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April  2021, 26(4): 2037-2053. doi: 10.3934/dcdsb.2020365

Asymptotic dynamics of hermitian Riccati difference equations

Yueh-Cheng Kuo 1, , Huey-Er Lin 2,, and  Shih-Feng Shieh 2,

1. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan
2. 

Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan

* Corresponding author: Huey-Er Lin

(dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas)

Received  October 2020 Revised  November 2020 Published  April 2021 Early access  December 2020

In this paper, we consider the hermitian Riccati difference equations. Analogous to a Riccati differential equation, there is a connection between a Riccati difference equation and its associated linear difference equation. Based on the linear difference equation, we can obtain an explicit representation for the solution of the Riccati difference equation and define the extended solution. Further, we can characterize the asymptotic state of the extended solution and the rate of convergence. Constant equilibrium solutions, periodic solutions and closed limit cycles are exhibited in the investigation of asymptotic behavior of the hermitian Riccati difference equations.

Keywords: Riccati difference equations, hermitian Riccati difference equations, Riccati differential equations, the extended solution, asymptotic behavior.
Mathematics Subject Classification: 39Axx, 15Axx, 37N35.
Citation: Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365
References:
[1]

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations: In Control and Systems Theory, Birkhauser, Basel, 2003. doi: 10.1007/978-3-0348-8081-7.

[2]

R. R. Bitmead and M. R. Gevers, Riccati Difference and Differential Equations: Convergence, Monotonicity and Stability, In: S. Bittanti et al. (Ed.) The Riccati Equation, Berlin, Springer Verlag, 1991.

[3]

P. E. Caines and D. Q. Mayne, On the discrete time matrix Riccati equation of optimal control, Int. J. Control, 12 (1970), 785-794.  doi: 10.1080/00207177008931892.

[4]

S. W. Chan, G. C. Goodwin and K. S. Sin, Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems, IEEE Trans. Automal. Contr., 29 (1984), 110-118.  doi: 10.1109/TAC.1984.1103465.

[5]

D. J. Clement and B. D. O. Anderson, Polynomial factorization via the Riccati equation, SIAM J. Appl. Math., 31 (1976), 179-205.  doi: 10.1137/0131017.

[6]

G. Freiling and V. Ionescu, Nonsymmetric discrete-time difference and algebraic Riccati equations: Some representation formulae and comments, Dynam Systems Appl., 8 (1999), 421-437. 

[7]

G. Freiling and A. Hochhaus, Convergence and existence results for continuous- and discrete-time Riccati equations, Result.Math., 42 (2002), 252-276.  doi: 10.1007/BF03322854.

[8]

A. Gorodnik, Dynamical Systems and Ergodic Theory, Lecture Notes.

[9] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371.
[10]

V. Kučera, The discrete Riccati equation of optimal control, Kybernetika, 8 (1972), 430-447. 

[11]

Y. C. Kuo, H. E. Lin and S. F. Shieh, Time-asymptotic dynamics of hermitian riccati differential equations, Taiwanese J. Math., 24 (2020), 131-158.  doi: 10.11650/tjm/190605.

[12] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford, Clarendon Press, 1995. 
[13]

W. W. Lin, V. Mehrmann and H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra and Appl., 302/303 (1999), 469-533.  doi: 10.1016/S0024-3795(99)00191-3.

[14]

H. Wimmer, On the existence of a least and negative-semidefinite solution of the discrete-time algebraic Riccati equation, J. Math. Est. and Contr., 5 (1995), 445-457. 

show all references

References:
[1]

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations: In Control and Systems Theory, Birkhauser, Basel, 2003. doi: 10.1007/978-3-0348-8081-7.

[2]

R. R. Bitmead and M. R. Gevers, Riccati Difference and Differential Equations: Convergence, Monotonicity and Stability, In: S. Bittanti et al. (Ed.) The Riccati Equation, Berlin, Springer Verlag, 1991.

[3]

P. E. Caines and D. Q. Mayne, On the discrete time matrix Riccati equation of optimal control, Int. J. Control, 12 (1970), 785-794.  doi: 10.1080/00207177008931892.

[4]

S. W. Chan, G. C. Goodwin and K. S. Sin, Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems, IEEE Trans. Automal. Contr., 29 (1984), 110-118.  doi: 10.1109/TAC.1984.1103465.

[5]

D. J. Clement and B. D. O. Anderson, Polynomial factorization via the Riccati equation, SIAM J. Appl. Math., 31 (1976), 179-205.  doi: 10.1137/0131017.

[6]

G. Freiling and V. Ionescu, Nonsymmetric discrete-time difference and algebraic Riccati equations: Some representation formulae and comments, Dynam Systems Appl., 8 (1999), 421-437. 

[7]

G. Freiling and A. Hochhaus, Convergence and existence results for continuous- and discrete-time Riccati equations, Result.Math., 42 (2002), 252-276.  doi: 10.1007/BF03322854.

[8]

A. Gorodnik, Dynamical Systems and Ergodic Theory, Lecture Notes.

[9] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371.
[10]

V. Kučera, The discrete Riccati equation of optimal control, Kybernetika, 8 (1972), 430-447. 

[11]

Y. C. Kuo, H. E. Lin and S. F. Shieh, Time-asymptotic dynamics of hermitian riccati differential equations, Taiwanese J. Math., 24 (2020), 131-158.  doi: 10.11650/tjm/190605.

[12] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford, Clarendon Press, 1995. 
[13]

W. W. Lin, V. Mehrmann and H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra and Appl., 302/303 (1999), 469-533.  doi: 10.1016/S0024-3795(99)00191-3.

[14]

H. Wimmer, On the existence of a least and negative-semidefinite solution of the discrete-time algebraic Riccati equation, J. Math. Est. and Contr., 5 (1995), 445-457. 

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