scholarly journals Global Carleman estimate for stochastic parabolic equations, and its application

2014 ◽  
Vol 20 (3) ◽  
pp. 823-839 ◽  
Author(s):  
Xu Liu
Keyword(s):  
Parabolic Equations ◽  
Carleman Estimate ◽  
Global Carleman Estimate

Inverse problems for stochastic parabolic equations with additive noise

2020 ◽  
Vol 29 (1) ◽  
pp. 93-108
Author(s):  
Ganghua Yuan
Keyword(s):  
Inverse Problems ◽  
Parabolic Equations ◽  
Additive Noise ◽  
Stability Result ◽  
Main Tool ◽  
Conditional Stability ◽  
Final Time ◽  
History Of ◽  
Global Carleman Estimate ◽  
Stochastic Parabolic Equation

Abstract In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.

Local state observation for stochastic hyperbolic equations

2020 ◽  
Vol 26 ◽  
pp. 79
Author(s):  
Qi Lü ◽  
Zhongqi Yin
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equations ◽  
Unique Continuation ◽  
Local State ◽  
Carleman Estimate ◽  
State Observation ◽  
Unique Continuation Property ◽  
Continuation Property ◽  
Global Carleman Estimate

In this paper, we solve a local state observation problem for stochastic hyperbolic equations without boundary conditions, which is reduced to a local unique continuation property for these equations. This result is proved by a global Carleman estimate. As far as we know, this is the first result in this topic.

scholarly journals Global Carleman estimate on a network for the wave equation and application to an inverse problem

2011 ◽  
Vol 1 (3) ◽  
pp. 307-330 ◽  
Author(s):  
Lucie Baudouin ◽  
◽  
Emmanuelle Crépeau ◽  
Julie Valein ◽  
◽  
...  
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Carleman Estimate ◽  
Global Carleman Estimate

scholarly journals On the determination of the principal coefficient from boundary measurements in a KdV equation

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Lucie Baudouin ◽  
Eduardo Cerpa ◽  
Emmanuelle Crépeau ◽  
Alberto Mercado
Keyword(s):  
Inverse Problem ◽  
Kdv Equation ◽  
Carleman Estimate ◽  
Lipschitz Stability ◽  
Single Solution ◽  
De Vries ◽  
Boundary Measurements ◽  
Global Carleman Estimate

AbstractThis paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

A NEW UNIQUE CONTINUATION PROPERTY FOR THE KORTEWEG–DE VRIES EQUATION

2014 ◽  
Vol 90 (1) ◽  
pp. 90-98 ◽  
Author(s):  
MO CHEN ◽  
PENG GAO
Keyword(s):  
Vries Equation ◽  
Finite Interval ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
De Vries ◽  
Continuation Property ◽  
Global Carleman Estimate

AbstractThe aim of this paper is to obtain a new unique continuation property (UCP) for the Korteweg–de Vries equation posed on a finite interval. Compared with the previous UCP, we need fewer conditions on the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg–de Vries equation.

scholarly journals Carleman estimates for time-discrete parabolic equations and applications to controllability

2020 ◽  
Vol 26 ◽  
pp. 12
Author(s):  
Franck Boyer ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Parabolic Equations ◽  
Linear Time ◽  
Parabolic Systems ◽  
Carleman Estimate ◽  
Carleman Estimates ◽  
Discretization Scheme ◽  
Parabolic Operator ◽  
Time Step ◽  
Coupled Parabolic Systems ◽  
Observability Estimates

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

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