We analyze the properties of the maximal set of initial conditions of the problem of weak practical stability of discrete inclusions with spatial components. We prove compactness and the properties of the boundary and interior of the maximal set of practical stability. In the linear case, we obtain the Minkowski function and the inverse Minkowski function of the optimal set of initial conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Halanay and D. Wexler, Qualitative Theory of Impulsive Systems [in Russian], Mir, Moscow (1971).
A. N. Bashnyakov, V. V. Pichkur, and I. V. Hitko, “On maximal initial data set in problems of practical stability of discrete system,” J. Automat. Inf. Sci., 43, No. 3, 1–8 (2011).
A. Michel, L. Hou, and D. Liu, Stability of Dynamical Systems, Birkhäuser, Boston (2008).
V. V. Pichkur and M. S. Sasonkina, “Maximum set of initial conditions for the problem of weak practical stability of a discrete inclusion,” J. Math. Sci., 194, No. 4, 414–425 (2013).
V. V. Pichkur and M. S. Sasonkina, “Practical stabilization of discrete control systems,” Int. J. Pure Appl. Math., 81, No. 6, 877–884 (2012).
P. Antsaklis and A. Michel, Linear systems, Birkhäuser, Boston (2006).
P. Agarwal Ravi, Difference Equations and Inequalities. Theory, Methods, and Applications, Nat. Univ. Singapore, Singapore (1968).
O. Galor, Discrete Dynamical Systems, Springer, Berlin (2007).
Ya. N. Linder and V. V. Pichkur, “Conditions of weak practical stability of differential inclusions with impulse impact,” J. Automat. Inf. Sci., 43, No. 11, 57–65 (2011).
O. V. Kapustyan, M. O. Perestyuk, and I. V. Romanyuk, “Stability of global attractors of impulsive infinite-dimensional systems,” Ukr. Math. J., 70, No. 1, 30–41 (2018).
O. D. Kichmarenko and M. L. Karpycheva, “General averaging scheme for discrete equations with variable delay,” J. Math. Sci., 226, No. 3, 270–284 (2017).
F. G. Garashchenko and V. V. Pichkur, “On optimal sets of practical stability,” J. Automat. Inf. Sci., 31, No. 4-5, 1–12 (1999).
F. G. Garashchenko and V. V. Pichkur, “Optimal sets of practical stability of differential inclusions,” J. Automat. Inf. Sci., 35, No. 1-4, 5–12 (2003).
F. G. Garashchenko and V. V. Pichkur, “On properties of maximal set of external practical stability of discrete systems,” J. Automat. Inf. Sci., 48, No. 3, 46–53 (2016).
A. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Basel (1997).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow (2004).
V. V. Pichkur and M. S. Sasonkina, “On the continuous dependence of discrete inclusions on the initial conditions solutions,” Tavr. Vesn. Inform. Mat., No. 6, 73–80 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Neliniini Kolyvannya, Vol. 22, No. 4, pp. 526–531, October–December, 2019.
Rights and permissions
About this article
Cite this article
Pichkur, V.V., Linder, Y.M. & Tairova, M.S. On the Practical Stability of Discrete Inclusions with Spatial Components. J Math Sci 254, 280–286 (2021). https://doi.org/10.1007/s10958-021-05304-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05304-7