The control problem for the heat equation in the case of a composite material

Abstract

In modern technology there are devices whose details are subject to thermal effects. As a result, the properties of materials may change in parts. In turn, this leads to the failure of the device. You can control surface temperature to protect components from excessive heat. The temperature measuring device cannot be placed at the point where the heating takes place, since the heating temperature is too high. On the other hand, the temperature of the outer side of the protective layer itself is of little interest: you need to know the temperature on the surface of the material. Therefore, one can either, knowing the temperature at the material-protective layer media section, solve the inverse problem, or solve the heating control problem. In this paper, we propose to consider the problem of controlling heating under the condition of controlling the temperature of the interface. We consider the problem on a half-line, and the following sections are identified in the problem: the heating region, a small thickness of the protective layer, the region occupied by the material, and the semi-infinite region in which the material interacts with the environment. At the same time, we consider such a coefficient of thermal conductivity that it is constant in each of the regions, but it experiences a first-order discontinuity at the interfaces. This makes it impossible to build a classical solution to the problem in the entire solution area, and we build solutions that are classic, each in its own subregion, and that meets the matching conditions at the interfaces. The control problem is as follows: the heating function, on the one hand, should be maximum all the time, and on the other, the temperature at the interface should not exceed a certain critical value (for example, the melting temperature of the material). The constructed solution of the problem is simple, but requires rather cumbersome proofs.