Show simple item record

dc.contributor.advisorZuazua Iriondo, Enrique
dc.contributor.authorBiccari, Umberto
dc.description.abstractIn 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 (¿ )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 L2 control supported in a neighbourhood ¿ of the boundary of a bounded C1,1 domain ¿ RN. More precisely, we show that both the Schr¿odinger and the wave equation are null-controllable, for s ¿ 1/2 and for s ¿ 1 respectively. Furthermore, these exponents are sharp and controllability fails for s < 1/2 (resp. s < 1) for the Schr¿odinger (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) := ¿/x2, whose singularity is localised at one extreme of the space interval (0, 1) in which the PDE is defined. For all 0 < ¿ < 1/4, we obtain the null controllability of the equation, acting with a L2 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 [76] for parabolic equations with variable degenerate coefficients. Finally, we study the interior controllability of a heat equation with the singular inversesquare potential (x) := ¿/¿2, involving the distance ¿ to the boundary of a bounded and C2 domain ¿ RN, N ¿ 3. For all ¿ ¿ 1/4 (the critical Hardy constant associated to the potential ), we obtain the null controllability employing a L2 control supported in an open subset ¿ ¿ . Moreover, we show that the upper bound ¿ = 1/4 is sharp. Our proof relies on a new Carleman estimate, obtained employing a weight properly designed for compensating the singularities of the
dc.description.sponsorshipBasque Center for Applied Mathematicses
dc.subjectdifferential equations in partial derivativeses
dc.subjectecuaciones diferenciales en derivadas parcialeses
dc.titleOn the controllability of Partial Differential Equations involving non-local terms and singular potentialses
dc.rights.holder(cc)2016 UMBERTO BICCARI (cc by-nc-sa 4.0)

Files in this item


This item appears in the following Collection(s)

Show simple item record

(cc)2016 UMBERTO BICCARI (cc by-nc-sa 4.0)
Except where otherwise noted, this item's license is described as (cc)2016 UMBERTO BICCARI (cc by-nc-sa 4.0)