On the Estimates in Various Spaces to the Control Function of the Extremum Problem for Parabolic Equation

Abstract

For the minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential, we formulate a result on the existence and uniqueness of a minimizer from a prescribed set. We use a weight quadratic cost functional showing the temperature deviation. We obtain estimates for the norm of control functions in terms of the value of the quality functional in different functional spaces. It gives us a possibility to estimate the required internal energy of the system. To prove these results we establish the positivity principle.