On the reachable set for the one-dimensional heat equation

International audienceThe goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions

Reachable states and holomorphic function spaces for the 1-D heat equation

The description of the reachable states of the heat equation is one of the central questions in control theory. The aim of this work is to present new results for the 1-D heat equation with boundary control on the segment [0, π]. In this situation it is known that the reachable states are holomorphic in a square D the diagonal of which is given by [0, π]. The most precise results obtained recently say that the reachable space is contained between two well known spaces of analytic function: the Smirnov space E^2(D) and the Bergman space A^2(D). We show that the reachable states are exactly the sum of two Bergman spaces on sectors the intersection of which is D. In order to get a more precise information on this sum of Bergman spaces, we also prove that it includes the Smirnov-Zygmund space E_{LlogL}(D) as well as a certain weighted Bergman space on D

Analytic properties of heat equation solutions and reachable sets

There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain (Ω⊂ℝ) at positive time all solutions are analytically extendable to a geometrically determined subdomain ℰ(Ω) of ℂ containing (Ω). This domain is sharp in the sense that there is no larger domain for which this is true. If (Ω) is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighbourhood of ℰ(Ω) is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case

Reachable states for the distributed control of the heat equation

We are concerned with the determination of the reachable states for the distributed control of the heat equation on an interval. We consider either periodic boundary conditions or homogeneous Dirichlet boundary conditions. We prove that for a $L^2$ distributed control, the reachable states are in the Sobolev space $H^1$ and that they have complex analytic extensions on squares whose horizontal diagonals are regions where no control is applied

Reachable states for the distributed control of the heat equation

We are concerned with the determination of the reachable states for the distributed control of the heat equation on an interval. We consider either periodic boundary conditions or homogeneous Dirichlet boundary conditions. We prove that for a $L^2$ distributed control, the reachable states are in the Sobolev space $H^1$ and that they have complex analytic extensions on squares whose horizontal diagonals are regions where no control is applied

From black box scattering to inverse problems, adventures in applied analysis

Meine Forschung liegt in Bereich der klassischen Analysis, der Spektraltheorie und der inversen Probleme. Meine Arbeit kann als reine Mathematik klassifiziert werden kann, sie hat jedoch auch Anwendungen in der medizinischen Bildgebung, der Geophysik, in Quantenmechanik und Elektromagnetismus. Diese Art von Forschung, theoretisch, aber auf physikalischen Fragestellungen beruhend, hoffe ich fortzusetzen. Derzeit bin ich in eine Vielzahl von Projekten eingebunden. In der angewandten Analysis studiere ich Kontrolltheorie und inverse Probleme. Auf der theoretischen Seite beende ich gerade Arbeiten zu Resolventenabschätzungen für topologische Mannigfaltigkeiten und Maxwell-Gleichungen