On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System

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.


Introduction
The dynamical behavior of a mechanical system was usually modeled via differential-algebraic equations (DAE) whose general form appears as (, , ẏ ) = 0, 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][2][3][4][5][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 (, ,  () ) = 0, where , ,  () could be vectors if necessary.Till now, the majority of these attempts concentrated on the algorithms for solving the FDAE [7][8][9][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.
For given z() ∈ R  , whether there exists unique solution y() ∈ R  for LFDAE (1) is a problem of great significance in application and it is our main concern as well.

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.
is a linear operator maps function y() into its derivative of order , moving the left hand term in ( * ) to the right, we have Because  is a matrix consisting of constant number, it is commutable with operator  0  ()  ; using Neumann series and taking  as the index of nilpotent matrix  into account, we obtain z () . ( Secondly, we prove function y z() into ( * ) yields Obviously, y() is really the solution of ( * ).
So equation Now let us elaborate on Theorem 5 below.
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,  =  1  2 ,  =  1  2 in system (8) could be obtained in the following Weierstrass canonical form: Hence system (8) appears as By setting y = [y 1 , y 2 ], 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 z 2 ().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 , 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].

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 ,  ∈ R × ( ̸ = ) 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.
Considering the structure of Ẽ, Ã, we now divide the vector ỹ() and vector z() 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 Ẽ, Ã 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  0 () are zeros; that is,  0 () = 0.In this case, (16) either has no solution while  0 () ̸ = 0 or has solutions of infinite number (as long as  0 () = 0, any function that is differentiable of order  could be considered as the solution of ( 16)).So there exists no unique solution for (16).
and an additional formula from which we get ( 2 6  ) Thus 5 could be rewritten in a recursive form of reverse order: (25  ) Obviously, to any given differential enough z L (), 5 has unique solution y L ().But the obtained  L (1) () hardly satisfy the equation  0  ()   L (1) () =  L (1) () in 5 except very special situation.Therefore, (18) has no feasible solution for any given z L ().

Mathematical Problems in Engineering
We further obtain the equivalence of (28) as below: Obviously, to any given z  (), solution y  () could be derived from (29) if z() is differentiable enough; that is, z is well defined.Thus, (19) always has unique solution towards any given suitable vector z  ().
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 (,  1 ) is regular.The above-mentioned summarization brings us to Theorem 9.
Then, the system has unique solution if there exist nonsingular matrices  ∈  × and  ∈  × , such that (, )∼ ( Ẽ, Ã), where Ẽ = diag(, ), Ã = diag( 1 , ), and matrix pair (,  1 ) is regular.Remark 10.Theorem 9 is also suitable when other fractional derivatives are involved.Since  0  ()  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.

Examples
Now let us take some examples on the theorems represented before.) ) ) ) ) ) .
(32)       = ( − 1) 2 , which is not a zero polynomial; hence the matrix pair (,  1 ) is regular.From Theorem 9, the fractional system has unique solution.Now let us find the solution.
By setting y() =  ỹ(), that is, ỹ() =  −1 y(), multiplying  on the left to (a), we obtain the equivalent transformation of (a) in the next form: