DOI: https://doi.org/10.15587/1729-4061.2019.178635

Finding of bounded solutions to linear impulsive systems

Farkhod Asrorov, Valentyn Sobchuk, Оlexandr Kurylko

Abstract


The problem of the existence of bounded on the entire real axis solutions to linear nonhomogeneous systems of differential equations undergoing impulsive perturbations at the fixed moments of time is investigated. Sufficient conditions for the hyperbolicity of solutions to the homogeneous multidimensional impulsive system are obtained. The derived conditions are applied to the study of the bounded solutions to the nonhomogeneous impulsive system. Sufficient conditions for the existence of a unique bounded solution to the nonhomogeneous system in the case of weak regularity of the corresponding homogeneous system are formulated. The advantage of such an approach is that the established conditions can be effectively tested for specific classes of impulse-perturbed systems, since they are formulated in terms of coefficients of initial problems. The obtained conditions allow applying classical solution methods of differential equations for the propositions on solvability and continuous dependence of solutions on parameters of the impulsive systems.

The theory of systems with impulsive actions has wide possibilities for its application. Many evolution processes in physics, engineering, automatic control, biology, economics are exposed to short-term perturbations during their evolution. For example, processes with abrupt changes are observed in mechanics (spring movement under shock influence, functioning of the clock mechanism, change of rocket speed at separation stages), in radio engineering (generation of impulses of various forms). Similar processes are also observed in biology (heart beat, cell division), biotechnology (growing biocomposites), and control theory (industrial robots).

Therefore, qualitative investigation of impulsive systems in this work is a  relevant challenge of the modern theory of mathematical modeling

Keywords


differential equations; impulsive system; bounded solution; Green-Samoilenko function; regular solution

References


Anashkin, O. V., Dovzhik, T. V., Mit'ko, O. V. (2010). Ustoychivost' resheniy differentsial'nyh uravneniy pri nalichii impul'snyh vozdeystviy. Dinamicheskie sistemy, 28, 3–10.

Wang, Y., Lu, J. (2020). Some recent results of analysis and control for impulsive systems. Communications in Nonlinear Science and Numerical Simulation, 80, 104862. doi: https://doi.org/10.1016/j.cnsns.2019.104862

Dashkovskiy, S., Feketa, P., Kapustyan, O., Romaniuk, I. (2018). Invariance and stability of global attractors for multi-valued impulsive dynamical systems. Journal of Mathematical Analysis and Applications, 458 (1), 193–218. doi: https://doi.org/10.1016/j.jmaa.2017.09.001

Asrorov, F., Perestyuk, Y., Feketa, P. (2017). On the stability of invariant tori of a class of dynamical systems with the Lappo–Danilevskii condition. Memoirs on Differential Equations and Mathematical Physics, 72, 15–25.

Kapustyan, O. V., Asrorov, F. A., Perestyuk, Y. M. (2019). On the Exponential Stability of a Trivial Torus for One Class of Nonlinear Impulsive Systems. Journal of Mathematical Sciences, 238 (3), 263–270. doi: https://doi.org/10.1007/s10958-019-04234-9

Kapustian, O. A., Sobchuk, V. V. (2018). Approximate Homogenized Synthesis for Distributed Optimal Control Problem with Superposition Type Cost Functional. Statistics, Optimization & Information Computing, 6 (2), 233–239. doi: https://doi.org/10.19139/soic.v6i2.305

Bonotto, E. M., Bortolan, M. C., Caraballo, T., Collegari, R. (2016). Impulsive non-autonomous dynamical systems and impulsive cocycle attractors. Mathematical Methods in the Applied Sciences, 40 (4), 1095–1113. doi: https://doi.org/10.1016/j.jde.2016.11.036

Bonotto, E. M., Gimenes, L. P., Souto, G. M. (2017). Asymptotically almost periodic motions in impulsive semidynamical systems. Topological Methods in Nonlinear Analysis, 49 (1), 133–163. doi: https://doi.org/10.12775/tmna.2016.065

Dashkovskiy, S., Feketa, P. (2017). Input-to-state stability of impulsive systems and their networks. Nonlinear Analysis: Hybrid Systems, 26, 190–200. doi: https://doi.org/10.1016/j.nahs.2017.06.004

Iovane, G., Kapustyan, A. V., Valero, J. (2008). Asymptotic behaviour of reaction–diffusion equations with non-damped impulsive effects. Nonlinear Analysis: Theory, Methods & Applications, 68 (9), 2516–2530. doi: https://doi.org/10.1016/j.na.2007.02.002

Perestyuk, M. O., Kapustyan, O. V. (2012). Long-time behavior of evolution inclusion with non-damped impulsive effects. Memoirs of Differential equations and Mathematical physics, 56, 89–113.

Kapustyan, O. V., Perestyuk, M. O. (2016). Global Attractors in Impulsive Infinite-Dimensional Systems. Ukrainian Mathematical Journal, 68 (4), 583–597. doi: https://doi.org/10.1007/s11253-016-1243-0

Izhikevich, E. M. (2006). Dynamical systems in neuroscience: The geometry of excitability and bursting. MIT Press. doi: https://doi.org/10.7551/mitpress/2526.001.0001


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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061