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Abstract
We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a matrix canonical form concerning ranks, and matrix partitioning. The procedure aims to reduce the descriptor system to a descriptor system of fractional differential equations. We also impose a condition on coefficient matrices, related to the symmetry of the solution for descriptor systems. It follows that an explicit form of its general solution is given in terms of matrix power series concerning Mittag–Leffler functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases. In particular, we obtain an alternative procedure to solve linear continuous-time descriptor systems via a matrix canonical form.
Highlights
A motivation of this work comes from the treatment of linear algebra and differential equations for control and system theory
The last step is to transform the new variables to the original ones, so that an explicit formula of the general solution is obtained in terms of Mittag–Leffler matrix functions
We investigate certain interesting special cases of the main system for both descriptor systems of coupled matrix equations, single descriptor matrix equations, and linear continuous-time descriptor system
Summary
A motivation of this work comes from the treatment of linear algebra and differential equations for control and system theory. To solve a regular linear continuous-time system (1), many authors used an idea of transforming the generalized state vector x (t) through a suitable matrix canonical form [5,6,7]. A general system of non-homogeneous coupled linear matrix differential equations takes the form. A simple system of non-homogeneous linear matrix fractional differential equations takes the form. A general system of non-homogeneous coupled linear matrix fractional differential equations takes the form. The last step is to transform the new variables to the original ones, so that an explicit formula of the general solution is obtained in terms of Mittag–Leffler matrix functions. We investigate certain interesting special cases of the main system for both descriptor systems of coupled matrix equations, single descriptor matrix equations, and linear continuous-time descriptor (vector) system.
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