Geometric conditions for the exact controllability of fractional free and harmonic Schrödinger equations

Abstract

We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko–Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel–Rotach formula.

Appendix

This appendix contains the proof of two instrumental results. The first part is devoted to the proof of a technical lemma used in the proof of Proposition 2.1. The purpose of the second part is to recall the proof of Miller’s result stated in Proposition 1.3.

1.1 A technical lemma

The following technical result is used in the proof of Proposition 2.1:

Lemma 5.1

If A is a measurable subset of $[-\frac{\pi }{2}, \frac{\pi }{2}]$ and $\delta =|A|$ denotes its Lebesgue measure, then

$$\begin{aligned} \int _A \sin ^2 x \ \mathrm{d}x \ge \int _{-\frac{\delta }{2}}^{\frac{\delta }{2}} \sin ^2 x \ \mathrm{d}x. \end{aligned}$$

Proof

By using that the function $x \mapsto \sin ^2 x$ is even and increasing on $[0,\frac{\pi }{2}]$, we observe that

$$\begin{aligned}&\int _A \sin ^2 x \ \mathrm{d}x = \int _{A \cap \big [-\frac{\delta }{2}, \frac{\delta }{2} \big ]} \sin ^2 x \ \mathrm{d}x + \int _{A \cap \big (\big [-\frac{\pi }{2},\frac{\pi }{2}\big ] {\setminus } \big [-\frac{\delta }{2}, \frac{\delta }{2}\big ]\big )} \sin ^2 x \ \mathrm{d}x \nonumber \\&\quad \ge \int _{A \cap \big [-\frac{\delta }{2}, \frac{\delta }{2} \big ]} \sin ^2 x \ \mathrm{d}x+ \Big |A \cap \Big (\Big [-\frac{\pi }{2},\frac{\pi }{2}\Big ] {\setminus } \Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ]\Big ) \Big | \sin ^2\Big (\frac{\delta }{2} \Big ) \nonumber \\&\quad \ge \int _{A \cap [-\frac{\delta }{2}, \frac{\delta }{2} ]} \sin ^2 x \ \mathrm{d}x+ \frac{\big |A \cap \big ([-\frac{\pi }{2},\frac{\pi }{2}] {\setminus } [-\frac{\delta }{2}, \frac{\delta }{2}]\big ) \big |}{\big | \big [-\frac{\delta }{2}, \frac{\delta }{2}\big ] \cap \big (\big [-\frac{\pi }{2},\frac{\pi }{2}\big ]{\setminus } A\big ) \big |}\int _{\big [-\frac{\delta }{2}, \frac{\delta }{2}\big ] \cap \big (\big [-\frac{\pi }{2},\frac{\pi }{2}\big ]{\setminus } A\big )} \sin ^2 x \ \mathrm{d}x. \end{aligned}$$

(5.1)

By using that $\delta =|A|=|[-\frac{\delta }{2}, \frac{\delta }{2}]|$, we notice that

$$\begin{aligned}&\Big |A \cap \Big (\Big [-\frac{\pi }{2},\frac{\pi }{2}\Big ] {\setminus } \Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ]\Big ) \Big | = |A| -\Big |A \cap \Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ]\Big | \nonumber \\&\quad = \Big |\Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ]\Big |-\Big |A \cap \Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ]\Big | = \Big | \Big [-\frac{\delta }{2}, \frac{\delta }{2}\Big ] \cap \Big (\Big [-\frac{\pi }{2},\frac{\pi }{2}\Big ]{\setminus } A\Big ) \Big |. \end{aligned}$$

(5.2)

It follows from (5.1) and (5.2) that

$$\begin{aligned} \int _A \sin ^2 x \ \mathrm{d}x \ge \int _{A \cap [-\frac{\delta }{2}, \frac{\delta }{2} ]} \sin ^2 x \ \mathrm{d}x+ \int _{[-\frac{\delta }{2}, \frac{\delta }{2}] \cap ([-\frac{\pi }{2},\frac{\pi }{2}]{\setminus } A)} \sin ^2 x \ \mathrm{d}x = \int _{-\frac{\delta }{2}}^{\frac{\delta }{2}} \sin ^2 x \ \mathrm{d}x. \end{aligned}$$

This ends the proof of Lemma 5.1. $\square$

1.2 Spectral inequalities and exact controllability

This section is devoted to recall the proof of Miller’s result [13, Corollary 2.17] stated in Proposition 1.3 which provides necessary and sufficient spectral estimates for the observability of system (1.3) to hold. The proof of Proposition 1.3 is based on another Miller’s result [13, Theorem 2.4] characterizing the observability with resolvent conditions which is recalled below and whose proof is omitted:

Proposition 5.2

(Miller [13, Theorem 2.4]). Let $(A,\mathcal {D}(A))$ be a self-adjoint operator on $L^2(\mathbb {R}^d)$, which is the infinitesimal generator of a strongly continuous group $(e^{itA})_{t \in \mathbb {R}}$ on $L^2(\mathbb {R}^d)$. The system (1.3) is exactly observable from a measurable subset $\omega \subset \mathbb {R}^d$ if and only if there exist some positive constants $M>0$ and $m>0$ such that

$$\begin{aligned} \forall f \in \mathcal {D}(A), \, \forall \lambda \in \mathbb {R}, \quad \Vert f\Vert ^2_{L^2(\mathbb {R}^d)} \le M \Vert (A- \lambda ) f \Vert ^2_{L^2(\mathbb {R}^d)} + m \Vert f\Vert ^2_{L^2(\omega )}. \end{aligned}$$

(5.3)

When condition (5.3) is satisfied, exact observability holds in any time $T > \pi \sqrt{M}$.

