ABOUT ONE INVARIANT MULTIVALUED MAPPING IN THE TASK OF THERMAL CONDUCTIVITY WITH TIME DELAY

Аннотация

This paper considers a matter on the strong and weak invariance of the constant multi-valued mapping towards thermal conductivity equation with boundary control in the presence of time delay. Sufficient conditions for the strong and weak invariance of this multi-valued mapping were obtained for the heat transfer control task with a delayed argument with boundary and initial conditions. The concept of "invariant sets" is applied to a system with distributed parameters, the physical meaning of which is to "keep" the object in the desired state as long as possible by controlling it. At the same time, here object retention is understood not in the geometric sense, but in the sense of holding the average value relative to the volume of the object. The necessary conditions for keeping the object in the desired state are proposed. To solve the equations, we first expand the definition of the elliptic operator and the operator itself to a self-adjoint operator, and then consider the existence of a solution that belongs to the energy space of this operator. This uses the fact that the extended operator has generalized eigenvalues and generalized eigenfunctions that make up the complete system both in the energy space of the operator and in each space. In this case, a generalized solution is understood as a solution, because it is represented as a Fourier series whose coefficients satisfy infinite ordinary differential equations. Just in this system of differential equations there is a control parameter. An essential point in considering this task is that the controls are located on the border of the area. In this case, the control area is a convex compact polyhedron, and the restriction area and the terminal set are half-spaces. This, in certain conditions, allows the possibility of applying the obtained results in solving practical tasks.