Necessary conditions for similarity between two lattice differential equations

Xiaoying Wang; Yong Li

doi:10.3934/dcdsb.2024101

Article Contents

2025, Volume 30, Issue 2: 554-576. Doi: 10.3934/dcdsb.2024101

This issue Previous Article Theoretical and numerical study of the decay in a viscoelastic Bresse System Next Article Carleman linearization of nonlinear systems and its finite-section approximations

Necessary conditions for similarity between two lattice differential equations

Xiaoying Wang^1, and
Yong Li^{1, 2, ,}

1.
School of Mathematics, Jilin University, Changchun, 130012, China
2.
Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China

^*Corresponding author: Yong Li
^*Corresponding author: Yong Li

Received: June 2024

Revised: July 2024

Early access: August 8, 2024

Published online: August 8, 2024

The work was supported by NSFC Grant (12071175; 11901056; 11571065), Jilin Science and Technology Development Program (20190201302JC; 20180101220JC).

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Abstract

This paper aims to study the similarity between two lattice differential equations. Motivated by the the conjugate theory of dynamical systems, we present the definition of conjugacy and similarity for the lattice differential equations and introduce a functional to measure the degree of their similarity. We prove the maximum principle for lattice control systems. Furthermore, for two lattice differential equations, we determine the necessary conditions for the minimizer of the functional. Then we apply the definitions and results to provide the conditions that the linear minimizer satisfies.

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Mathematics Subject Classification: Primary: 34C41, 34A33, 37L60.

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References

[1]	P. W. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50. doi: 10.1016/j.physd.2014.08.004.
[2]	H. Bessaih, M. J. Garrido-Atienza, X. Han and B. Schmalfuss, Stochastic lattice dynamical systems with fractional noise, SIAM J. Math. Anal., 49 (2017), 1495-1518. doi: 10.1137/16M1085504.
[3]	A. Calvia, S. Federico and F. Gozzi, State constrained control problems in Banach lattices and applications, SIAM J. Control Optim., 59 (2021), 4481-4510. doi: 10.1137/20M1376959.
[4]	S. N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.
[5]	X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467. doi: 10.3934/dcds.2011.31.445.
[6]	X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009. doi: 10.1016/j.jde.2016.05.015.
[7]	X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12. doi: 10.1016/j.physd.2018.09.004.
[8]	X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.
[9]	P. Hartman, Ordinary Differential Equations, John Wiley, New York, 1964.
[10]	X. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.
[11]	X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems. Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.
[12]	W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339. doi: 10.1016/S0362-546X(03)00065-8.
[13]	M. van Noort, M. A. Porter, Y. Yi and S.-N. Chow, Quasiperiodic dynamics in Bose-Einstein condensates in periodic lattices and superlattices, J. Nonlinear Sci., 17 (2007), 59-83. doi: 10.1007/s00332-005-0723-4.
[14]	B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.
[15]	X. Wang, Y. Han and Y. Li, Similarity between two stochastic differential systems, preprint, 2023, arXiv: 2310.10901.
[16]	X. Wang, Y. Li and Y. Han, Similarity between two dynamical systems, preprint, 2023, arXiv: 2310.03383.
[17]	X. Wang, J. Shen, K. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516. doi: 10.1016/j.jde.2021.01.026.
[18]	Z. Wu, Y. Liu and H. Wang, Optimal Control Theory: A Concise Introduction, Higher Education Press, Beijing, 2017.
[19]	W. Yan, Y. Li and S. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 032702, 17 pp. doi: 10.1063/1.3319566.
[20]	X. Zhou, Y. Li and X. Jiang, Periodic solutions in distribution of stochastic lattice differential equations, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 1300-1322. doi: 10.3934/dcdsb.2022123.