부산대학교 도서관

This paper deals with the interior controllability of a fractional Schrödinger equation: $$ \cases i \partial_t u + (-\Delta)^s u = h \chi_{\omega}, & t >0, \ x \in \Omega,\\ u \equiv 0, &t >0, \ x \in \Bbb R^N \setminus \Omega,\\ u(0,x) = u_0(x), & x \in \Omega, \endcases $$ where $T>0$ is fixed, $\Omega$ is a bounded $C^{1,1}$ domain of $\Bbb R^N$, $u_0 \in L^2(\Omega)$ is a given initial datum, and the control $h \in L^2((0,T)\times\omega)$ is active in a neighborhood $\omega$ of the boundary of $\Omega$, namely $$ \omega \coloneq \Cal O_\varepsilon \cap \Omega, \quad \Cal O_\varepsilon \coloneq \bigcup_{x \in \Gamma_0} B(x,\varepsilon), \quad \Gamma_0\coloneq \{x \in \partial \Omega: (x \cdot \nu) >0\}, $$ in which $\nu$ denotes the outward-point unit normal vector to $\partial \Omega$. The fractional Laplacian operator $(-\Delta)^s$ is defined, for $s \in (0,1)$, by the singular integral $$ (-\Delta)^s u(x) \coloneq c_{N,s} \ \roman{p.v.}\int_{\Bbb R^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} \, \roman d y \coloneq c_{N,s} \lim_{\varepsilon \to 0^+} \int_{|x-y|>\varepsilon} \frac{u(x)-u(y)}{|x-y|^{N+2s}} \, \roman d y, $$ with $c_{N,s}\coloneq \frac{s2^{2s}\Gamma(\frac{N+2s}{2} )}{\pi^{N/2}\Gamma(1-s)}$. \par For $s \in (1/2,1)$, exact null-controllability holds for any $T>0$; for $s=1/2$, it holds for a sufficiently large time $T\ge T_0 >0$; on the other hand, for $s<1/2$, at least in one space dimension, the equation fails to be controllable. \par For $N=1$, the controllability results are established by employing spectral techniques and Ingham-type estimates; this approach does not actually require assuming $\omega$ to be a neighborhood of the boundary of $\Omega$ and provides an explicit lower bound on $T_0$. For $N\ge 2$, the results are obtained by applying the multiplier method and a fractional Pohozaev-type identity.