For $s\in [\tfrac {1}{2},\, 1)$, let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$. We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$

\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \]

$u \equiv 0$

$\mathbb {R}^{n} \times [-T,\, 0]$

This paper deals with the numerical computation of boundary null controls for the 1D waveequation with a potential. The goal is to compute approximations of controls that drivethe solution from a prescribed initial state to zero at a large enough controllabilitytime. We do not apply in this work the usual duality arguments but explore instead adirect approach in the framework of global Carleman estimates. More precisely, we considerthe control that minimizes over the class of admissible null controls a functionalinvolving weighted integrals of the state and the control. The optimality conditions showthat both the optimal control and the associated state are expressed in terms of a newvariable, the solution of a fourth-order elliptic problem defined in the space-timedomain. We first prove that, for some specific weights determined by the global Carlemaninequalities for the wave equation, this problem is well-posed. Then, in the framework ofthe finite element method, we introduce a family of finite-dimensional approximate controlproblems and we prove a strong convergence result. Numerical experiments confirm theanalysis. We complete our study with several comments.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

Exact controllabilityresults for a multilayer plate system are obtained from the method of Carleman estimates.The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwichplate” system due to Rao and Nakra. The multilayer version involves a number ofLamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controlsare assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for thecoupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lamé system.

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a boundeddomain. The damping is located on a neighborhood of a suitable portion of theboundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

In this paper, we prove a controllabilityresult for a fluid-structure interaction problem. In dimension two,a rigid structure moves into an incompressible fluid governed byNavier-Stokes equations. The control acts on a fixed subset of thefluid domain. We prove that, for small initial data, this system isnull controllable, that is, for a given T > 0, the system can bedriven at rest and the structure to its reference configuration attime T. To show this result, we first consider a linearizedsystem. Thanks to an observability inequality obtained from aCarleman inequality, we prove an optimal controllability result witha regular control. Next, with the help of Kakutani's fixed pointtheorem and a regularity result, we pass to the nonlinear problem.

Let L be a parabolic second order differential operator on the domain $ \bar{\Pi}=\left[ 0,T\right] \times {\mathbb R}.$ Given a function $\hat{u}: {\mathbb R\rightarrow R}$ and $\hat{x}>0$ such that the support of û is contained in $(-\infty ,-\hat{x}]$, we let $\hat{y}:\bar{\Pi}\rightarrow {\mathbb R}$ be the solution to the equation: \[ L\hat{y}=0,\text{\quad }\hat{y}|_{\{0\}\times {\mathbb R}}=\hat{u} . \] Given positive bounds $0<x_{0}<x_{1},$ we seek a function u with support in $\left[ x_{0},x_{1}\right] $ such that the corresponding solution y satisfies: \[ y(t,0)=\hat{y}(t,0)\quad \quad \forall t\in \left[ 0,T\right] . \] We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that $\hat{y}|_{[0,T]\times \{0\}}$ can be C0-approximated, but an exact solution does not exist in general. This result solves the problem of almost replicating a barrier option in the generalised Black–Scholes framework with a combination of European options, as stated by Carr et al. in [6].


The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in Lk .