Optimal control of thermal and wave processes in composite materials

The paper indicates the approach and the corresponding him method of penalty functions for analyzing the problems of optimal control of thermal and wave processes in structural elements made of composite materials (composites). An object that is quite common in the industrial sphere, the structure of which is a set of layers (phases) of unidirectional composites — layered composites, is considered. When solving problems related to the analysis and description of the states of composites, quantitative characteristics of layers that are not functions of the coordinates of the points of the medium are usually used in order not to solve the corresponding problems for an inhomogeneous medium. Such functions are elements of Sobolev spaces, first of all, functions summable with a square. The convenience lies in the fact that when finding the conditions for solvability of initial-boundary value problems of various types (in most cases, such problems are the basis of mathematical models of many physical processes), it is possible to reduce to operator-difference systems, for which it is easy to construct a priori estimates of weak solutions. The next step after establishing the weak solvability of the initial-boundary value problem of the thermal or wave process in composites is the formulation and solution of the problem of optimal control of these processes. The proposed method of penalty functions on the example of solving such problems is a general method. It is applicable with slight modifications also not only in the case of elliptic, parabolic and other problems (including nonlinear) for scalar functions, but also for vector functions. An example of the latter is the Navier — Stokes system, widely used in the description of network-like hydrodynamic processes, considered in Sobolev spaces, the elements of which are functions with carriers on $n$-dimensional network-like domains, $n\geq 2$.