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.
Highlights
The dynamical behavior of a mechanical system was usually modeled via differential-algebraic equations (DAE) whose general form appears as F(t, y, y)̇ = 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
Some investigators tried using fractional differential-algebraic equations (FDAE), which denote the combination of DAE and fractional differential equations (FDE), in dealing with the studied system
The general form of FDAE appears as F(t, y, y(α)) = 0, where F, y, y(α) could be vectors if necessary
Summary
Researchers had effectively solved engineering problems with fractional differential equations (FDE), which involves fractional derivatives y(α) in the model [1,2,3,4,5,6]. Some investigators tried using fractional differential-algebraic equations (FDAE), which denote the combination of DAE and FDE, in dealing with the studied system. 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. 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
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