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An Application Of The Theory Of Scale Of Banach Spaces

100%
Dawidowski Ł.
Annales Mathematicae Silesianae
|
2015
|
tom 29
|
nr 1
51-59
EN
The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.
2
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On an optimal control problem for a quasilinear parabolic equation

80%
Farag S. H., Farag M. H.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 2
239-250
EN
An optimal control problem governed by a quasilinear parabolic equation with additional constraints is investigated. The optimal control problem is converted to an optimization problem which is solved using a penalty function technique. The existence and uniqueness theorems are investigated. The derivation of formulae for the gradient of the modified function is explainedby solving the adjoint problem.
3
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Source localization and sensor placement in environmental monitoring

61%
Khapalov A.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 3
445-458
EN
In this paper we discuss two closely related problems arising in environmental monitoring. The first is the source localization problem linked to the question How can one find an unknown "contamination source"? The second is an associated sensor placement problem: Where should we place sensors that are capable of providing the necessary "adequate data" for that? Our approach is based on some concepts and ideas developed in mathematical control theory of partial differential equations.
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