We consider the system (1.3). If this system is exactly observable from a measurable subset $\omega \subset \mathbb {R}^d$ at some time $T>0$, Proposition 5.2 proves that there exist some positive constants $M>0$ and $m>0$ such that the resolvent estimate (5.3) holds. Let $\lambda \in \mathbb {R}$, $0< D < \frac{1}{M}$ and $f \in \mathbb {1}_{\{|A-\lambda | \le \sqrt{D}\}}\big (\mathcal {D}(A)\big )$. The functional calculus shows that

$$\begin{aligned} \Vert (A-\lambda )f \Vert _{L^2(\mathbb {R}^d)} \le \sqrt{D} \Vert f \Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(5.4)

It follows from (5.3) and (5.4) that

$$\begin{aligned} \forall \lambda \in \mathbb {R}, \forall f \in \mathbb {1}_{\{|A-\lambda | \le \sqrt{D}\}}\big (\mathcal {D}(A)\big ), \quad \Vert f\Vert ^2_{L^2(\mathbb {R}^d)} \le M D \Vert f\Vert ^2_{L^2(\mathbb {R}^d)} +m \Vert f\Vert ^2_{L^2(\omega )}, \end{aligned}$$

(5.5)

that is

$$\begin{aligned} \forall \lambda \in \mathbb {R}, \forall f \in \mathbb {1}_{\{|A-\lambda | \le \sqrt{D}\}}\big (\mathcal {D}(A)\big ), \quad \Vert f\Vert ^2_{L^2(\mathbb {R}^d)} \le \frac{m}{1-MD} \Vert f\Vert ^2_{L^2(\omega )}. \end{aligned}$$

It establishes the spectral estimates (1.4) for all $f \in \mathbb {1}_{\{|A-\lambda | \le \sqrt{D}\}}\big (L^2(\mathbb {R}^d)\big )$ since the domain $\mathcal {D}(A)$ is dense in $L^2(\mathbb {R}^d)$.

Conversely, let us assume that there exist some positive constants $D>0$ and $k>0$ such that the spectral estimates (1.4) hold. Let $\lambda \in \mathbb {R}$ and $f \in \mathcal {D}(A)$. With $f_{\lambda } = \mathbb {1}_{\{|A-\lambda | \le \sqrt{D} \}} f$ and $f_{\perp }=f-f_{\lambda }$, we deduce from (1.4) and the functional calculus that for all $\lambda \in \mathbb {R}$ and $f \in \mathcal {D}(A)$,

$$\begin{aligned}&\Vert f\Vert ^2_{L^2(\mathbb {R}^d)} = \Vert f_{\lambda } \Vert ^2_{L^2(\mathbb {R}^d)} + \Vert f_{\perp }\Vert ^2_{L^2(\mathbb {R}^d)} \le k \Vert f_{\lambda }\Vert ^2_{L^2(\omega )} + \frac{1}{D} \Vert (A- \lambda ) f_{\perp }\Vert ^2_{L^2(\mathbb {R}^d)} \nonumber \\&\quad \le k \Vert f_{\lambda }\Vert ^2_{L^2(\omega )} + \frac{1}{D} \Vert (A- \lambda ) f \Vert ^2_{L^2(\mathbb {R}^d)}. \end{aligned}$$

(5.6)

Let $T> \pi \sqrt{\frac{1+k}{D}}$ and $\varepsilon >0$ such that

$$\begin{aligned} T^2 >\pi ^2 \frac{1+(1+\varepsilon ^2)k}{D}. \end{aligned}$$

The functional calculus shows that for all $\lambda \in \mathbb {R}$ and $f \in \mathcal {D}(A)$,

$$\begin{aligned}&\Vert f_{\lambda }\Vert ^2_{L^2(\omega )} =\Vert f-f_{\perp }\Vert ^2_{L^2(\omega )} \le (1+\varepsilon ^{-2}) \Vert f\Vert ^2_{L^2(\omega )} + (1+\varepsilon ^2) \Vert f_{\perp } \Vert ^2_{L^2(\omega )} \nonumber \\&\quad \le (1+\varepsilon ^{-2}) \Vert f\Vert ^2_{L^2(\omega )} + (1+\varepsilon ^2) \Vert f_{\perp } \Vert ^2_{L^2(\mathbb {R}^d)} \le (1+\varepsilon ^{-2}) \Vert f\Vert ^2_{L^2(\omega )}\nonumber \\&\quad + \frac{1+\varepsilon ^2}{D} \Vert (A-\lambda ) f_{\perp } \Vert ^2_{L^2(\mathbb {R}^d)} \nonumber \\&\quad \le (1+\varepsilon ^{-2}) \Vert f\Vert ^2_{L^2(\omega )} + \frac{1+\varepsilon ^2}{D} \Vert (A-\lambda ) f \Vert ^2_{L^2(\mathbb {R}^d)}. \end{aligned}$$

(5.7)

It follows from (5.6) and (5.7) that

$$\begin{aligned} \forall \lambda \in \mathbb {R}, \forall f \in \mathcal {D}(A), \quad \Vert f\Vert ^2_{L^2(\mathbb {R}^d)} \le k(1+\varepsilon ^{-2}) \Vert f\Vert ^2_{L^2(\omega )} + \frac{1+k(1+\varepsilon ^2)}{D} \Vert (A- \lambda ) f \Vert ^2_{L^2(\mathbb {R}^d)}. \end{aligned}$$

We can deduce from Proposition 5.2 that the system (1.3) is exactly observable from $\omega$ in time T.