# American Institute of Mathematical Sciences

February  2005, 12(2): 243-250. doi: 10.3934/dcds.2005.12.243

## Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients

 1 Mathematics Department, University of Nevada, Reno, Reno, NV 89511, United States 2 Transmag Research Institute, Academy of Sciences of Ukraine, 49005 Dnepropetrovsk, Piesarzhevsky 5, Ukraine

Received  October 2003 Revised  September 2004 Published  December 2004

Linear systems with partially uncertain time-dependent coefficients naturally arise in various areas of applied sciences. The Lyapunov function method became the prevailing approach to stability analysis of these systems, where it implies sufficient conditions for asymptotic stability. However, this method is not applicable to Hamiltonian systems, because they are at the utmost neutrally (but not asymptotically) stable. This paper describes stability analysis of linear Hamiltonian systems with partially uncertain periodic coefficients resulting in a generalization of a famous Yakubovich's stability theorem which significantly reduces its computational burden. We assume that the Hamiltonian is described as a sum of known and uncertain matrices, and for the latter only bilateral bounds are known. We propose a definition of such a system's stability and derive a necessary and sufficient stability criterion determined by positions of the multipliers of the corresponding marginal systems. Systems with a more general definition of uncertain matrices are also considered.
Citation: Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243
 [1] Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017 [2] Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 [3] Philippe Jouan, Said Naciri. Asymptotic stability of uniformly bounded nonlinear switched systems. Mathematical Control & Related Fields, 2013, 3 (3) : 323-345. doi: 10.3934/mcrf.2013.3.323 [4] Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069 [5] Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059 [6] José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357 [7] Liming Wang. A passivity-based stability criterion for reaction diffusion systems with interconnected structure. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 303-323. doi: 10.3934/dcdsb.2012.17.303 [8] Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707 [9] Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939 [10] Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 [11] Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 [12] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [13] Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 [14] B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 [15] Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983 [16] Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 [17] Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 [18] Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222 [19] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [20] 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

2018 Impact Factor: 1.143

## Metrics

• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar