ﻻ يوجد ملخص باللغة العربية
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 $(-1,1)timesmathbb{T}timesmathbb{R}$.
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
We give necessary and sufficient conditions for the controllability of a Schrodinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of n
Observability inequalities on lattice points are established for non-negative solutions of the heat equation with potentials in the whole space. As applications, some controllability results of heat equations are derived by the above-mentioned observability inequalities.
In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the par
In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We