type operator. We also formulate the similar family of functions that are used in solutions of systems with Riemann-Liouville-type operator. Let us define the family of functions parameterized by and by with the following values:
Proposition 19. Let be the function defined by (26). Then for such that .
Proof. By simple calculations we have where the summation exists for .
Note that for any and the following relations hold:(1), , and .(2) and .(3).(4), .(5) for .(6)For , we have .
Taking particular value of and considering the family of functions the particular case of Mittag-Leffler function from Definition 7 can be written as , where . Note that in fact , so the right hand side is finite and consequently; values always exist. For there is a delta exponential function: .
For the family of functions we have similar behaviour for fractional summations as for so we can state the following proposition.
Proposition 20. Let and . Then for , one has
Proof. In the proof we use the fact that . Then using item 5 from properties of functions , we see that . Hence taking the inverse transform we get the assertion.
The next presented operator is called the Riemann-Liouville-type fractional -difference operator. The definition of the operator can be found, for example, in [16] (for ) or in [21, 22] (for any ).
Definition 21. Let . The Riemann-Liouville-type fractional -difference operator of order for a function is defined by where .
For one can get where . Moreover for one has .
The next propositions give useful identities of transforming fractional difference equations into fractional summations.
Proposition 22 (see [33]). Let , , , and be a real-valued function defined on . The following formula holds , .
In [21] the authors show that similarly as in Lemmas 6 and 15 there exists the transition formula for the Riemann-Liouville-type operator between the cases for any and .
Lemma 23. Let and . Then , where and .
For the case , we write . Then the result from Proposition 22 can be stated as where , and with .
Proposition 24. For , , let one define , where and . Then where and .
Proof. Let . Then the assertion for comes from Lemma 23. Let . Then , where and . Then we easily see equality (33).
For we have the same as for the Caputo-type operator , which also agrees with the transform of difference of .
Using the -transform of the Riemann-Liouville-type operator for we can calculate that while for we get and consequently, .
Now, let us consider systems with the Riemann-Liouville-type operator for , given by the following form: where , , , and is real matrix and with initial condition . System (35) with can be rewritten as follows: where and , . We also use the notation , what helps to avoid the problem with domains and allows to see the variable vector as a sequence.
Proposition 25 (see [4]). Let and . The linear initial value problem, has the unique solution given by the formula where and .
Proposition 26. Let and . Then the -transform of the Mittag-Leffler function is given by
Proof. Let us consider problem (36a)-(36b) and take the -transform on both sides of (36a). Then we get from (33) for , , where and . And after calculations we see that ; hence is the solution of the considered initial value problem. And then taking initial conditions as vectors from standard basis from we get the fundamental matrix , whose transform is given by (39).
Let . Then for we have , . Taking into account the image of the scalar Mittag-Leffler function in the -transform we get , where and are substituted into (39). Hence using the power series expansion the scalar Mittag-Leffler function can be written as .
5. Grünwald-Letnikov-Type Operator
The third type of the operator, which we take under our consideration, is the Grünwald-Letnikov-type fractional -difference operator, see, for example, [2, 6–14] for cases and also for general case .
Definition 27. Let . The Grünwald-Letnikov-type -difference operator of order for a function is defined by where
Proposition 28. For , , let one define , where and , . Then where and .
Proof. Note that .
The following comparison has been proven in [4].
Proposition 29. Let . Then , where , for , or for and .
Proposition 30. Let . Then , where for .
Proof. Let . Then by Proposition 28 we get , where , and , . Hence taking the inverse transform we get the assertion.
From Proposition 30 we have the possibility of stating exact formulas for solutions of initial value problems with the Grünwald-Letnikov-type difference operator by the comparison with parallel problems with the Riemann-Liouville-type operator. And we can use the same formula for the -transform of as for the Riemann-Liouville-type operator. Then many things could be easily done for systems with the Grünwald-Letnikov-type difference operator.
The next proposition is proved in [36] for the case .
Proposition 31. The linear initial value problem, has the unique solution given by the formula for any and dimensional state transition matrices are determined by the recurrence formula with .
Taking into account Propositions 25 and 30 we can state the following result.
Proposition 32. The linear initial value problem, has the unique solution given by the formula where .
As a simple conclusion we have that
6. Comparisons and Examples
Here we present some comparisons for the -transforms of Caputo- and Riemann-Liouville-type operators as the third one used in systems with the left shift gives the same solutions as the Riemann-Liouville-type operator. Then we present the example with mixed operators in two equations. In this chapter we also state the formula in the complex domain of the solution to the initial value problem for semilinear systems like for control systems.
