Geometric control in the presence of a black box

By Nicolas Burq, Maciej Zworski. 2004

219 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5c81990d9df533177ac20b637e1c5a3c6ed9f6b7

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Citations in the corpus

The Wave Equation: Control and Numerics (2012, methodology): “The proof of this statement can be derived using resolvent estimates [16, 79] (see also [88] for a similar estimate) but this method does not yield sharp estimates on the observability time.”
A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator. (2005, methodology): “We first recall this result (see [8,21] for the proof).”
Resolvent conditions for the control of unitary groups and their approximations (2012, methodology): “The proof of sufficiency using the Fourier transform is close and was inspired by the proof of Theorem 7 in [3], where the resolvent is assumed compact.”
From resolvent bounds to semigroup bounds (2010, methodology): “u = w in exponentially weighted spaces are related via Fourier-Laplace transform and we can use Plancherel’s formula. Variants of this simple idea have also been used in more concrete situations. See [3, 10, 17, 20]. Note that we can improve a little the conclusion of (a). If the property (a) is tru”
Observation and Control for Operator Semigroups (2009, methodology): “The approach based on microlocal analysis, without explicit reference to the wave equation, has been further developed in Burq and Zworski [27].”
Exact boundary controllability of the nonlinear Schrödinger equation (2009, result): “[1,3,10,12,13,15–21,23] and the references therein), there are only a few results for the nonlinear Schrödinger equation, a situation in sharp contrast with other nonlinear dispersive equations, e.”
Quantum decay rates in chaotic scattering (2007, unknown): “To see how results of this type imply dynamical estimates see [ 6 ] and [8].”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, background): “More recently an interesting characterization of the observability inequality in terms of the spectrum of the underlying operator has been derived in [134], [20], [149] and [167].”
Sharp polynomial decay rates for the damped wave equation on the torus (2012, methodology): “Proof of Theorem 2.3 assuming Proposition 2.4 To prove Theorem 2.3, we express the observability condition as a resolvent estimate (also known as the Hautus test), as introduced by Burq and Zworski [2004], and further developed by Miller [2005] and Ramdani, Takahashi, Tenenbaum and Tucsnak [Ramdani ”
Non-uniform stability for bounded semi-groups on Banach spaces (2008, background): “Such spectral criteria have also proved efficient in control theory [29, 7, 21, 27].”
Semiclassical non-concentration near hyperbolic orbits (2006, methodology): “d hyperbolic geodesic γ, we show that Z X\U |u|2dx≥ c log|λ| , that is, if uconcentrates near γ, the rate is logarithmic. This generalizes results of Colin de Verdi`ere-Parisse [CVP] and Burq-Zworski [BuZw]. As anotherapplicationof ourmain results we considerthe damped waveequation ˆ ∂2 t−∆ +2a(x)∂ ”
Price's Law on Nonstationary Space-Times (2011, unknown): “For more information we refer the reader to [16], [14], [53], [70], [10].”
On the observability of time-discrete conservative linear systems (2008, methodology): “It was proved in [2] and [17] that system (1.”
Indirect controllability of locally coupled wave-type systems and applications (2013, background): “However, in some cases (see [18,12]), these conditions are not necessary.”
Uniformly exponentially stable approximations for a class of damped systems (2009, background): “4) is the so-called resolvent estimate, which has been proved in [4,22] to be equivalent to the continuous observability inequality (1.”