Internal control for non-local Schrodinger and wave equations involving the fractional Laplace operator

We analyse the interior controllability problem for a non-local Schr\"odinger equation involving the fractional Laplace operator $(-\Delta)^s$, $s\in(0,1)$, on a bounded $C^{1,1}$ domain $\Omega\subset\mathbb{R}^n$. The controllability from a neighbourhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. As a consequence of that, we obtain the controllability for a non-local wave equation involving the higher order fractional Laplace operator $(-\Delta)^{2s}=(-\Delta)^s(-\Delta)^s$, $s\in[1/2,1)$. The results follow from a new Pohozaev-type identity for the fractional Laplacian recently proved by X. Ros-Oton and J. Serra and from an explicit computation of the spectrum of the operator in the one dimensional case

On the controllability of Partial Differential Equations involving non-local terms and singular potentials

In this thesis, we investigate controllability and observability properties of Partial Differential Equations describing various phenomena appearing in several fields of the applied sciences such as elasticity theory, ecology, anomalous transport and diffusion, material science, porous media flow and quantum mechanics. In particular, we focus on evolution Partial Differential Equations with non-local and singular terms. Concerning non-local problems, we analyse the interior controllability of a Schr\"odinger and a wave-type equation in which the Laplace operator is replaced by the fractional Laplacian $(-\Delta)^s$. Under appropriate assumptions on the order $s$ of the fractional Laplace operator involved, we prove the exact null controllability of both equations, employing a $L^2$ control supported in a neighbourhood $\omega$ of the boundary of a bounded $C^{1,1}$ domain $\Omega\subset\mathbb{R}^N$. More precisely, we show that both the Schrodinger and the wave equation are null-controllable, for $s\geq 1/2$ and for $s\geq 1$ respectively. Furthermore, these exponents are sharp and controllability fails for $s<1/2$ (resp. $s<1$) for the Schrödinger (resp. wave) equation. Our proof is based on multiplier techniques and the very classical Hilbert Uniqueness Method. For models involving singular terms, we firstly address the boundary controllability problem for a one-dimensional heat equation with the singular inverse-square potential $V(x):=\mu/x^2$, whose singularity is localised at one extreme of the space interval $(0,1)$ in which the PDE is defined. For all $0<\mu<1/4$, we obtain the null controllability of the equation, acting with a $L^2$ control located at $x=0$, which is both a boundary point and the pole of the potential. This result follows from analogous ones presented in \cite{gueye2014exact} for parabolic equations with variable degenerate coefficients. Finally, we study the interior controllability of a heat equation with the singular inverse-square potential $\Lambda(x):=\mu/\delta^2$, involving the distance $\delta$ to the boundary of a bounded and $C^2$ domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$. For all $\mu\leq 1/4$ (the critical Hardy constant associated to the potential $\Lambda$), we obtain the null controllability employing a $L^2$ control supported in an open subset $\omega\subset\Omega$. Moreover, we show that the upper bound $\mu=1/4$ is sharp. Our proof relies on a new Carleman estimate, obtained employing a weight properly designed for compensating the singularities of the potential

Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

This article is devoted to the analysis of control properties for a heat equation with a singular potential μ/δ2, defined on a bounded C2 domain Ω⊂RN, where δ is the distance to the boundary function. More precisely, we show that for any μ≤1/4 the system is exactly null controllable using a distributed control located in any open subset of Ω, while for μ>1/4 there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Near-field imaging of single walled carbon nanotubes emitting in the telecom wavelength range

International audienceHybrid systems based on carbon nanotubes emitting in the telecom wavelength range and Si-photonic platforms are promising candidates for developing integrated photonic circuits. Here, we consider semiconducting single walled carbon nanotubes (s-SWNTs) emitting around 1300 nm or 1550 nm wavelength. The nanotubes are deposited on quartz substrate for mapping their photoluminescence in hyperspectral near-field microscopy. This method allows for a sub-wavelength resolution in detecting the spatial distribution of the emission of single s-SWNTs at room temperature. Optical signature delocalized over several micrometers is observed, thus denoting the high quality of the produced carbon nanotubes on a wide range of tube diameters. Noteworthy, the presence of both nanotube bundles and distinct s-SWNT chiralities is uncovered

On the controllability of Partial Differential Equations involving non-local terms and singular potentials

In this thesis, we investigate controllability and observability properties of Partial Differential Equations describing various phenomena appearing in several fields of the applied sciences such as elasticity theory, ecology, anomalous transport and diffusion, material science, porous media flow and quantum mechanics. In particular, we focus on evolution Partial Differential Equations with non-local and singular terms. Concerning non-local problems, we analyse the interior controllability of a Schr\"odinger and a wave-type equation in which the Laplace operator is replaced by the fractional Laplacian $(-\Delta)^s$. Under appropriate assumptions on the order $s$ of the fractional Laplace operator involved, we prove the exact null controllability of both equations, employing a $L^2$ control supported in a neighbourhood $\omega$ of the boundary of a bounded $C^{1,1}$ domain $\Omega\subset\mathbb{R}^N$. More precisely, we show that both the Schrodinger and the wave equation are null-controllable, for $s\geq 1/2$ and for $s\geq 1$ respectively. Furthermore, these exponents are sharp and controllability fails for $s<1/2$ (resp. $s<1$) for the Schrödinger (resp. wave) equation. Our proof is based on multiplier techniques and the very classical Hilbert Uniqueness Method. For models involving singular terms, we firstly address the boundary controllability problem for a one-dimensional heat equation with the singular inverse-square potential $V(x):=\mu/x^2$, whose singularity is localised at one extreme of the space interval $(0,1)$ in which the PDE is defined. For all $0<\mu<1/4$, we obtain the null controllability of the equation, acting with a $L^2$ control located at $x=0$, which is both a boundary point and the pole of the potential. This result follows from analogous ones presented in \cite{gueye2014exact} for parabolic equations with variable degenerate coefficients. Finally, we study the interior controllability of a heat equation with the singular inverse-square potential $\Lambda(x):=\mu/\delta^2$, involving the distance $\delta$ to the boundary of a bounded and $C^2$ domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$. For all $\mu\leq 1/4$ (the critical Hardy constant associated to the potential $\Lambda$), we obtain the null controllability employing a $L^2$ control supported in an open subset $\omega\subset\Omega$. Moreover, we show that the upper bound $\mu=1/4$ is sharp. Our proof relies on a new Carleman estimate, obtained employing a weight properly designed for compensating the singularities of the potential

GLOBAL NON-NEGATIVE APPROXIMATE CONTROLLABILITY OF PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

In this work, we consider the linear 1 − d heat equation with some singular potential (typically the so-called inverse square potential). We investigate the global approximate controllability via a multiplicative (or bilinear) control. Provided that the singular potential is not super-critical, we prove that any non-zero and non-negative initial state in L 2 can be steered into any neighborhood of any non-negative target in L 2 using some static bilinear control in L ∞. Besides the corresponding solution remains non-negative at all times