Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in ...finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.
This paper is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary ...conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.
In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index ...materials. Using the reflecting technique introduced in 26, we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. Three approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped with complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work 1. The second approach, which complements the first one, is variational and based on the Dirichlet principle. The last approach, which complements the second one, is also variational and uses the multiplier technique. Using these approaches, we are able to obtain new results on the well-posedness of these equations for which the conditions on the coefficients are imposed “partially” or “not strictly” on the interfaces of sign changing coefficients. This allows us to rediscover and extend known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is not in Hloc1(Rd) as usual and possibly not even in Lloc2(Rd). The optimality of our results is also discussed.
Dans cet article, on étudie le principe d'absorption limite et le caractère bien posé des équations de Helmholtz avec changements de signe des coefficients, ce qui modélise des matériaux d'indice négatif. En utilisant la technique de réflexion introduite dans 26, on dérive d'abord des problèmes de Cauchy. Le principe d'absorption limite et le caractère bien posé sont ensuite obtenus grâce à des estimations a priori pour ces problèmes. Trois approches sont proposées pour obtenir ces estimations. La première utilise les estimations a priori des systèmes elliptiques pour des conditions aux limites complémentaires dans l'ouvrage classique 1 d'Agmon, Douglis et Nirenberg. La deuxième approche, qui complète la première, est variationnelle et utilise le principe de Dirichlet. La dernière approche, qui complète la seconde, est également variationnelle et utilise la technique du multiplicateur. Utilisant ces approches, on peut obtenir des nouveaux résultats sur le caractère bien posé de ces équations, pour lesquelles les conditions sur les coefficients sont imposées “partiellement” ou “pas strictement” sur les interfaces où les coefficients changent la signe. Cela permet de redécouvrir et d'étendre les résultats connus obtenus par la méthode intégrale, la théorie des opérateurs pseudo differentiels, et l'approche T-coercivité. La solution unique, obtenue par le principe d'absorption limite, n'est pas dans Hloc1(Rd) comme d'habitude et n'est peut-être même pas dans Lloc2(Rd). L'optimalité de nos résultats est également discutée.
The nonlinear KdV equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right is considered. It is known that there is a set of ...critical lengths for which the solutions of the linearized system conserve the L2-norm if their initial data belong to a finite dimensional space M. We show that all solutions of the nonlinear system decay to 0 at least with the rate 1/t1/2 when dimM=1 or when dimM is even and a specific condition is satisfied, for sufficiently small initial data. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. Consequently, we rediscover all known results by a different approach and obtain new results. We also show that the decay rate is not slower than ln(t+2)/t for all critical lengths.
We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λδ in Lp(Ω) for p>1. We show that the limits are multiples of ∫Ω|∇u|p. This is a continuation ...of our previous work where the case p=1 was considered.
We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is ...to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach involves the complementing conditions for the Cauchy problems associated with two elliptic equations, which were studied in a general setting by Agmon, Douglis, and Nirenberg. The variational approach explores the variational structure of the Cauchy problems of the Maxwell equations. As a result, we obtain general conditions on the coefficients for which the limiting absorption principle and the well-posedness hold. Moreover, these
new
conditions are of a local character and easy to check. Our work is motivated by and provides general sufficient criteria for the stability of electromagnetic fields in the context of negative-index metamaterials.
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering ...theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the Lp regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with C∞-coefficients.
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