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Dynamical generalized functions and the multiplication problem. (English. Russian original) Zbl 1442.46030

Russ. Math. 51, No. 5, 32-43 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 5, 33-45 (2007).
The problem considered in the paper is the construction of a product of discontinuous and generalized functions suitable to applications to nonlinear differential equations. Functions called dynamical by the authors are introduced as transformations of an interval to a set of measurable functions. The introduced functions are taken as test functions for constructing dynamical generalized functions that extend functions from the {L. Schwartz} space \(\mathscr{D}'\).
Linear operations, differentiation, and some type of multiplicators are defined for the generalized dynamical functions, especially, continuous and associative multiplications of discontinuous and generalized functions.
Applications to optimal control problems for differential equations are given. A comparison with some known results on multiplication is made, but among them there are no results of {V. K. Ivanov} on algebras of some generalized functions.

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F10 Operations with distributions and generalized functions
49N25 Impulsive optimal control problems
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