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Register a new userObservability inequalities for the heat equation with bounded potentials on the whole space
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In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat equation with bounded potentials on the whole space.
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We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Omega = (-1,1)timesmathbb{T}timesmathbb{T}$$ taking as observation regions slices of the form $omega=(a,b) times mathbb{T} times mathbb{T}$ or tubes $omega = (a,b) times omega_y times mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $omega$ and the boundary of $Omega$: $$T_{min} geqslant frac{1}{8} min{(1+a)^2,(1-b)^2}.$$ Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain $(-1,1)timesmathbb{T}timesmathbb{R}$.
This article is dedicated to insensitization issues of a quadratic functional involving the solution of the linear heat equation with respect to domains variations. This work can be seen as a continuation of [P. Lissy, Y. Privat, and Y. Simpore. Insensitizing control for linear and semi-linear heat equations with partially unknown domain. ESAIM Control Optim. Calc. Var., 25:Art. 50, 21, 2019], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider boundary variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate insensitization, (ii) approximate insensitization combined with an exact insensitization for a finite-dimensional subspace, and (iii) exact insensitization. We provide positive answers to questions (i) and (ii) and partial results to question (iii).
We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behavior.
We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.
We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carath{e}odory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.
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