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Some exact controllability results for the linear KDV equation and uniform controllability in the zerodispersion limit
 1/2): 61–100. CONTROL AND STABILIZATION OF THE KDV EQUATION 681
"... In this paper, we deal with controllability properties of linear and nonlinear Kortewegde Vries equations in a bounded interval. First, we establish the null controllability of the linear equation via the left Dirichlet boundary condition, and its exact controllability via both Dirichlet boundary c ..."
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Cited by 19 (5 self)
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In this paper, we deal with controllability properties of linear and nonlinear Kortewegde Vries equations in a bounded interval. First, we establish the null controllability of the linear equation via the left Dirichlet boundary condition, and its exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain local exact controllability results for the nonlinear KdV equation. Finally, we prove a result of uniform controllability of the linear KdV equation in the limit of zerodispersion. 1
The initialboundary value problem for the Kortewegde Vries equation
"... Abstract. We prove local wellposedness of the initialboundary value problem for the Kortewegde Vries equation on right halfline, left halfline, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators. ..."
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Abstract. We prove local wellposedness of the initialboundary value problem for the Kortewegde Vries equation on right halfline, left halfline, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.
Controllability of the Kortewegde Vries equation from the right Dirichlet boundary condition
, 2009
"... In this paper, we consider the controllability of the Kortewegde 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 c ..."
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Cited by 13 (0 self)
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In this paper, we consider the controllability of the Kortewegde 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. hal00455223, version 1 9 Feb 2010 1
Nonhomogeneous Boundary Value Problems for the Kortewegde Vries and the Kortewegde VriesBurgers Equations in a Quarter Plane
, 2007
"... Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0, ..."
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Cited by 9 (3 self)
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Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0,
Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete Contin
 Dyn. Syst
"... Abstract. Studied here is the largetime behavior and eventual periodicity of solutions of initialboundaryvalue problems for the BBM equation and the KdV equation, with and without a Burgerstype dissipation appended. It is shown that the total energy of a solution of these problems grows at an al ..."
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Cited by 8 (5 self)
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Abstract. Studied here is the largetime behavior and eventual periodicity of solutions of initialboundaryvalue problems for the BBM equation and the KdV equation, with and without a Burgerstype dissipation appended. It is shown that the total energy of a solution of these problems grows at an algebraic rate which is in fact sharp for solutions of the associated linear equations. We also establish that solutions of the linear problems are eventually periodic if the boundary data are periodic. 1. Introduction. Initialboundaryvalue
Global wellposedness and asymptotic behavior of a class of initialboundaryvalue problem of the Kortewegde Vries equation on a finite domain
 Math. Control Related Fields
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Scattering for the Forced NonLinear Schrodinger Equation with a Potential on the HalfLine
"... In this paper we construct the scattering operator for the forced nonlinear Schrodinger equation with a potential on the halfline. Moreover, in the case where the force is zero, and the solutions satisfy the homogeneous Dirichlet boundary condition at zero, we prove that the scattering operator de ..."
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Cited by 7 (3 self)
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In this paper we construct the scattering operator for the forced nonlinear Schrodinger equation with a potential on the halfline. Moreover, in the case where the force is zero, and the solutions satisfy the homogeneous Dirichlet boundary condition at zero, we prove that the scattering operator determines uniquely the potential and the nonlinearity and we give a method for the reconstruction of both. # 2000 AMS classification 35P25, 35R30 and 81U40. Research partially supported by proyecto PAPIIT, IN 105799, DGAPAUNAM.
Explicit solutions to the Kortewegde Vries equation on the half line
 INVERSE PROBLEMS
, 2006
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Uniform controllability of a transport equation in zero diffusiondispersion limit
 Math. Models Methods Appl. Sci
"... In this paper, we consider the controllability of a transport equation perturbed by small di®usion and dispersion terms. We prove that for a su±ciently large time, the cost of the nullcontrollability tends to zero exponentially as the perturbation vanishes. For small times, on the contrary, we prov ..."
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Cited by 5 (0 self)
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In this paper, we consider the controllability of a transport equation perturbed by small di®usion and dispersion terms. We prove that for a su±ciently large time, the cost of the nullcontrollability tends to zero exponentially as the perturbation vanishes. For small times, on the contrary, we prove that this cost grows exponentially.
UNIFORM STABILIZATION IN WEIGHTED SOBOLEV SPACES FOR THE KDV EQUATION POSED ON THE HALFLINE
"... Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in ..."
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Cited by 3 (1 self)
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Abstract. Studied here is the largetime behavior of solutions of the Kortewegde Vries equation posed on the right halfline under the effect of a localized damping. Assuming as in [19] that the damping is active on a set (a0, +∞) with a0> 0, we establish the exponential decay of the solutions in the weighted spaces L 2 ((x + 1) m dx) for m ∈ N ∗ and L 2 (e 2bx dx) for b> 0 by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived. 1. Introduction. The Kortewegde Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel [8, 14, 15]. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a socalled nonlinear Fourier transform,