The control transmutation method and the cost of fast controls. (English) Zbl 1109.93009

Miller, Luc

SIAM J. Control Optim. 45, No. 2, 762-772 (2006).

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.

MSC:

93B05	Controllability
93B17	Transformations
47D09	Operator sine and cosine functions and higher-order Cauchy problems

Keywords:

controllability; fast controls; control cost; transmutation; cosine operator function; heat equation; linear Ginzburg-Landau equation