The Control Transmutation Method and the Cost of Fast Controls

In this paper, the null controllability in any positive time T of the first-order equation (1) ẋ(t) = eAx(t)+Bu(t) (|θ| < π/2 fixed) is deduced from the null controllability in some positive time L of the second-order equation (2) z̈(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) = ∫ R 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 x0 to zero grows at most like ‖x0‖ exp(α∗L/T ) as the control time T tends to zero. (The rate α∗ 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.