Stabilization of linear parabolic boundary control systems is studied. While the system consists of a pair of standard linear differential operators (L, τ) of the Dirichlet type, it generally admits no Riesz basis associated with it. In this sense the system has enough generality as a prototype of general systems. A difficulty arises when applying existing procedures - via fractional powers of the associated elliptic operator - to our problem. The paper proposes a new algebraic approach to stabilization which has a substantial application to a variety of boundary control systems.