Proposition 33. Let , where . Then
Proof. It is enough to compare Propositions 16 and 24. We can also equivalently take under consideration (31) in the real domain and observe that .
Comparing the -transform of two types of the Mittag-Leffler functions we get the following relation.
Proposition 34. Let , . Then
Proof. In fact if we take the -transform of the right side, then we have Using formula (39) we get and consequently, we reach the assertion.
In the following example we consider control systems with the Caputo- and Riemann-Liouville-type operators. We get the same transfer function for initial condition .
Example 35. Let , , and . Let us consider linear initial value problems of the following forms: (a)for the Caputo-type operator:
(b)for the Riemann-Liouville-type operator:
with initial condition , and constant matrices , , , .
We solve the problems using the -transform. Since , then applying the -transform to the first equations in both systems, we get the following equation: . Then , where , , and . Consequently, for the output of the considered systems in the complex domain, we have the following formula: , where . Then the solution for the output in the real domain is as follows: , where . We can easily check the value at the beginning that , as .
7. Towards Stability Analysis
Let us introduce the stability notions for fractional difference systems involving the Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type operators.
Firstly, let us recall that the constant vector is an equilibrium point of fractional difference system (21) if and only if (and in the case of the Riemann-Liouville difference systems of the form (35)) for all .
Assume that the trivial solution is an equilibrium point of fractional difference system (21) (or (35)). Note that there is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables.
Definition 36. The equilibrium point of (21) (or (35)) is said to be(a) stable if, for each , there exists such that implies , for all ,(b) attractive if there exists such that implies (iv) asymptotically stable if it is stable and attractive. The fractional difference systems (21), (35) are called stable/asymptotically stable if their equilibrium point is stable/asymptotically stable.
Example 37. Let us consider systems (23a)-(23b) or (37a)-(37b) but with . Then for system (23a)-(23b) we have that , while for system (37a)-(37b) the solution satisfies the following . Hence, then systems with the Caputo-type operator are stable and systems with the Riemann-Liouville operator are attractive hence asymptotically stable.
Proposition 38. Let be the set of all roots of the equation Then the following items are satisfied. (a)If all elements from are strictly inside the unit circle, then systems (21) and (35) are asymptotically stable.(b)If there is such that , both systems (21) and (35) are not stable.
Proof. The main tool in the proof is Proposition 9 and the method of decomposition of the matrix of systems into Jordan’s blocks. Some basic tool is similar to those used in [30]. However in the proof we do not use the same tools as in the cited paper. Only the beginning could be read as similar. The -transform of is , where for systems with the Caputo-type operator and for systems with the Riemann-Liouville-type operator. Moreover the condition from Proposition 9 that is satisfied. Let , where is invertible and and are Jordan’s blocks of order . Let be algebraic multiplicities of eigenvalues ; let be their geometric multiplicities, . The matrix is a diagonal block matrix. The number of blocks corresponding to is and their form is given by the upper triangular matrix where and is the size of the block. Because and is formed by a linear combination of convolutions of with the rational functions from (54), is the set of all poles of as . (a)If all elements from are strictly inside the unit circle, then .(b)If there is an element from that , then and systems (21) and (35) are not stable.
Now we can easily derive some basic corollary.
Corollary 39. If there is , such that , then system (21) is not asymptotically stable. Or equivalently, if system (21) is asymptotically stable, then for all .
Example 40. Let us consider the family of planar systems with order described by the matrix . Observe that so the considered planar systems have the unique equilibrium point . Since condition (53) does not depend on the type of an operator, we take into account systems (21) with the Caputo-type difference operator. In our case we get for (i) , that elements from are inside the unit circle and the system is stable, trajectories are presented at Figure 1 but for (ii) , we get that the system is unstable, see Figure 2. In this case we can have stability for smaller step . For the given matrix and , the stability of the systems depends also on the length of the step. It is easy to see that , and , . Then for we have and consequently, if and , then by Corollary 39 system (21) is not asymptotically stable. Additionally, one can see that for and we get and the equilibrium point is not stable. Therefore, the condition given for eigenvalues of the matrix describing the system is not sufficient for asymptotic stability of the system.
(a) The graph of
(b) The graph of
(c) Phase trajectory
8. Conclusions
The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type fractional-order difference initial value problems for linear systems are discussed. We stress the formula for the -transform of the discrete Mittag-Leffler matrix function. Additionally, the stability problem of the considered systems is studied. The next natural step is to consider the semilinear control systems and develop the controllability property via the complex domain.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are very grateful to anonymous reviewers for helping them to improve the paper. The project was supported by the founds of National Science Centre granted on the bases of the decision no. DEC-2011/03/B/ST7/03476. The work was supported by Bialystok University of Technology Grant G/WM/3/2012.