Based on the notion of generalized homogeneity, a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. A method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with nonhomogeneous operators. The design scheme is demonstrated for the heat equation with the control input distributed on the segment [0,1].