Abstract
We execute the complete integration of the simplest matrix Riccati equation in the two- and three-dimensional cases for an arbitrary linear differential operator. The solution is constructed in terms of the Jordan form of an unknown matrix and the corresponding similarity matrix. We show that a similarity matrix is always representable as the product of two matrices one of which is an invariant of the differential operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. I. Zelikin, Homogeneous Spaces and the Riccati Equation in the Calculus of Variations (Faktorial, Moscow, 1998) [in Russian].
F. A. Chernous’ko and V. B. Kolmanovskii, Optimal Control under Random Perturbations (Nauka, Moscow, 1978) [in Russian].
L. V. Ovsyannikov, Lectures on the Fundamentals of Gas Dynamics (Izhevsk Institute of Computer Science, Moscow–Izhevsk, 2003) [in Russian].
A. P. Chupakhin, “Barochronous Gas Motions: General Properties and Submodels of Types \((1,2)\) and \((1,1) \),” Preprint No. 4–98 (Lavrentyev Institute of Hydrodynamics, Novosibirsk, 1998).
A. P. Chupakhin, “Nonbarochronous Submodels of Types \((1,2) \) and \((1,1) \) of the Gas Dynamics Equations,” Preprint No. 1–99 (Lavrentyev Institute of Hydrodynamics, Novosibirsk, 1999).
A. A. Cherevko and A. P. Chupakhin, “The Stationary Ovsyannikov Vortex,” Preprint No. 1–2005 (Lavrentyev Institute of Hydrodynamics, Novosibirsk, 2005).
E. Miller, “A Regularity Criterion for the Navier—Stokes Equation Involving Only the Middle Eigenvalue of the Strain Tensor,” Arch. Rational Mech. Anal. 235, 99–139 (2020).
C. A. Stephen and M. W. Gary, “Hierarchies of New Invariants and Conserved Integrals in Inviscid Fluid Flow,” Phys. Fluids 32, 086104 (2020) [https://doi.org/10.1063/5.0011649].
A. G. Fat’yanov, “Some Semianalytical Method for Solution of the Direct Dynamical Problems in Stratified Media,” Dokl. Akad. Nauk SSSR 310 (2), 323–327 (1990).
A. L. Karchevsky, “Analytical Solution of the Maxwell Equations in the Frequency Domain for Horizontal Layered Anisotropic Media,” Geologiya i Geofizika 48 (8), 889–898 (2007).
A. L. Karchevsky and B. R. Rysbayuly, “Analytical Expressions for a Solution of Convective Heat and Moisture Transfer Equations in the Frequency Domain for Layered Media,” Euras. J. Math. Comp. Appl. 3 (4), 55–67 (2015).
A. L. Karchevsky, “Analytical Solutions of the Differential Equation of Transverse Vibrations of a Piecewise Homogeneous Beam in the Frequency Domain for the Boundary Conditions of Various Types,” Sibir. Zh. Ind. Mat. 23 (4), 48–68 (2020) [J. Appl. Ind. Math. 14 (4), 648–665 (2020)].
J. Peyrière, “On an Article by W. Magnus on the Fricke Characters of Free Groups,” J. Algebra 228, 659–673 (2020).
W. Magnus, “Rings of Fricke Characters and Automorphism Groups of Free Groups,” Math. Zh. 170, 91–103 (1980).
C. Procesi, “The Invariant Theory of \(n\times n \) Matrices,” Adv. Math. 19, 306–381 (1976).
Yu. P. Razmyslov, “Trace Identities of Complete Matrix Algebras over a Field of Zero Characteristic,” Izv. Akad. Nauk SSSR Ser. Mat. 38 (4), 723–756 (1974) [Math. USSR, Izv. 8, 727–760 (1975)].
A. Whittemore, “On Special Linear Characters of Free Groups of Rank \(n\geq 4 \),” Proc. Amer. Math. Soc. 40, 383–388 (1973).
Y. Avishai, D. Berend, and V. Tkachenko, “Trace Maps,” Internat. J. Modern Phys. B. 11, 3525–3542 (1997).
M. Bresara, C. Procesi, and S. Spenko, “Quasi-Identities on Matrices and the Cayley–Hamilton Polynomial,” Adv. Math. 280, 439–471 (2015).
M. D. Cvetković and L. S. Velimirović, “Application of Shape Operator under Infinitesimal Bending of Surface,” Filomat 33 (4), 1267–1271 (2019).
L. S. Velimirović, M. D. Cvetković, M. S. Najdanović, and N. M. Velimirović, “Variation of Operator under Infinitesimal Bending of Surface,” Appl. Math. Comput. 225, 480–486 (2013).
K. K. Brustad, “Total Derivatives of Eigenvalues and Eigenprojections of Symmetric Matrices,” arXiv: 1905.06045v1 [math.AP] 15 May 2019.
N. J. Rose, “On the Eigenvalues of a Matrix Which Commutes with Its Derivative,” Proc. Amer. Math. Soc. 4, 752–754 (1965).
I. J. Epstein, “Conditions for a Matrix to Commute with Its Integral,” Proc. Amer. Math. Soc. 14, 266–270 (1963).
M. Hausner, “Eigenvalues of Certain Operators on Matrices,” Comm. Pure Appl. Math. 14, 155–156 (1961).
F. R Gantmakher, The Theory of Matrices (Fizmatlit, Moscow, 2010; AMS Chelsea Publ., Providence, 1998).
Funding
The authors were supported by the Programs of Basic Research nos. III.22.4.1 and I.1.5 (project no. 0314–2019–0011) of the Siberian Branch of the Russian Academy of Sciences.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by Ya.A.Kopylov
Rights and permissions
About this article
Cite this article
Neshchadim, M.V., Chupakhin, A.P. On Integration of a Matrix Riccati Equation. J. Appl. Ind. Math. 14, 732–742 (2020). https://doi.org/10.1134/S1990478920040110
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478920040110