Email this article ON WELL-POSEDNESS OF GENERALIZED NEURAL FIELD EQUATIONS WITH IMPULSIVE CONTROL
- Authors: Burlakov E.O.1, Zhukovskiy E.S.2
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Affiliations:
- Norwegian University of Life Sciences
- Tambov State University named after G.R. Derzhavin
- Issue: Vol 21, No 1 (2016)
- Pages: 16-27
- Section: Articles
- URL: https://ogarev-online.ru/2686-9667/article/view/362906
- DOI: https://doi.org/10.20310/1810-0198-2016-21-1-16-27
- ID: 362906
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Abstract
We formulate and prove the theorem on well-posedness of abstract Volterra equations in metric spaces. We consider nonlinear nonlocal integral Volterra equation involving generalizing equations typically used in mathematical neuroscience. We investigate solutions that tend to zero at any moment with unbounded growth of the spatial variable. In the literature such solutions are called «spatially localized solutions» or «bumps». They correspond to normal brain functioning. For the main equation, we consider an impulsive control problem, where the control parameters are moments, when the solution discontinue, and corresponding jumps’ values. Such control systems model electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric (not linear) space. Using the aforementioned theorem, we obtain conditions for existence and uniqueness of solution to this equation and continuous dependence of this solution on the control.
About the authors
Evgeniy Olegovich Burlakov
Norwegian University of Life Sciences
Email: eb_@bk.ru
Post-graduate Student As, Norwegian
Evgeny Semenovich Zhukovskiy
Tambov State University named after G.R. Derzhavin
Email: zukovskys@mail.ru
Doctor of Physics and Mathematics, Professor, Director of the Research Institute of Mathematics, Physics and Informatics Tambov, the Russian Federation
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