In this paper, the null-controllability in any positive time T of the first-order equation (1) x(t) = e(i theta) Ax(t)+ Bu(t) (vertical bar theta vertical bar < pi/2 fixed) is deduced from the null-controllability in some positive time L of the second-order equation (2) z(t) = Az(t)+ Bv(t). The differential equations ( 1) and ( 2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method makes explicit the input function u of (1) in terms of the input function v of (2): u(t) = integral(R) k(t, s) v(s) ds, where the compactly supported kernel k depends only on T and L. This method proves roughly that the norm of a u steering the system (1) from an initial state x(0) = x(0) to the final state x(T) = 0 grows at most like parallel to x(0)parallel to exp(alpha L-*(2)/T) as the control time T tends to zero. (The rate alpha(*) is characterized independently by a one-dimensional controllability problem.) In applications to the cost of fast controls for the heat equation, L is roughly the length of the longest ray of geometric optics which does not intersect the control region.