Carleman estimates and observability inequalities for parabolic equations with interior degeneracy

We consider a parabolic problem with degeneracy in the interior of the spatial domain and Neumann boundary conditions. In particular, we focus on the well-posedness of the problem and on Carleman estimates for the associated adjoint problem. The novelty of the present paper is that, for the first time, the problem is considered as one with an interior degeneracy and Neumann boundary conditions, so no previous result can be adapted to this situation. As a consequence, new observability inequalities are established.

This article proposes to study a class of parabolic problems with interior degeneracy in a set of positive measure in order to establish well-posedness and obtain Carleman estimates for such problems. Observability inequalities are obtained as a consequence of these estimates.

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

We consider a parabolic problem with degeneracy in the interior of the spatial domain and we focus on the well-posedness of the problem and on inverse source problems. The novelties of the present paper are two. First, the degeneracy point is in the interior of the spatial domain. Second, we consider Neumann boundary conditions so that no previous result can be adapted to this situation.

We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

In this work, we establish a null controllability property of a dispersive age‐structured population dynamics model with finite‐time delay on the mortality rate term. We assume that the dispersion term includes an interior degeneracy satisfying some suitable assumptions. To achieve our main results, we act according to the classical procedure, which is the Carleman estimates of a suitable adjoint system, and then its associated observability inequality. It is well known that the basis of Carleman's estimates is an adjusted weight function and one of the originalities of our article is the expression of the function as a function of age and time variables.

We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.

ABSTRACTThis paper concerns the null controllability of stochastic degenerate parabolic equation with convection term in weakly degenerate case. Due to the degeneracy, we first transfer to study an approximate nondegenerate system. Next, we establish the Carleman estimate for backward stochastic degenerate parabolic equation with convection term. In order to deal with the convection term, the Carleman estimate is established by introducing an auxiliary function to transfer the diffusion and the convection term into a whole complex divergence‐like form. By this Carleman estimate, we then obtain an observability inequality for the adjoint system of the approximate system. Based on the observability inequality and an approximate argument, we proved our null controllability result.

In this paper we study the null controllability property for a single population model in which the population y depends on time t, space x, age a and size $$\tau $$. Moreover, the diffusion coefficient k is degenerate at a point of the domain or both extremal points. Our technique is essentially based on Carleman estimates. The $$\tau $$ dependence requires us to modify the weight for the Carleman estimates, and accordingly the proof of the observability inequality. Thanks to this observability inequality we obtain a null controllability result for an intermediate problem and finally for the initial system through suitable cut off functions.

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C2-boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H1-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H2-solutions, and ultimately for H1-solutions. The passage from H2- to H1-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H1-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ0,Γ1} that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L2(Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L2(Ω), by an attractive boundary feedback.

We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time-dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time-dependent lower-order coefficients, and (c) they allow for significantly smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. As a standard application, we establish exact controllability for general linear waves, again in the setting of time-dependent domains and regions of control.

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 following class of degenerate/singular parabolic operators:$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters$\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities

We consider the nonstationary linearized Navier–Stokes equations in a bounded domain and first we prove a Carleman estimate with a regular weight function. Second we apply the Carleman estimate to a lateral Cauchy problem for the Navier–Stokes equations and prove the Hölder stability in determining the velocity and pressure field in an interior domain. In the final section, we apply the results for the linearized Navier–Stokes equations to the fully nonlinear Navier–Stokes equations and establish a similar Hölder stability estimate within sufficiently smooth solutions, and prove the uniqueness of Leray–Hopf weak solutions by surface stresses on an arbitrarily chosen sub-boundary.

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.