The optimal control problem with quadratic cost criteria is solved for the class of systems described by a scalar parabolic partial differential equation where the spatial differential operator is linear, is of arbitrary order, and is defined on the real linex≥ 0. The control is applied at the boundaryx= 0 and throughout the interiorx> 0. The Hamilton-Jacobi equation for the problem is solved by applying elementary matrix methods to obtain an expression of Greens formula which involves the boundary control explicitly. The Riccati equation is formulated and, for the ease where the spatial operator is self-adjoint, the spectral representation of the Riccati equation is derived. Previous results have shown that the spectral representation of the Riccati equation for a self-adjoint differential operator on a finite spatial interval, possessing only a point spectrum, consists of an infinite set of bilinear ordinary differential equations. The new results presented here for the case of an operator on a semi-infinite spatial interval show that the spectral representation of the Riccati equation for a self-adjoint differential operator on a semi-infinite spatial interval, possessing a continuous spectrum of multiplicitym, consists ofmbilinear partial differential equations.