Non-structural controllability of linear elastic systems with structural damping

Abstract

13 pages, a4paper, no figures. Note added in proof: After our article was accepted for publication, we became aware of a paper to appear in Asymptotic Analysis: "Internal null-controllability for a structurally damped beam equation" by Julian Edward and Louis Tebou. This paper concerns (18) when $M$ is a segment and focuses on the limit $\rho\to 0$. We claim that our Theorem~2 generalizes the main result of this paper (n.b. $L_{\Omega}<\infty$ always hold when the dimension of $M$ is one). Since Theorem 1 of this paper says that the cost does not depend on $\rho<2$ and our Theorem~2 estimates the cost by $C_{2} \exp \left(C_{\rho}L^{2}/T \right)$, we need only explain why the constants $C_{2}$ and $C_{\rho}$ do not depend on $\rho$. By the control transmutation method used in section~2.4, this reduces to proving that the constants $C_{1}$ and $C_{2}$ in our theorem~3 do not depend on $\rho$. But these constants come from applying theorem~1 of [23], so that these constants only depend on the function denoted $\nu$ there. Moreover, section~5.2 of [23] proves that $\nu$ depends only on $\abs{c}$ when $\lambda_{k}=a+ck^{2}$. But the formula $\lambda_{\pm}=\left( \rho\pm i\sqrt{2^{2}-\rho^{2}}\right) \mu/2$ (lemma~3 with $\alpha=1/2$) implies $\abs{\lambda_{\pm}}=\mu$ so that $\abs{c}$ does not depend on $\rho$ here, which completes the proof of our claim.This paper proves that any initial condition in the energy space for the plate equation with square root damping z''- r Delta z' + Delta^2 z' = u on a smooth bounded domain, with hinged boundary conditions z=Delta z=0, can be steered to zero by a square integrable input function u supported in arbitrarily small time interval [0,T] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like exp(C_p /T^p) for any real p>1 and some C_p>0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p=1 and the dependence of C_p on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow

Carleman estimate for an adjoint of a damped beam equation and an application to null controllability

“…To the best of our knowledge our article is the first one proving a Carleman estimate for the adjoint of the damped beam equation (1.1). The null controllability problem (1.1) without the potential term is already studied in the articles [23], [26] and [12] using spectral methods. Their technique is completely different from ours which is based in proving a Carleman estimate for the adjoint to the system (1.1).…”

“…The approach of both the articles [23] and [12] is based on proving an observability estimate by using Fourier decomposition and suitably using Bassel's inequality and Ingham-type inequalities for complex frequencies. In [26] the author explicitly obtains the cost of the control as T −→ 0, by tracking the constants in the observability estimate using spectral methods. The main focus of the present article is to derive a new Carleman estimate for the dual to the problem (1.1).…”

“…Next one can exploit the gap property in between two consecutive eigenvalues in order to apply spectral methods to prove null controllability of the system (1.1) with a = 0. For details we refer to the articles [12], [23] and [26]. In the present article we will further not discuss about the spectral methods and will rely on a new Carleman estimate which is the base of our analysis.…”

Observability and stabilization of magnetic Schrödinger equations

“…Exact observability inequalities for the (magnetic) wave equation can transferred to observability inequalities for the (magnetic) Schrödinger equation and vice versa via a transmutation method (see Miller [33] and references therein) or by an abstract framework consisting in transforming a second order evolution equation into a first order evolution equation (see [37,Theorem 6.7.5 and Proposition 6.8.2] for more details).…”

Approximate controllability of infinite dimensional system with internal damping dependent on fractional powers of system operator

“…Using the frequency‐domain method, approximate controllability in finite time of the second‐order system is shown to be equivalent to the approximate controllability for the corresponding simplified first‐order system. In [20], controllability of the plate equation with square root damping and locally distributed control is investigated. Controllability condition is derived using duality principle.…”