2003, 2003(Special): 734-741. doi: 10.3934/proc.2003.2003.734

On the central stability zone for linear discrete-time Hamiltonian systems

1. 

Department of Automatic Control, University of Craiova, A.I. Cuza Str. No. 13, RO-1100 Craiova, Romania

Received  September 2002 Revised  March 2003 Published  April 2003

In this paper we start from the discrete version of linear Hamiltonian systems with periodic coefficients

$_y_(k+1) - _y(_k) = \lambda B(__k)_y(_k) + \lambda D_(k^z_(k+1))$
$_z_(k+1) - _z(_k) = -\lambda A(__k)_y(_k) - \lambda B*_(k^z_(k+1))$


where $A_k$ and $D_k$ are Hermitian matrices, $A_k$, $B_k$, $D_k$ define $N$-periodic sequences, and $\lambda$ is a complex parameter. For this system a Krein-type theory of the $\lambda$-zones of strong (robust) stability may be constructed. Within this theory the side $\lambda$-zones’ width may be estimated using the multipliers’ “traffic rules” of Krein while the central stability zone (centered around $\lambda$ = 0) is estimated using the eigenvalues of a certain boundary value problem which is self-adjoint. In the discrete-time there occur some specific differences with respect to the continuous time case due to the fact that the transition matrix (hence the monodromy matrix also) is not entire with respect to $\lambda$ but rational. During the paper we consider some specific cases (the matrix analogue of the discretized Hill equation, the J-unitary and symplectic systems, real scalar systems) for which the results on the eigenvalues are complete and obtain some simplified estimates of the central stability zones.

Citation: Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734
[1]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010

[2]

Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253

[3]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[4]

Sofian De Clercq, Koen De Turck, Bart Steyaert, Herwig Bruneel. Frame-bound priority scheduling in discrete-time queueing systems. Journal of Industrial and Management Optimization, 2011, 7 (3) : 767-788. doi: 10.3934/jimo.2011.7.767

[5]

Haijun Sun, Xinquan Zhang. Guaranteed cost control of discrete-time switched saturated systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4515-4522. doi: 10.3934/dcdsb.2020300

[6]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129

[7]

Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial and Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175

[8]

Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control and Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017

[9]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213

[10]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[11]

Elena K. Kostousova. On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty. Conference Publications, 2011, 2011 (Special) : 864-873. doi: 10.3934/proc.2011.2011.864

[12]

Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197

[13]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. Permanence and universal classification of discrete-time competitive systems via the carrying simplex. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1621-1663. doi: 10.3934/dcds.2020088

[14]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111

[15]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489

[16]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015

[17]

Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393

[18]

Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153

[19]

Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial and Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016

[20]

Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277

 Impact Factor: 

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]