P. P. Tkachenko, D. V. Balandin, T. V. Ryabikova

Sirius University of Science and Technology (Sirius, Russian Federation)

Abstract. An analytical framework for synthesizing worst-case external disturbances for linear dynamical systems described by ordinary differential equations is presented in the paper. The study is conducted for three classical functional spaces $L_2, L_{\infty}, L_1$ over a fixed time interval, which corresponds to identifying disturbances with bounded energy, bounded amplitude, and bounded impulse, respectively. Linear elastic mechanical systems are chosen as a illustrative object of analysis, thus providing an intuitive interpretation of the results. A unified performance metric is introduced for quantitative assessment of solutions. This metrics is the ratio of a system's target output (e.g., maximum deviation) to the $L_p$-norm of the disturbance (i.e. the normalized system response). Explicit analytical expressions for the worst-case disturbances and their corresponding performance indices are derived. The interrelations between the indices obtained for different disturbance classes are examined. Numerical simulation results are provided for single- and multiple-degree-of-freedom systems, represented as chains of point masses interconnected by elastic and damping elements, and connected to a movable base.

Key Words: multi-mass elastic system, maximal deformation, worst-case disturbance, linear ODE system, indicators of oscillatory activity, $L_p$-norm

For citation: P. P. Tkachenko, D. V. Balandin, T. V. Ryabikova. The problems of the worst-case disturbances acting on multi-mass elastic system. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:4(2025), 451–470. DOI: https://doi.org/10.15507/2079-6900.27.202504.451-470

Information about the authors:

Polina P. Tkachenko, Research Assistant at the Department of «Mathematical robotics and artificial intelligence», Sirius University of Science and Technology (1 Olympic Ave., Sirius Federal Territory 354340, Russia), ORCID: https://orcid.org/0000-0001-5132-234X, PTkachen@gmail.com

Dmitry V. Balandin, Dr. Sci. (Phys.-Math.), Professor at the Department of «Mathematical robotics and artificial intelligence», Sirius University of Science and Technology (1 Olympic Ave., Sirius Federal Territory 354340, Russia), ORCID: https://orcid.org/0000-0001-7727-5924, dbalandin@yandex.ru

Tatiana V. Ryabikova, PhD. Sci. (Phys.-Math.), Research Assistant at the Department of «Mathematical robotics and artificial intelligence», Sirius University of Science and Technology (1 Olympic Ave., Sirius Federal Territory 354340, Russia), ORCID: https://orcid.org/0000-0003-0302-2064, tanya.dovid@gmail.com