On the cost of fast controls for thermoelastic plates

7 pages, no figures. Submitted November 28, 2005. Accepted October 6, 2006. To appear in Asymptotic Analysis.International audienceThis paper proves that any initial condition in the energy space for the system of thermoelastic plates without rotatory inertia on a smooth bounded domain with hinged mechanical boundary conditions and Dirichlet thermal boundary condition can be steered to zero by a square integrable input function, either mechanical or thermal, supported in arbitrarily small sub-domain and time interval [0,T]. As T tends to zero, for initial states with unit energy norm, the norm of this input function grows at most like exp( C_p / T^p ) for any real p > 1 and some C_p > 0. These results are analogous to the optimal ones known for the heat flow and the proof uses the heat control strategy of Lebeau and Robbiano

On exponential observability estimates for the heat semigroup with explicit rates

13 pages, a4 paper, no figures, some references and an appendix added. To appear in Rendiconti Lincei: Matematica e Applicazioni.This note concerns the final time observability inequality from an interior region for the heat semigroup, which is equivalent to the null-controllability of the heat equation by a square integrable source supported in this region. It focuses on exponential estimates in short times of the observability cost, also known as the control cost and the minimal energy function. It proves that this final time observability inequality implies four variants (an integrated inequality with singular weights, an integrated inequality in infinite times, a sharper inequality and a Sobolev inequality) with roughly the same exponential rate everywhere and some control cost estimates with explicit exponential rates concerning null-controllability, null-reachability and approximate controllability. A conjecture and open problems about the optimal rate are stated. This note also contains a brief review of recent or to be published papers related to exponential observability estimates: boundary observability, Schrödinger group, anomalous diffusion, thermoelastic plates, plates with square root damping and other elastic systems with structural damping

Spectral inequalities for elliptic pseudo-differential operators on closed manifolds

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $\nu>0,$ in the H\"ormander class $\Psi^\nu_{\rho,\delta}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $\omega\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral inequality for an elliptic operator and the null-controllability for its diffusion model) extend in the setting of closed manifolds, classical results of the control theory, as the spectral inequality due to Lebeau and Robbiano and their result on the null-controllability of the heat equation giving a complete picture of the subject in the setting of closed manifolds. For the proof of the spectral inequality we introduce a periodization approach in time inspired by the global pseudo-differential calculus due to Ruzhansky and Turunen.Comment: 31 Page

Estimates for sums of eigenfunctions of elliptic pseudo-differential operators on compact Lie groups

We extend the estimates proved by Donnelly and Fefferman and by Lebeau and Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in the control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.Comment: 40 Pages, 5 Figures, New subsubsection 3.9. arXiv admin note: text overlap with arXiv:2209.1069