Related papers: Observability inequalities for heat equations with…
We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding…
We consider the Kelvin-Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply the Carleman estimate to prove the Lipschitz stability in determining a spatial varying function…
We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-\Delta u\leq v^\lambda\] \[0\leq v_t-\Delta v\leq u^\sigma\] in $\Omega\times (0,1)$, where $\lambda$ and…
In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control system $w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control.…
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed,…
Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a…
This article is devoted to analyze control properties for the heat equation with singular potential $-\mu/|x|^2$ arising at the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$. This problem was firstly studied by Vancostenoble…
In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ \mathbb{R}^m_t \times \mathbb{R}^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with…
This paper deals with the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so called Boussinesq approximation. In the zero heat diffusion case, by adapting…
In the paper, we show a global Carleman estimate for the non-local heat equation. To be more precise, let $\Omega\subset\RR^d$ be a bounded domain and $\CO\subset\Omega$ an open subdomain, $s\in(0,1)$. We show that there exist constants…
We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition…
Smooth solutions of the forced incompressible Euler equations satisfy an energy balance, where the rate-of-change in time of the kinetic energy equals the work done by the force per unit time. Interesting phenomena such as turbulence are…
We build up an asymptotic observability identity for the heat equation in the whole space. It says that one can approximately recover a solution, through observing it over some countable lattice points in the space and at one time. This…
We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$ N *), assumed to be an unknown perturbation of a reference domain. We are interested…
We discuss the identification of a time-dependent potential in a time-fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem.…
We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-\Delta u+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper,…
In this article, we prove null-controllability results for the heat equation associated tofractional Baouendi-Grushin operators $$\partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u= \mathbb{1}_\Omega h$$ where $V$ is a potential that…
Energy decay is established for the damped wave equation on compact Riemannian manifolds where the damping coefficient is allowed to depend on time. Using a time dependent observability inequality, it is shown that the energy of solutions…
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$,…
We consider a transport equation by a gradient vector field with a small viscous perturbation --$\epsilon\Delta_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit…