Finite-difference method for solving non-stationary problems of convection-diffusion control

The work is related to the study of the control of heat propagation on the rod. The magnitude of the power and the location of the heat sources are controlled in such a way that, at a given time interval, the temperature of the rod is within predetermined limits, and the total power of the heat sources is minimal. After approximation by the finite-difference method, this control problem passes to the linear programming problem and is solved by the M-method of linear programming. The objective functional is linear and due to the absence of the coercively property, significant difficulties arise in establishing the existence of a continuous exact solution. One of the distinctive features of this work is that the non-stationary problem is considered, i.e. the change in temperature depends not only on spatial variables, but also on time. It should be noted that in this case, we consider a separate problem of optimizing the linear functional at each layer in time. Although it is easy to get similar results for a set of time layers. By moving to the numerical implementation and solving the resulting system of linear algebraic equations, in fact, we actually have the numerical value of the Green's function in the form of a matrix. Then, by the M-method of linear programming, a numerical solution of the problem is constructed. The description of the algorithms based and the results of numerical experiments is given. To illustrate the result, a numerical example is considered. Software has been created for carrying out numerical experiments to solve the problem.

Keywords

Authors

References