Optimal control of the stress-deformed states of a composite layered medium

The proposed study considers the compositional medium, which is a set of a finite number of volumetric layers with clearly defined surfaces of mutual adjacency. The mathematical description of such a medium is carried out by means of a layered domain, which defines a model of a layered elastic compositional medium in three-dimensional Euclidean space. Functions describing the quantitative characteristics of the material of the compositional medium belong to the class of bounded summable functions that have generalized derivatives and are elements of Sobolev space. At the same time, the following hypothesis is adopted: the elements of the surfaces of mutual adjoining layers are not subject to tension and compression during deformation (bending) (analogous to one of the well-known Kirchhoff hypotheses). The work consists of three parts: the first part presents a mathematical description of a layered medium with the terminology of layered domains, classical function spaces with a carrier in these domains, a description of phenomena near the surfaces of adjoining layers of a compositional medium; the second part is devoted to the description of deformations of the compositional medium and contains the formulation of the problem of the stress-deformed state of the compositional layered medium in a weak formulation, the definitions of auxiliary spaces and the classical statements used to analyze the problem, sufficient conditions for the weak solvability of the boundary value problem are established; the third (main) part is devoted to solving the problem of optimal distributed control of stress-deformed states of a compositional layered medium. The results of the study can be effectively used to solve the problems of optimal control of deformation processes of complexly structured continuous media. At the same time, the approaches used to analyze boundary value problems of continuum mechanics extend to more general representations of the components of the tensor function of deformations, which means that they can significantly expand the possibilities of analyzing more general problems of optimizing deformable composite materials.