Some generalizations of the Showalter–Sidorov problem for Sobolev-type models

At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very significant. The initial Showalter–Sidorov condition is not simply a generalization of the Cauchy condition for Sobolev-type models. It allows to find an approximate solution without checking the coordination of initial data. This article presents an overview of some results of the Chelyabinsk mathematical school on Sobolev type equations obtained using either directly Showalter–Sidorov condition or its generalizations.
The article consists of seven sections. The first one includes results on investigation of solvability of an optimal measurement problem for the Shestakov–Sviridyuk model. The second section provides an overview of the currently existing approaches to the concept of white noise. The third section contains results on solvability of a weakened Showalter–Sidorov problem for the Leontief type system with additive “white noise”. In the fourth section we present results on the unique solvability of multipoint initial-final value problem for the Sobolev type equation of the first order. A study of optimal control of solutions to this problem is discussed in the fifth section. The sixth and the seventh sections contain results related to research of optimal control of solutions to the Showalter–Sidorov problem and initial-final value problem for the Sobolev-type equation of the second order, respectively.