Extremum problems and control function estimates for a parabolic equation

We consider an extremum problem associated with a mathematical model of the temperature control. It is based on a one-dimensional non-self-adjoint parabolic equation of general form. Determining the optimal control as a function minimizing the weighted quadratic functional, we prove the existence of a solution to the problem of the double minimum by control and weight functions. We also obtained upper estimates for the norm of the control function in terms of the value of the functional. These estimates are used to prove the existence of the minimizing function for unbounded sets of control functions.