Closed Forms of General Solutions for Rectangular Systems of Coupled Generalized Sylvester Matrix Differential Equations

Pattrawut Chansangiam

Abstract


We investigate a rectangular system of coupled generalized Sylvester matrix differential equations in both nonhomogeneous and homogeneous cases. In order to obtain a closed form of its general solution, we transform it to an equivalent vector differential equation. This is done by using the vector operator and the Kronecker product. An explicit form of its general solution is given in terms of matrix series concerning Mittag-Leffler functions, exponentials, and hyperbolic functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases.


Keywords


Matrix differential equation; Vector operator; Kronecker product; Mittag-Leffler function; Matrix exponential

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References


Z. Al-Zhour, A computationally-efficient solutions of coupled matrix differential equations for diagonal unknown matrices, Journal of Mathematical Sciences: Advances and Applications 1(2) (2008), 373 – 387, https://pdfs.semanticscholar.org/eba7/ae799aec881f3baa2d7e3978641a3ddbc5c7.pdf.

Z. Al-Zhour, Efficient solutions of coupled matrix and matrix differential equations, Intelligent Control and Automation 3(2) (2012), 176 – 187, DOI: 10.4236/ica.2012.32020.

Z. Al-Zhour, The general (vector) solutions of such linear (coupled) matrix fractional differential equations by using Kronecker structures, Applied Mathematics and Computation 232 (2014), 498 – 510, DOI: 10.1016/j.amc.2014.01.079.

Z. Al-Zhour, New techniques for solving some matrix and matrix differential equations, Ain Shams Engineering Journal 6 (2015), 347 – 354, DOI: 10.1016/j.asej.2014.08.009.

Z. Al-Zhour, The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense, Alexandria Engineering Journal 55 (2016), 1675 – 1681, DOI: 10.1016/j.aej.2016.02.024.

S. Barnett, Matrix differential equations and Kronecker products, SIAM Journal on Applied Mathematics 24(1) (1973), 1 – 5, DOI: 10.1137/0124001.

G. N. Boshnakov, The asymptotic covariance matrix of the multivariate serial correlations, Stochastic Processes and their Applications 65 (1996), 251 – 258, DOI: 10.1016/S0304-4149(96)00104-4.

T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems, Springer, London (1995), DOI: 10.1007/978-1-4471-3037-6.

H.-W. Cheng and S. S.-T. Yau, More explicit formulas for the matrix exponential, Linear Algebra and its Applications 262 (1997), 131 – 163, DOI: 10.1016/S0024-3795(97)80028-6.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York (1991), DOI: 10.1017/CBO9780511840371.

A. D. Karageorgos, A. A. Pantelous and G. I. Kalogeropoulos, Transferring instantly the state of higher-order linear descriptor (regular) differential systems using impulsive inputs, Journal of Control Science and Engineering 2009 (2009), Article ID 562484, 1 – 32, DOI: 10.1155/2009/562484.

A. Kilicman and Z. Al-Zhour, The general common exact solutions of coupled linear matrix and matrix differential equations, The 2nd International Conference on Research and Education in Mathematics (ICREM 2), 2005, pp. 75 – 87, URL: https://www.researchgate.net/publication/266167634_The_general_common_exact_solutions_of_coupled_linear_matrix_and_matrix_differential_equations.

R. Kongyaksee and P. Chansangiam, Solving system of nonhomogeneous coupled linear matrix differential equations in terms of Mittag-Leffler matrix functions, Journal of Computational Analysis and Applications 26(1) (2019), 1150 – 1160, http://www.eudoxuspress.com/images/JOCAAA-VOL-27-2019-ISSUE-7.pdf#page=78.

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometries, 3rd edition, John Wiley & Sons (2019), https://books.google.co.in/books?hl=en&lr=&id=8sOKDwAAQBAJ&oi=fnd&pg=PR13&ots=wg6z7KEYzj&sig=x70aNyZu4vmzeLOwmzoLcbqPgTU#v=onepage&q&f=false.

C. Moler and C. V. Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Review 45(1) (2003), 3 – 000, DOI: 10.1137/S00361445024180.

S.G. Mouroutsos and P.D. Sparis, Taylor series approach to system identification, analysis and optimal control, Journal of the Franklin Institute 319(3) (1985), 359 – 371, DOI: 10.1016/0016-0032(85)90056-0.

A. A. Pantelous, A. D. Karageorgos and G. I. Kalogeropoulos, A new approach for second-order linear matrix descriptor differential equations of Apostol–Kolodner type, Mathematical Methods in Applied Sciences 37 (2014), 257 – 264, DOI: 10.1002/mma.2824.

C. R. Rao and M. B. Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore (1998), DOI: 10.1142/9789812779281_0010.

B. Ross, Fractional Calculus and Its Applications, Springer-Verlag, Berlin (1975), DOI: 10.1007/BFb0067096.

S. Saechai and P. Chansangiam, Solving non-homogeneous coupled linear matrix differential equations in terms of matrix convolution product and Hadamard product, Journal of Informatics and Mathematical Sciences 10 (2018), 237 – 245, DOI: 10.26713/jims.v10i1-2.647.

W. H. Steeb and Y. Hardy, Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra, World Scientific, Singapore (2011), DOI: 10.1142/9789814335324_0005.

K. Tansri and P. Chansangiam, General solutions for descriptor systems of coupled generalized Sylvester matrix fractional differential equations via canonical forms, Symmetry 12 (2020), 283, DOI: 10.3390/sym12020283.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i3.1324

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