부산대학교 도서관

This paper deals with the interior controllability of a fractionalSchrödinger 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 $\BbbR^N$, $u_0 \in L^2(\Omega)$ is a given initial datum, and the control$h \in L^2((0,T)\times\omega)$ is active in a neighborhood $\omega$ ofthe boundary of $\Omega$, namely$$\omega \coloneq \Cal O_\varepsilon \cap \Omega, \quad \CalO_\varepsilon \coloneq \bigcup_{x \in \Gamma_0} B(x,\varepsilon), \quad\Gamma_0\coloneq \{x \in \partial \Omega: (x \cdot \nu) >0\},$$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 $T\ge T_0 >0$; onthe other hand, for $s<1/2$, at least in one space dimension, theequation fails to be controllable.\par For $N=1$, the controllability results are established by employingspectral techniques and Ingham-type estimates; this approach does notactually require assuming $\omega$ to be a neighborhood of the boundaryof $\Omega$ and provides an explicit lower bound on $T_0$. For $N\ge2$, the results are obtained by applying the multiplier method and afractional Pohozaev-type identity.