Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 1, Pages 1-10
doi:10.1155/S1048953396000019
On an infinite-dimensional differential equation in vector
distribution with discontinuous regular functions in a right hand side
1Institute of Control Science, Moscow, Russia
236-1-135,Matveevskaya ul., Moscow 119517, Russia
Received 1 April 1995; Revised 1 September 1995
Copyright © 1996 Michael V. Basin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
An infinite-dimensional differential equation in vector distribution in a Hilbert space is studied in case of an unbounded operator and discontinuous regular
functions in a right-hand side. A unique solution (vibrosolution) is defined for
such an equation, and the necessary and sufficient existence conditions for a vibrosolution are proved. An equivalent equation with a measure, which enables us to
directly compute jumps of a vibrosolution at discontinuity points of a distribution function, is also obtained. The application of the obtained results to control
theory is discussed in the conclusion.