The Control Transmutation Method and the Cost of Fast Controls

Abstract

In this paper, the null-controllability in any positive time T of the first-order equation (1) ${\dot{x}}(t)=e^{i\theta}Ax (t)+Bu(t)$ ($|{\theta}| < \pi/2$ fixed) is deduced from the null-controllability in some positive time L of the second-order equation (2) $\ddot{z}(t)=Az(t)+Bv(t)$. 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)=\int_{\mathbb{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 $\|{x_{0}}\|\exp(\alpha_{*} L^{2}/T)$ as the control time T tends to zero. (The rate $\alpha_{*}$ is characterized independently by a one-dimensional controllability problem.) In 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.