Abstract
The technical (practical) stability problem for a set of trajectories of discrete systems on a metric space of nonempty convex compact sets in \(\Bbb R ^ n\) is considered. On the basis of known results of convex geometry and comparison method, an approach of constructing the auxiliary Lyapunov functionals for the study of technical stability in terms of two measures of evolutionary equations with Hukuhara difference operator is proposed. The problem of estimating the solutions of equations is reduced to the study of finite-dimensional difference equations of comparison. Examples of technical stability study are given to illustrate the constructiveness of this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Blasi, F.S., Iervolino, F. “Equazioni differentiali con soluzioni a valore compatto convesso”, Bull. Unione mat. Ital. 4 (2), 491–501 (1969).
Tolstonogov, A.A. Differential Inclusions in a Banach Space (Nauka, Novosibirsk, 1986) [in Russian].
Lakshmikantham, V., Gnana Bhaskar, T., Vasundhara Devi, J. Theory of Set Differential Equations in Metric Spaces (Cambridge Sci. Publ., London, 2006).
Park, J.Y., Han, H.K. “Existence and Uniqueness Theorem for a Solution of Fuzzy Differential Equations”, Int. J. Math. and Math. Sci. 22 (2), 271–279 (1999).
Plotnikov, A.V. Differential Inclusions with Hukuhara Derivative and Some Control Problems (Odessa, dep. v VINITI, N 2036-82.12, 1982) [in Russian].
Moiseev, M.D. “On Some Methods of the Theory of Technical Stability”, Trudy Voen.-vozd. Academii im. Zhukovskogo 135 (1945).
Moiseev, M.D. Essays on the Development of the Theory of Stability (Gostekhizdat, Moscow–Leningrad, 1949) [in Russain].
Četaev, N.G. “On the Choice of Parameters of a Stable Mechanical System”, Akad. Nauk SSSR. Prikl. Mat. Meh. 15, 371–-372 (1951).
Karacharov, K.A., Pilyutik, A.G. Introduction to the Technical Theory of Motion Stability (Fizmatgiz, Moscow, 1962) [in Russian].
Martynyuk, A.A. Practical Stability of Motion (Naukova Dumka, Kiev, 1983) [in Russian].
Lakshmikantham, V., Leela, S., Martynyuk, A.A. Practical Stability of Nonlinear Systems (World Scientific, Singapore, 1990).
Ocheretnyuk, E.V., Slyn'ko, V.I. “Estimates of the Volume of Solutions of Some Differential Equations with a Hukuhara Derivative”, Math. Notes 97 (3–4), 431–437 (2015).
Ocheretnyuk, E.V., Slyn'ko, V.I. “Qualitative Analysis of Solutions of Nonlinear Differential Equations with the Hukuhara Derivative in the Space \({\operatorname{conv\,}}(\Bbb R^2)\) ”, Differ. Equ. 51 (8), 998–-1013 (2015).
Slyn'ko, V.I. “Stability in Terms of Two Measures for Set Difference Equations in Space \({\operatorname{conv\,}}(\Bbb R^n)\)”, Appl. Anal. 96(2), 278–292 (2017).
Aleksandrov, A.D. Selected works. Ch. 1: Geometry and Applications (Nauka, Novosibirsk, 2006) [in Russian].
Leichtweiss, K. Konvexe Mengen (Convex sets) (VEB, Berlin, 1980; Nauka, Moscow, 1985).
Chernous'ko, F.L. Estimation of the Phase State of Dynamic Systems (Nauka, Moscow, 1988) [in Russian].
Funding
This work was partially supported by the Ministry of Education and Science of Ukraine (project no. 0116U004691).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 7, pp. 63–75.
About this article
Cite this article
Denysenko, V.S. On Technical Stability for Sets of Trajectories of Discrete Systems. Russ Math. 64, 54–65 (2020). https://doi.org/10.3103/S1066369X20070075
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X20070075