The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Jean-michel Coron - One of the best experts on this subject based on the ideXlab platform.
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Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet Boundary Condition
IEEE Transactions on Automatic Control, 2013Co-Authors: Eduardo Cerpa, Jean-michel CoronAbstract:This paper deals with the stabilization problem for the Korteweg-de Vries equation posed on a bounded interval. The control acts on the left Dirichlet Boundary Condition. At the right end-point, Dirichlet and Neumann homogeneous Boundary Conditions are considered. The proposed feedback law forces the exponential decay of the system under a smallness Condition on the initial data. Moreover, the decay rate can be tuned to be as large as desired. The feedback control law is designed by using the backstepping method.
Sergio Guerrero - One of the best experts on this subject based on the ideXlab platform.
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controllability of the korteweg de vries equation from the right Dirichlet Boundary Condition
Systems & Control Letters, 2010Co-Authors: Olivier Glass, Sergio GuerreroAbstract:In this paper, we consider the controllability of the Korteweg–de Vries equation in a bounded interval when the control operates via the right Dirichlet Boundary Condition, while the left Dirichlet and the right Neumann Boundary Conditions are kept to zero. We prove that the linearized equation is controllable if and only if the length of the spatial domain does not belong to some countable critical set. When the length is not critical, we prove the local exact controllability of the nonlinear equation.
Marian Slodicka - One of the best experts on this subject based on the ideXlab platform.
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full discretization of a nonlinear parabolic problem containing volterra operators and an unknown Dirichlet Boundary Condition
Numerical Methods for Partial Differential Equations, 2015Co-Authors: Marijke Grimmonprez, Marian SlodickaAbstract:The reconstruction of an unknown solely time-dependent Dirichlet Boundary Condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet Condition is eliminated using a given integral overdetermination. A time-discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time-discrete approximation scheme in space. Finally, full-discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015
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a nonlinear parabolic integro differential problem with an unknown Dirichlet Boundary Condition
Journal of Computational and Applied Mathematics, 2015Co-Authors: Marijke Grimmonprez, Marian SlodickaAbstract:A nonlinear parabolic integro-differential equation ? t g ( u ) - Δ u = F + ? 0 t f ( s , u ( s ) ) d s with a known Neumann Boundary Condition on a part of the Boundary and an unknown Dirichlet Boundary Condition α ( t ) on the other part of the Boundary is studied. The inverse problem of identifying the unknown time-dependent function α ( t ) from an additional integral measurement E ( t ) = ? ? g ( u ( t , x ) ) d x is investigated. The well-posedness of the problem in suitable function spaces is shown and a numerical time-discrete scheme for approximations is designed. Convergence of the proposed scheme is supported by a numerical experiment.
Toshikazu Kuniya - One of the best experts on this subject based on the ideXlab platform.
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a reaction diffusion susceptible vaccinated infected recovered model in a spatially heterogeneous environment with Dirichlet Boundary Condition
Mathematics and Computers in Simulation, 2021Co-Authors: Jinliang Wang, Ran Zhang, Toshikazu KuniyaAbstract:Abstract In this paper, we study a Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model in a spatially heterogeneous environment under the Dirichlet Boundary Condition. We define the basic reproduction number ℜ 0 by the spectral radius of the next generation operator, and show that it is a threshold parameter. The disease extinction and persistence in the case of a bounded domain are considered. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ℜ 0 1 ; the system is uniformly persistent and an endemic equilibrium exists if ℜ 0 > 1 . To verify our theoretical results, we perform some numerical simulations, using the Fredholm discretization method to identify ℜ 0 .
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global threshold dynamics of an infection age structured sir epidemic model with diffusion under the Dirichlet Boundary Condition
Journal of Differential Equations, 2020Co-Authors: Abdennasser Chekroun, Toshikazu KuniyaAbstract:Abstract In this paper, we are concerned with the global asymptotic behavior of an infection age-structured SIR epidemic model with diffusion in a general n-dimensional bounded spatial domain under the homogeneous Dirichlet Boundary Condition. By using the method of characteristics, we reformulate the model into a system of a reaction-diffusion equation and a Volterra integral equation. We define the basic reproduction number R 0 by the spectral radius of a compact positive linear operator and show that if R 0 1 , then the disease-free steady state is globally attractive, whereas if R 0 > 1 , then a positive endemic steady state exists and the system is uniformly persistent. By numerical simulation for the 2-dimensional case, we show that R 0 depends on the shape of the spatial domain. This result is in contrast with the case of the homogeneous Neumann Boundary Condition, in which R 0 is independent of the spatial domain.
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an infection age space structured sir epidemic model with Dirichlet Boundary Condition
Mathematical Modelling of Natural Phenomena, 2019Co-Authors: Abdennasser Chekroun, Toshikazu KuniyaAbstract:In this paper, we are concerned with the global asymptotic behavior of an SIR epidemic model with infection age-space structure. Under the homogeneous Dirichlet Boundary Condition, we first reformulate the model into the coupled reaction-diffusion and difference system by using the method of characteristics. We then obtain the spatially heterogeneous disease-free steady state and define the basic reproduction number ℛ 0 by the spectral radius of the next generation operator. We then show the existence and uniqueness of the global classical solution by constructing suitable upper and lower solutions. As a threshold result, we establish that the disease-free steady state is globally attractive if ℛ 0 0 > 1. Finally, numerical simulations are exhibited to illustrate our theoretical results together with how to compute ℛ 0 .
Alexander Ukhlov - One of the best experts on this subject based on the ideXlab platform.
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Spectral Stability Estimates of Dirichlet Divergence Form Elliptic Operators
Analysis and Mathematical Physics, 2020Co-Authors: V. M. Gol'dshtein, Valerii Pchelintsev, Alexander UkhlovAbstract:We study spectral stability estimates of elliptic operators in divergence form $$-\text {div} [A(w) \nabla g(w)]$$ with the Dirichlet Boundary Condition in non-Lipschitz domains $${\widetilde{\varOmega }} \subset {\mathbb {C}}$$ . The suggested method is based on the theory of quasiconformal mappings, weighted Sobolev spaces theory and its applications to the Poincare inequalities.
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Spectral Stability Estimates of Dirichlet Divergence Form Elliptic Operators
arXiv: Analysis of PDEs, 2019Co-Authors: V. M. Gol'dshtein, Valerii Pchelintsev, Alexander UkhlovAbstract:We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet Boundary Condition in non-Lipschitz domains $\widetilde{\Omega} \subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincare inequalities.