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Mathematical Problems in Engineering
Volume 8 (2002), Issue 4-5, Pages 367-387

Schwartz' distributions in nonlinear setting: Applications to differential equations, filtering and optimal control

CICESE Research Center, Electronics and Telecommunication Department, P.O. Box 434944 San Diego, CA 92143-4944, USA

Received 14 January 2002; Revised 12 March 2002

Copyright © 2002 Hindawi Publishing Corporation. 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.


The paper is intended to be of tutorial value for Schwartz' distributions theory in nonlinear setting. Mathematical models are presented for nonlinear systems which admit both standard and impulsive inputs. These models are governed by differential equations in distributions whose meaning is generalized to involve nonlinear, non single-valued operating over distributions. The set of generalized solutions of these differential equations is defined via closure, in a certain topology, of the set of the conventional solutions corresponding to standard integrable inputs. The theory is exemplified by mechanical systems with impulsive phenomena, optimal impulsive feedback synthesis, sampled-data filtering of stochastic and deterministic dynamic systems.