Observability and control of Schrödinger equations (Q2719160)

The author considers the control system NEWLINE\[NEWLINE \left\{\begin{matrix} i\partial_t w+\Delta w=v_{|\omega}&{\text{ in}}& \Omega\times (0,t),\cr w=0 &{\text{ on}}&\partial \Omega\times (0,t),\cr w(\cdot,0)=w_0 &{\text{ in}}&\Omega \end{matrix}\right.NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(n\geq 1\), \(\omega\) is a nonempty open subset of \(\Omega\) and \(w_0\) is taken in an appropriate space \(X\). He addresses the problem of null controllability in time \(t\) by controls \(v\in L^1 (0,t; X)\). By the Hilbert uniqueness method, the controllability results are obtained by solving the dual observability problem. Further, the results on observability and controllability for the Schrödinger equation are obtained from known results for parabolic and hyperbolic problems. The first result gives a logarithmic observability estimate in case no geometrical conditions are requested on \(\omega\). The second result concerns the particular one-dimensional situation. The third result is about the case \(n>1\) under the Bardos-Lebeau-Rauch geometric control condition.