In this work, we consider boundary control problem associated with a parabolic equation on a interval. On the part of the border of the considered segment, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered interval gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation of the second kind. The control parameter is defined on one. The estimate of a minimal time for achieving the given average and at the interval temperature is found.
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Alimov, Shavkat and Dekhkonov, Farrukh
"On a control problem associated with fast heating of a thin rod,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 2
, Article 1.
Available at: https://uzjournals.edu.uz/mns_nuu/vol2/iss1/1