On the control problem associated with the heating process

In the paper, we consider the initial-boundary problem for the heat conduction equation inside a bounded domain. On the part of the border of the considered domain, 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 domain gets a given value. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton’s law. The control parameter is equal to the magnitude of output of hot or cold air and is de ned on a given part of the boundary, and the weight function is not assumed to be strictly positive in the given domain. Then, we found the dependence $T(\theta)$ on the parameters of the temperature process when $ \theta $ is close to critical value.