Optimal control of discrete heat flow on a homogeneous half-line
B.A. Markov1, A.I. Sidikova2, I.A. Gainova3
1South Ural State University, Higher School of Electronics and Computer Science, Chelyabinsk, Russia
2South Ural State University, Institute of Natural and Exact Sciences, Chelyabinsk, Russia
3Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: optimal heating control, heat equation, semi-infinite line, solution with bounded upper value
Abstract
The article studies a problem of optimal control of heating for a homogeneous half-line. The heating problem is set for a heat conduction equation defined on a half-line where the temperature tends to zero at infinity. At the origin of the spatial coordinate a heat flux, i. e. a non-homogeneous boundary condition of the second kind, is given. The heat flux is modeled using a heating function which is a continuous broken line. This choice of the function is explained by the properties of the technical device under study. The article proves the existence of a solution to such a problem and the uniqueness of its classical solution with a certain error. The optimality of the heating control in this paper means that at the boundary x=0 the temperature at any time is maximum permissible (or, in the first time interval, maximum possible), and at the same time does not exceed a certain critical value, which is chosen to be equal to 1. For the optimal control, a recurrence formula is found at different times, it is proven that this is exactly the optimal solution. That is, at large values of the heat flux the critical temperature at the boundary will be exceeded at some point in time, and at the smaller values the temperature will be lower than that allowed by the material. It is also proven that the heat flux found is the exact upper bound of all admissible heat fluxes for a given discrete control and that such a flux is unique.
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