On heat transfer control problems with pointwise observation

In this paper, we consider a control problem with pointwise observation for a one-dimensional parabolic equation, which arises in mathematical models of climate control in industrial greenhouses. This problem is described by a general parabolic equation with a variable diffusion coefficient, convection coefficient, and depletion potential. For an extremal problem of minimizing integral weighted quadratic quality functionals of two different types, we prove the existence and uniqueness of the minimizing function. The exact and dense controllability of the problems are analyzed, the qualitative properties of the minimizing function are investigated, and estimates of the control functions are obtained. A special form of the maximum principle is obtained to prove the results.