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

##### Abstract

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.