Abstract
The existence and uniqueness of the solution of a new kind of system—linear fractional differential-algebraic equations (LFDAE)—are investigated. Fractional derivatives involved are under the Caputo definition. By using the tool of matrix pair, the LFDAE in which coefficients matrices are both square matrices have unique solution under the condition that coefficients matrices make up a regular matrix pair. With the help of equivalent transformation and Kronecker canonical form of the coefficients matrices, the sufficient condition for existence and uniqueness of the solution of the LFDAE in which coefficients matrices are both not square matrices is proposed later. Two examples are given to justify the obtained theorems in the end.
1. Introduction
The dynamical behavior of a mechanical system was usually modeled via differential-algebraic equations (DAE) whose general form appears as , including both differential and algebraic equations to describe the corresponding constraints, for example, by Newton’s laws of motion or by position constraints such as the movement on a given surface. On the other hand, researchers had effectively solved engineering problems with fractional differential equations (FDE), which involves fractional derivatives in the model [1–6].
Recently, some investigators tried using fractional differential-algebraic equations (FDAE), which denote the combination of DAE and FDE, in dealing with the studied system. The general form of FDAE appears as , where , , could be vectors if necessary. Till now, the majority of these attempts concentrated on the algorithms for solving the FDAE [7–10], while fundamental problems such as the existence and uniqueness of the solution were neglected.
Existence and uniqueness of the solution of the model are of great significance, since the existence of the solution guarantees the practicability of the model, while the uniqueness of the solution guarantees the validity of the obtained solution.
In this paper, the existence and uniqueness of the solution of linear fractional differential-algebraic systems are discussed to lay the groundwork for the further studies and applications.
2. Linear Fractional Differential-Algebraic System
Consider the initial value problem for linear fractional differential-algebraic equations (LFDAE) in the formwhere , , , , , . are constant coefficients matrices. denotes Caputo’s fractional derivative operator [11–13], , since Caputo’s fractional derivative allows us to couple the fractional differential equations with initial conditions in the traditional form . And it is linked to other fractional definitions under certain conditions [11]. For convenience, we write system (1) as () in short.
For given , whether there exists unique solution for LFDAE (1) is a problem of great significance in application and it is our main concern as well.
System (1) is also valid when . In this case, by a simple coordinate transformation, we can obtain , where
It is worth pointing out that algebraic constraints are represented while rows in whose elements are all 0 appear.
3. Equivalent Transformation of LFDAE
To discuss the theorems on the existence and uniqueness of the solution of the LFDAE introduced in Section 2, we give definition of equivalent matrix pair [14] as a necessary preparation.
Definition 1. Two pairs of matrices and , , are called (strongly) equivalent if there exist nonsingular matrices and such that , . If this is the case, one writes ()~().
Multiplying nonsingular matrix on the left and setting , is a nonsingular matrix, LFDAE (1), where , , is transformed intowhich is again a linear fractional differential-algebraic equation with constant coefficients. System (2) could be written as and ~. Because of the linearity of the operator and the nonsingularity of square matrices , , it is evident that system (2) has the same solution property as system (1). Thus, by equivalent transformation, we can consider the transformed system (2) instead of (1) with respect to solvability and related questions.
4. Regular Matrix Pair and the Solution of LFDAE
To simplify the problem, we now focus on the LFDAE in which , are both square matrices; that is, . Recall Definition 2 [14].
Definition 2. Let , the matrix pair is called regular matrix pair if , and the characteristic polynomial is not the zero polynomial. A matrix pair which is not regular is called singular.
Lemma 3 allows us to gain an equivalent but more simple form of the matrix pair ()—Weierstrass canonical form [14].
Lemma 3. Let and be regular. Then, one has the Weierstrass canonical formwhere is a matrix in Jordan canonical form and is a nilpotent matrix also in Jordan canonical form. Moreover, it is allowed that one or the other block is not present.
Let be the index of nilpotent matrix ; that is, and (in fact, is always less than ); we have Theorem 4 which is fundamental on the solvability of LFDAE (1).
Theorem 4. Let be the index of nilpotent matrix ; is differentiable enough; that is, is well defined, where denotes sequential fractional derivatives [11, 15–17]. Then linear fractional differential-algebraic equationhas unique solution .
Proof. Firstly, we prove solutions of have the form of .
Since is a linear operator maps function into its derivative of order , moving the left hand term in to the right, we haveBecause is a matrix consisting of constant number, it is commutable with operator ; using Neumann series and taking as the index of nilpotent matrix into account, we obtainSecondly, we prove function is the solution of .
Substituting into yieldsObviously, is really the solution of .
So equation has the unique solution .
Now let us elaborate on Theorem 5 below.
Theorem 5. Let and is differentiable enough; then there exists unique solution for the linear fractional differential-algebraic equationsif the matrix pair is regular.
