Inverse problem for a parabolic system with two components by measurements of one component

We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $\theta$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at $\theta$ for the two components and for it, we can approximately control the two components of the $2 \times 2$ system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally we establish a Carleman estimate for a $3\times 3$ system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component

Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.Dans cet article nous étudions le lien entre la contrôlabilité à zéro d'un problème parabolique abstrait et la validité d'une inégalité qui est une version quantifiée du test de Fattorini–Hautus. Nous prouvons que cette inégalité permet de caractériser l'existence d'un temps minimal pour le problème de contrôlabilité à zéro et, selon les hypothèses considérées, d'obtenir la valeur de ce temps minimal. Nous prouvons aussi que dans la plupart des exemples connus de problèmes paraboliques ayant un temps minimal de contrôle à zéro, cette inégalité est une condition nécessaire et suffisante de contrôlabilité.Ministerio de Economía y Competitivida

Carleman Estimates for Some Non-Smooth Anisotropic Media

International audienceLet B be a n × n block diagonal matrix in which the first block C τ is an hermitian matrix of order (n − 1) and the second block c is a positive function. Both are piecewise smooth in Ω, a bounded domain of R n. If S denotes the set where discontinuities of C τ and c can occur, we suppose that Ω is stratified in a neighborhood of S in the sense that locally it takes the form Ω × (−δ, δ) with Ω ⊂ R n−1 , δ > 0 and S = Ω × {0}. We prove a Carleman estimate for the elliptic operator A = −∇ · (B∇) with an arbitrary observation region. This Carleman estimate is obtained through the introduction of a suitable mesh of the neighborhood of S and an associated approximation of c involving the Carleman large parameters

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications

International audienceWe study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise \Con^1). The observability, for a {\em linear} equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for {\em semilinear} equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient

On the controllability of linear parabolic equations with an arbitrary control location for stratified media

We prove a null controllability result with an arbitrary control location in dimension greater than or equal to two for a class of linear parabolic operators with non-smooth coefficients. The coefficients are assumed to be smooth in all but one directions

Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation

International audienceWe consider the heat equation with a discontinuous diffusion coefficient and give uniqueness and stability results for both the diffusion coefficient and the initial condition from a measurement of the solution on an arbitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diffusion coefficient is assumed to be discontinuous across interfaces with a monotonicity condition

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

International audienceWe study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise \Con^1). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient

Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A = --c$\Delta$ acting in domains $\Omega$ := $\Omega$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $\epsilon$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $\epsilon$ $\rightarrow$ 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of ''layered media'' enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of ''flattening out'' of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a ''minimal amplitude hypothesis''. However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem

Controllability to trajectories for some parabolic systems of three and two equations by one control force

International audienceWe present a controllability result for a class of linear parabolic systems of 3 equations. To prove the result, we establish a global Carleman estimate for the solutions of a system of 2 coupled parabolic equations with first order terms. We also obtain stability results for the identification of coefficients of the systems

A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems

In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n, with t0 ∈ [0,T], is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A|B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem.Agence Nationale de la rechercheDirección General de Enseñanza Superio