Taiwanese Journal of Mathematics

Time-asymptotic Dynamics of Hermitian Riccati Differential Equations

Yueh-Cheng Kuo, Huey-Er Lin, and Shih-Feng Shieh

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The matrix Riccati differential equation (RDE) raises in a wide variety of applications for science and applied mathematics. We are particularly interested in the Hermitian Riccati Differential Equation (HRDE). Radon's lemma gives a solution representation to HRDE. Although solutions of HRDE may show the finite escape time phenomenon, we can investigate the time asymptotic dynamical behavior of HRDE by its extended solutions. In this paper, we adapt the Hamiltonian Jordan canonical form to characterize the time asymptotic phenomena of the extended solutions for HRDE in four elementary cases. The extended solutions of HRDE exhibit the dynamics of heteroclinic, homoclinic and periodic orbits in the elementary cases under some conditions.

Article information

Taiwanese J. Math., Advance publication (2019), 28 pages.

First available in Project Euclid: 5 July 2019

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Primary: 15Axx: Basic linear algebra 65L20: Stability and convergence of numerical methods 93C15: Systems governed by ordinary differential equations [See also 34H05] 31B35: Connections with differential equations

Riccati differential equation Hermitian Riccati differential equation Radon's lemma finite escape time phenomenon extended solutions Hamiltonian Jordan canonical form


Kuo, Yueh-Cheng; Lin, Huey-Er; Shieh, Shih-Feng. Time-asymptotic Dynamics of Hermitian Riccati Differential Equations. Taiwanese J. Math., advance publication, 5 July 2019. doi:10.11650/tjm/190605. https://projecteuclid.org/euclid.twjm/1562313624

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  • H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Systems & Control: Foundations & Applications, Birkhäuser, Basel, 2003.
  • U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Englewood Cliffs, NJ, 1988.
  • I. Babu\uska and V. Majer, The factorization method for the numerical solution of two point boundary value problems for linear ODE's, SIAM J. Numer. Anal. 24 (1987), no. 6, 1301–1334.
  • F. M. Callier and J. L. Willems, Criterion for the convergence of the solution of the Riccati differential equation, IEEE Trans. Automat. Control 26 (1981), no. 6, 1232–1242.
  • E. Davison and M. Maki, The numerical solution of the matrix Riccati differential equation, IEEE Trans. Automat. Control 18 (1973), no. 1, 71–73.
  • L. Dieci, M. R. Osborne and R. Russell, A Riccati transformation method for solving linear BVPs I: Theoretical aspects, SIAM J. Numer. Anal. 25 (1988), no. 5, 1055–1073.
  • ––––, A Riccati transformation method for solving linear BVPs II: Computational aspects, SIAM J. Numer. Anal. 25 (1988), no. 5, 1074–1092.
  • R. Hermann and C. Martin, Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations I: General Lie-theoretic methods, Math. Systems Theory 15 (1982), no. 3, 277–284.
  • ––––, Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations II, Math. Systems Theory 16 (1983), no. 4, 297–306.
  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
  • T.-M. Huang, R.-C. Li and W.-W. Lin, Structure-preserving Doubling Algorithms for Nonlinear Matrix Equations, Fundamentals of Algorithms 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018.
  • C. S. Kenney and R. B. Leipnik, Numerical integration of the differential matrix Riccati equation, IEEE Trans. Automat. Control 30 (1985), no. 10, 962–970.
  • H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972.
  • D. G. Lainiotis, Partitioned Riccati solutions and integration-free doubling algorithms, IEEE Trans. Automatic Control 21 (1976), no. 5, 677–689.
  • W.-W. Lin, V. Mehrmann and H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra Appl. 302–303 (1999), 469–533.
  • J. Radon, Über die Oszillationtheoreme der kunjugierten Punkte beim Probleme von Lagrange, Münchener Sitzungsberichte 57 (1927), 243–257.
  • ––––, Zum Problem von Lagrange, Hamburger Math. Einzelschr. 6, 1928.
  • M. A. Shayman, Phase portrait of the matrix Riccati equation, SIAM J. Control Optim. 24 (1986), no. 1, 1–65.