Abstract
This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chow, W. Uber systeme von linearen partiellen differentialgleichungen erster ordnung. Math. Ann., 117: 98–105 (1939)
Gaines, R., Mawhin, J. Coincidence degree and nonlinear differential equations. Springer-Verlag, Berlin, 1977
Haynes, G., Hermes, H. Nonlinear controllability via Lie theory. SIAM Journal on Control and Optimization, 1970, 8: 450–460 (1970)
Hermann, R. On the accessibility problem in control theory. International Symposium on Non-linear Differential Equations and Nonlinear Mechanics. Academic Press, New York, 1963, 325–332
Hermann, R., Krener, A. Nonlinear controllability and observability. IEEE Transactions on Automatic Control, AC-22(5): 728–740 (1977)
Kalman, R. On the general theory of control systems. IRE Transactions on Automatic Control., 4(3): 110 (1960)
Krener, A. A generalization of Chow’s theorem and the Bang-Bang theorem to nonlinear control problems. SIAM Journal on Control and Optimization, 12: 43–52 (1974)
Lobry, C. Controlabilit des systmes non linaires. SIAM Journal on Control and Optimization, 8: 573–605 (1970)
Schwartz, J. Nonlinear Functional Analysis. Notes, Courant Inst. of Math. Sci., 1965
Sussmann, H., Jurdevic, V. Controllability of nonlinear systems. Journal of Differential Equations, 1972, 12: 95–116 (1972)
Tan, X., Li Y. The null controllability of nonlinear discrete control systems with degeneracy. IMA J. Math. Control Inform., 34(3): 999–1010 (2017)
Tie, Lin. On near-controllability, nearly controllable subspaces, and near-controllability index of a class of discrete-time bilinear systems: a Root Locus Approach. SIAM Journal on Control and Optimization, 52(2): 1142–1165 (2014)
Tie, Lin. On small-controllability and controllability of a class of nonlinear systems. International Journal of Control, 87(10): 2167–2175 (2014)
Tie, Lin. On controllability, near-controllability, controllable subspaces, and nearly-controllable subspaces of a class of discrete-time multi-input bilinear systems. Systems Control Lett., 97: 36–47 (2016)
Zhao, S., Sun J. A geometric method for observability and accessibility of discrete impulsive nonlinear systems. International Journal of Systems Science, 44(8): 1522–1532 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by National Natural Science Foundation of China (grant No. 41874132), The third author is supported by National Natural Science Foundation of China (grant No. 11201173), Science and Technology Developing Plan of Jilin Province (grant No. 20180101220JC), and the fourth author is supported by National Basic Research Program of China (grant No. 2013CB834100), National Natural Science Foundation of China (grant No. 11171132, grant No. 11571065), Jilin DRC (grant No.2017C028-l).
Rights and permissions
About this article
Cite this article
Lyu, Y., Tan, Xl., Yang, X. et al. Controllability of Nonlinear Discrete Systems with Degeneracy. Acta Math. Appl. Sin. Engl. Ser. 39, 293–305 (2023). https://doi.org/10.1007/s10255-023-1047-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-023-1047-6