Proof. As claimed in Section 3, system (7) has the same solution property as system (8),We now discuss the solvability of system (8). Using Lemma 3, , in system (8) could be obtained in the following Weierstrass canonical form:Hence system (8) appears as By setting , system (8) is separated into two subsystems:Subsystem (11) is a normal fractional differential system, which has unique solution with given initial value [11]. As discussed in Theorem 4, subsystem (12) has unique solution .
Hence, system (8) has unique solution; accordingly system (7) has unique solution as a result. And the proof of Theorem 5 is completed.
Remark 6. Since the solution in Theorem 4 is obtained without specifying the initial value of , initial value is consistent when , where is the index of nilpotent matrix .
Remark 7. As derivative of fractional order is the generalization of derivative of integer order, the condition “the matrix pair is regular” plays the same role as in theory of differential-algebraic equation [18].
5. Kronecker Canonical Form and the Solution of LFDAE
We have investigated the existence and uniqueness of the solution of LFDAE (1) in which are both square matrices. Nevertheless, the general form of LFDAE (1), where are both not square matrices, is frequently modeled in mechanical systems. We go on to investigate this case by tools of equivalent transformation and the Kronecker canonical form of the system.
Kronecker canonical form which is referred to as , in Lemma 8 plays an important role in analyzing the existence and uniqueness of the solution of LFDAE. In 1974, Gantmacher pointed out Lemma 8 [19, 20].
Lemma 8. Let ; then there exist nonsingular matrices and , such that ()~() (), where , take the next form:Dimension of each matrix satisfies
According to Lemma 8, setting , in system (2) transformed from system (1) have the Kronecker canonical form through equivalent transformation: , .
Considering the structure of , , we now divide the vector and vector as below:
Hence, LFDAE (2) could be transformed into the following equivalent equations:and the equation
Now, let us discuss (16)–(20) to investigate the existence and uniqueness of the solution of LFDAE (2) in which , take the Kronecker canonical form.
With regard to (16), it is easy to see that the equation is solvable if and only if all components of are zeros; that is, . In this case, (16) either has no solution while or has solutions of infinite number (as long as , any function that is differentiable of order could be considered as the solution of (16)). So there exists no unique solution for (16).
With regard to (17), considering the particular form of , , (17) appears as
Denoting components of as and as , the above equation is precisely equal to We further have
Since in (23) is free of all restrictions, any function differentiable of order could be considered as ; thus (17) has solutions of infinite number.
With regard to (18), considering the particular form of matrices , , (18) appears as
Denoting components of as and as , the above equation is precisely equal toand an additional formula from which we get
Thus (25) could be rewritten in a recursive form of reverse order:
Obviously, to any given differential enough , (25) has unique solution . But the obtained hardly satisfy the equation in (25) except very special situation. Therefore, (18) has no feasible solution for any given .
With regard to (19), considering the particular form of matrices , (19) appears as
Denoting components of as and as , the above equation is equal to
We further obtain the equivalence of (28) as below:
Obviously, to any given , solution could be derived from (29) if is differentiable enough; that is, is well defined. Thus, (19) always has unique solution towards any given suitable vector .
With regard to (20), where , are square matrices, because of Theorem 5, (20) has unique solution if the matrix pair is regular.
Bringing the analysis above together, LFDAE (2) has unique solution if (16) to (18) do not appear in their Kronecker canonical form, while matrix pair is regular. The above-mentioned summarization brings us to Theorem 9.
Theorem 9. Let . The linear fractional differential-algebraic equationsin which , , satisfy the following:(1);(2) is well defined, and .Then, the system has unique solution if there exist nonsingular matrices and , such that ()~(), where , , and matrix pair () is regular.
Remark 10. Theorem 9 is also suitable when other fractional derivatives are involved. Since in the sequential fractional derivatives in Theorem 4 can mean the Riemann-Liouville, the Grünwald-Letnikov, the Caputo, or even any other definition of fractional derivatives [11], Theorem 5 is applicable to these fractional derivatives. Meanwhile, analysis in Section 5 is still valid for other definitions of fractional derivatives, so it is the same as Theorem 9 as a result.
6. Examples
Now let us take some examples on the theorems represented before.
Example 1. LFDAE in which is a nilpotent matrix and is as below (example of Theorem 4): In the example, is a nilpotent matrix with index , . So according to Theorem 4, the unique solution of Example 1 is Figure 1 shows the solution of Example 1. The correctness and the initial value consistency of the solution could be verified by computation.
Example 2. LFDAE with Kronecker canonical form is as below (example of Theorem 5):with the initial value .
In this example, , ; there exist two nonsingular matrices and :satisfying Obviously, , ; we further get , which is not a zero polynomial; hence the matrix pair () is regular.
From Theorem 9, the fractional system has unique solution. Now let us find the solution.
By setting , that is, , multiplying on the left to , we obtain the equivalent transformation of in the next form:It is easy to obtain the unique solution for by Laplace transform [11]: Accordingly, the solution for LFDAE is And the solution satisfies . The solution of Example 2 is shown in Figure 2.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This project is supported by the National Natural Science Foundation of China (no. 11272159).