  This paper is concerned with the optimal control problem governed by a linear
parabolic equation and subjected to box constraints on control variables. This
type of problem has important applications in heating and cooling systems. By
applying the scheme of Fenchel duality, we derive the dual problem explicitly
where the control constraints in primal problem are embedded in the dual
problem's objective functional. The existence and uniqueness of the solution to
the dual problem are proved and the first-order optimality conditions are also
derived. In addition, we discuss the saddle point property between solution of
the primal problem and the dual problem. The solution of primal problem can be
readily obtained by the solution of the dual problem. To solve the dual problem
numerically, we design two implementable method: conjugate gradient method and
semismooth Newton method. Three example problems are solved, numerical results
show that the proposed method is efficient and accurate.
