Explicit observability estimate for the wave equation with potential and its application

By Xu Zhang. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2000

96 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/597321b4189971d9723dff1d9d978de7923e1707

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods (2005, methodology): “These techniques need to be combined with Carleman or multiplier inequalities (see [33, 108]) allowing one to estimate the dependence of the observability constants on the potential of the linearized equation.”
Carleman estimates for parabolic equations and applications (2009, background): “For other types of equations, such pointwise Carleman estimates were proved in [112] and see also [5, 59, 105, 145].”
A unified controllability/observability theory for some stochastic and deterministic partial differential equations (2010, methodology): “As for the time-variant case, in principle one needs to employ Carleman estimates, see [17] for the parabolic equation and [47] for the hyperbolic equation.”
Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: H 1(Ω)-estimates (2004, methodology): “Moreover, the approach pursued here, as well as in [10, 11, 20, 46, 48, 49], for second-order hyperbolic equations, is inspired by [25] and accordingly is di¬erent from the pseudo-di¬erential analysis for general evolution equations in [41].”
Carleman Estimates with No Lower-Order Terms for General Riemann Wave Equations. Global Uniqueness and Observability in One Shot (2002, background): “Knowledge of explicit constants in these estimates plays a positive role in dealing with semilinear problems [LT9], [Z1], [ Z2 ], [Z3] as well as in estimating the minimal norm of the steering controls.”
The Wave Equation: Control and Numerics (2012, background): “The proof of observability for T ≥ 2 can be carried out in several ways, including Fourier series (and generalizations to non-harmonic Fouri er series, see [105]), multipliers (Komornik [60]; Lions [68, 69]), sidewise energy est imates ([110]), Carleman inequalities (Zhang [106]), and microlocal 3 t”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, methodology): “The existing ones require either some analyticity properties of the coefficients [182] and [172] or some geometric constraints in ω to apply the Carleman inequalities techniques [194].”
An Introduction to the Controllability of Partial Differential Equations (2004, result): “Similar results can be proved for more general potentials depending on x and t by multiplier methods and sidewiese energy estimates [73] and also using Carleman inequalities [66], [67].”
CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, methodology): “The proof of observability for T ≥ 2 can be carried out in several ways, including Fourier series, multipliers (Komornik, [126]; Lions, [142, 143]), Carleman inequalities (Zhang, [227]), and microlocal tools (Bardos et al.”