In this thesis, we give a mathematical background of solving a linear quadratic control problem for the heat equation, which involves noise on the boundary, in a concise way. We use the semigroup approach for the solvability of the problem. To obtain optimal controls, we use optimization techniques for convex functionals. Finally we give a feedback form for the optimal control. In order to enhance understanding of linear quadratic problem, we first present the methods in deterministic cases and then extend to noisy systems.