The ‘window problem’ for series of complex exponentials

Seidman, Thomas I.; Avdonin, Sergei; Иванов, С. А.

doi:10.1007/bf02511154

Cited by 34 publications

(75 citation statements)

References 13 publications

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“…Finally, in Section 4, we briefly comment on the multi-dimensional case and discuss some other extensions of the results of this paper and open problems. In particular, we describe why the results we prove are in agreement with those of [7], [11] and [24], in which it is shown that, in the case of bounded domains, the control needed to drive the system to rest blows up exponentially as the control time tends to zero.…”

Section: Introduction Problem Formulationsupporting

confidence: 85%

“…And this estimate is sharp. This sharp estimate may be found in [11] and [24], where the problem is addressed analyzing the biorthogonal family associated to the sequence of real exponentials {e −j 2 t } j≥1 in L 2 (0, δ) as δ → 0. In [7] sharp observability estimates were proved by means of Carleman inequalities in the context of the internal control problem for general bounded domains in any space dimension.…”

Section: 4mentioning

confidence: 96%

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On the lack of null-controllability of the heat equation on the half-line

Micu

Zuazua

2000

Trans. Amer. Math. Soc.

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Abstract. We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the L 2 boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials {e jt } j≥1 in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.

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Section: Introduction Problem Formulationsupporting

confidence: 85%

Section: 4mentioning

confidence: 96%

On the lack of null-controllability of the heat equation on the half-line

Micu

Zuazua

2000

Trans. Amer. Math. Soc.

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“…For many equations, the controllability on a segment [0, L] from one end can be formulated as a window problem for series of complex exponentials as in section 4.1 (note that in this case 2L is the length of the longest generalized geodesic in [0, L] which does not intersect one of the ends). In [Sei86] [Sei88] and [SY96], and generalized the window problem to a larger class of complex exponentials in [SG93] and [SAI00].…”

Section: Resultsmentioning

confidence: 99%

How Violent are Fast Controls for Schr�dinger and Plate Vibrations?

Miller

2004

Archive for Rational Mechanics and Analysis

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Abstract. Given a time T > 0 and a region Ω on a compact Riemannian manifold M , we consider the best constant, denoted C T,Ω , in the observation inequality for the Schrödinger evolution group of the Laplacian ∆ with Dirichlet boundary condition:. We investigate the influence of the geometry of Ω on the growth of C T,Ω as T tends to 0.By duality, C T,Ω is also the controllability cost of the free Schrödinger equation on M with Dirichlet boundary condition in time T by interior controls on Ω. It relates to hinged vibrating plates as well. We analyze the effects of wavelengths which are greater and lower than the control time T separately. We emphasize a tool of wider scope: the control transmutation method.We prove that C T,Ω grows at least like exp(d 2 /4T ), where d is the largest distance of a point in M from Ω, and at most like exp(α * L 2 Ω /T ), where L Ω is the length of the longest generalized geodesic in M which does not intersect Ω, and α * ∈]0, 4[ is the best constant in the following inequality for the Schrödinger equation on the segment [0, L] observed from the left end:} and the inequality holds with B = 1 and with B = ∂x. We also deduce such upper bounds on product manifolds for some control regions which are not intersected by all geodesics.

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“…When M is a Euclidean segment and Γ is one endpoint, (8) is an inequality on sums of exponentials coined a "window problem" in [SAI00]. A well trodden path in the harmonic analysis of this problem is to construct a Riesz basis of bi-orthogonal functions.…”

Section: 1mentioning

confidence: 99%

On exponential observability estimates for the heat semigroup with explicit rates

Miller¹

2006

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Abstract. This note concerns the final time observability inequality from an interior region for the heat semigroup, which is equivalent to the nullcontrollability 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 minimum energy function. It proves that this final time observability inequality implies four variants with roughly the same exponential rate everywhere (an integrated inequality with singular weights, an integrated inequality in infinite times, a sharper inequality and a Sobolev inequality) 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. IntroductionThe natural setting for the problem to be discussed is on manifolds, but all the statements can be understood, and are already interesting, when the domain M is a smooth bounded open set of R d with the flat metric so that the distance is dist(x, y), always considered with Dirichlet condition on the boundary ∂M . We shall refer to this setting as the Euclidean case.Although it can be skipped, for completeness we now describe the general setting. Let (M, g) be a smooth connected compact d-dimensional Riemannian manifold with metric g and smooth boundary ∂M . When ∂M = ∅, M denotes the interior andUnless mentioned otherwise, the range of the time T is (0, ∞) and the range of the initial state u 0 is L 2 (M ). The corresponding solution of the Cauchy problem for the (forward) heat equation is denoted by u(T, x) = (e T ∆ u 0 )(x), in short: u = e T ∆ u 0 is the (relative) temperature on R + × M .In this note, we make some remarks about the following observability inequality from Ω of the final state at time T : for any T ,

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The ‘window problem’ for series of complex exponentials

Cited by 34 publications

References 13 publications

On the lack of null-controllability of the heat equation on the half-line

On the lack of null-controllability of the heat equation on the half-line

How Violent are Fast Controls for Schr�dinger and Plate Vibrations?

On exponential observability estimates for the heat semigroup with explicit rates

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