Observation estimate for the heat equations with Neumann boundary conditions via logarithmic convexity

Abstract

We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…For instance, in switched control systems, the control input can change instantaneously at a specific instant. In this context, the impulse approximate controllability was studied for a linear heat equation with homogeneous Dirichlet and Neumann boundary conditions in [5,13], using a new strategy combining the logarithmic convexity method and the Carleman commutator approach. In [4], the authors have established a Lebeau-Robbiano-type spectral inequality for a degenerate one-dimensional elliptic operator with application to impulse control and finite-time stabilization.…”

“…where x 0 ∈ ω and ρ > 0 is suitably chosen. Then, one can prove the following impulse controllability for the equation ( 5): Theorem 2.6 ( [5]). The heat equation ( 5) is null approximate impulse controllable at time T .…”