American Institute of Mathematical Sciences

Advanced
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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

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

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

[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.  Google Scholar

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

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

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

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

[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.  Google Scholar

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

[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.   Google Scholar

[1]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278
[2]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[3]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
[4]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093
[5]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[6]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[7]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[8]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[9]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[10]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[11]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[13]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[14]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[15]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[16]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[17]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[18]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[19]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[20]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (10)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences