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We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the "gene type" of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed by one control force. To reach our goal, we develop first a Carleman type inequality for its adjoint system and consequently the pertinent observability inequality. Note that such a system is obtained via the original paradigm using the Lagrangian method. Afterwards, with the help of a cost function we will be able to deduce the existence of a control acting on a subset of the gene type domain and which steers both populations of a certain class of age to extinction in a finite time.\looseness-2

The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.

Abstract In this paper, we study the indirect stabilization of a system of plate equations which are weakly coupled and locally damped. By virtue of the general results due to Burq in the study of asymptotic behavior of solutions, we prove that the semigroup associated to the system is logarithmically stable under some assumptions on the damping and the coupling terms. For this purpose, we adopt an approach based on the growth of the resolvent on the imaginary axis, which can be obtained by some Carleman estimates.

This article concerns the null controllability of a nonlinear control system governed by coupled degenerate parabolic equations. We first establish the Carleman estimate and the observability inequality for solutions to the conjugate problem. Then we can prove the observability inequality and the null controllability of the linear control system. Finally, the nonlinear control system is shown to be null controllable by a fixed point argument and compact estimates.

This article concerns the null controllability of a control system governed by coupled degenerate parabolic equations with lower order terms. For these equations, the convection terms cannot be controlled by the diffusion terms. We establish a Carleman estimate and an observability inequality by using Carleman estimates for single degenerate parabolic equations with lower order term and some energy estimates. Then we prove that the system with two controls is null controllable. Finally, we show the null controllability of the system with one control, by constructing suitable controls.

This article shows Carleman estimate and null controllability of a cascade control system governed by the semilinear degenerate parabolic equations with the general convection terms. The semilinear parabolic equations are weakly degenerate on the boundary and the convection terms cannot be controlled by the diffusion terms. We establish the Carleman estimate and the observability inequality for the linear conjugate system, and prove that the control system is null controllable.

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time.

Abstract In this paper, we address an inverse problem for the Korteweg-de Vries equation posed on a bounded domain with boundary conditions proposed by Colin and Ghidaglia. More precisely, we retrieve the principal coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability of this inverse problem relies on a Carleman estimate for the linearized Korteweg-de Vries equation and the Bukhgeı̌m-Klibanov method.

This article concerns the inverse problem of the coupled age-structured population dynamics system with discontinuous diffusion coefficients. The internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of simultaneously retrieving two space dependent source terms in the given parabolic system. The proof of the result relies on Carleman estimates and certain energy estimates for parabolic system.

In this article, we establish a Carleman estimate for the plate equation in order to solve an inverse problem retrieving the zeroth-order term for a plate equation from boundary measurements. We prove the local stability result for this inverse problem. Our proof relies on Carleman estimate.

In this paper, we analyze the longtime behavior of the wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-diff-calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective.

In this article, we study the null controllability of a linear model with degenerate diffusion in population dynamics. We develop first a Carleman type inequality for the adjoint system of an intermediate model, and then an observability inequality. By a fixed point technique, we establish the existence of a control acting on a subset of the space domain that leads the population of a certain age to extinction in a finite time.

In this article, we study an inverse problem for weakly degenerate coupled parabolic systems with one force. We establish Lipschitz stability for the source term from measurements of one component of the solution at a positive time and on a subset of the space domain. The key ingredient is the derivation of a Carleman-type estimate.