Multiplicative Controllability for the Schrödinger Equation

In this chapter we discuss some recent results obtained for multiplicative controllability of the Schrödinger equation. In recent years a substantial progress has been made in investigating the controllability properties of the Schrödinger equation governed by multiplicative control. In this chapter we will discuss some of these results due to Beauchard [11, 12, 14], Beauchard and Coron [16], Beauchard and Mirrahimi [18], Chambrion, Mason, Sigalotti and Boscain [26], Nersesyan [124] and others. It should be noted that the Schrödinger equation that we study below has certain property which sets it apart from the other partial differential equations considered in this monograph. Namely, the L2-norms of its solutions are conserved,regardless of the value of real-valued multiplicative control applied. Therefore, all the results below deal with controllability properties on the unit L

2

-sphere S,

$$ S=\{ \varphi | \varphi \in L^{2}(\Omega, C), \int_{\Omega} | \varphi(x) |^{2} dx =1 \} \in L^{2}(\Omega, C), $$

where Ω is the system’s space domain. The Schrödinger equation with real-valued control is also complex-conjugate time-reversible in the sense that if control u(t), t ε [0,T] steers it from u

0

to u

1

at time t = T, then control u•(t) = u(T –t) steers this equation from ū

1

to ū

0

.