  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) (|\theta|<\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 explicits the input function u of (1) in terms of the
input function v of (2): u(t,x)=\int k(t,s)v(s)ds, where the compactly
supported kernel k depends on T and L only. It proves that the norm of a u
steering the system (1) from an initial state x_{0} to zero grows at most like
||x_{0}||\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 the applications to the cost of fast controls for the heat
equation, L is the length of the longest ray of geometric optics which does not
intersect the control region.
