Opuscula Math. 39, no. 1 (2019), 23-37
https://doi.org/10.7494/OpMath.2019.39.1.23

Opuscula Mathematica

# Dynamic system with random structure for modeling security and risk management in cyberspace

Miroslava Růžičková

Abstract. We deal with the investigation of $$L_{2}$$-stability of the trivial solution to the system of difference equations with coefficients depending on a semi-Markov chain. In our considerations, random transformations of solutions are assumed. Necessary and sufficient conditions for $$L_{2}$$-stability of the trivial solution to such systems are obtained. A method is proposed for constructing Lyapunov functions and the conditions for its existence are justified. The dynamic system and methods discussed in the paper are very well suited for use as models for protecting information in cyberspace.

Keywords: semi-Markov chain, random transformation of solutions, the Lyapunov function, $$L_{2}$$-stability, systems of difference equations, jumps of solutions, cybersecurity.

Mathematics Subject Classification: 34F05, 60J28.

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• https://orcid.org/0000-0003-3158-6844
• Kyiv National Economic University named after Vadym Hetman, Department of Computer Mathematics and Information Security, Kiev 03068, Peremogy 54/1, Ukraine
• Communicated by Josef Diblík.
• Revised: 2018-02-19.
• Accepted: 2018-03-02.
• Published online: 2018-08-07.