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On the stabilization and controllability for a third order linear equation
Abstract
We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation:
iu_t+i\gamma u_x+ \alpha u_{xx} + i\beta u_{xxx} =0,
where u=u(x,t) is a complex valued function defined in (0,L)\times(0,+\infty) and \alpha , \beta and \gamma are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths L , named \mathcal{X} , in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters \alpha , \beta and \gamma .
... then the L 2 −norm of the solution does not necessarily decay to zero. Here N is the set of critical lengths in the context of exact boundary controllability for the HLS (see [10,14] for the derivation of this set of critical lengths). For instance choosing the coefficients β = 1, α = 2 and δ = 8 with k = 1 and l = 2, we obtain L = π ∈ N . ...
... and use (14) to deduce thatŵ * solves target observer model given by ...
... Recall that the backstepping transformation (20) transforms (14) to (159) if p 1 , p 2 are chosen such that p 1 (x)iβp y (x, 0) − αp(x, 0) and p 2 (x) = −iβp(x, 0) where p is the backstepping kernel that solves (19). An example for the real and imaginary parts of the observer gains for a problem defined on [0, π] and the coefficients β = 1, α = 2, δ = 8, r = 0.05 are given in Figure 13. ...
Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the \begin{document}\end{document}-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.
... In section 2, for the sake of the completeness, we make a brief analysis of the linear case, that is, the system (1.1) without the term |u| 2 u. We use the work of Silva-Vasconcellos [10] and some multiplier techniques when we consider the additional damping term. ...
... In this section we analyze the linear system associated with system (1.1). Taking into account the work of Silva-Vasconcellos [10], we begin by analysing existence, uniqueness, regularity of solutions and exponential decay of the energy associated to the following system: ...
... We use semigroups theory to prove the existence and uniqueness and to show regularity of solutions we consider the multipliers techniques. The items i), ii), iii) and the dissipation law follow as in Silva-Vasconcellos [10], considering there the parameter γ = 0. ...
... As for the application of HUM in control problems, we can further refer to, e.g., [1,2,3,9,10,12,16,17,19,25,26,29,36] for control properties of classical linear partial differential equations, e.g., [7,8,15] for control properties of degenerate parabolic equations, e.g., [4,15] for control properties of degenerate hyperbolic equations, and e.g., [13,35] for control properties of heat, wave, and Schrödinger equations with singular potentials. Moreover, we can refer to, e.g., [21,32,33] for control properties of quasi-linear, semilinear, or nonlinear partial differential equations. ...
... By (1.14), (1.16), (A.8), and (A.9), we can arrive at (g 3 , g 4 ) 2 ≥ µ 2 − µβ (u, v) 2 (A. 10) for sufficiently large µ > ω max {µ 0 , β} > 0. Then we have, for µ > ω, ...
We consider control properties for second-order hyperbolic systems in anisotropic cases with variable coefficient matrices in a bounded domain in Rⁿ with C²-boundary. Assuming that the coefficient matrices satisfy suitable conditions, we establish observability inequalities for hyperbolic systems in anisotropic cases. Then, using the Hilbert uniqueness method, we deduce the exact controllability of the corresponding control problem for hyperbolic systems in anisotropic cases. The same results for linear elastodynamic systems in inhomogeneous and anisotropic media are provided to illustrate the application of the results.
... = 2πβ k 2 + kl + l 2 3βδ + α 2 : k, l ∈ Z + , (1.4) then the L 2 −norm of the solution does not necessarily decay to zero. Here N is the set of critical lengths in the context of exact boundary controllability for the HLS (see [8,12] for the derivation of this set of critical lengths). For instance choosing the coefficients β = 1, α = 2 and δ = 8 with k = 1 and l = 2, we obtain L = π ∈ N . ...
Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in Part I of this study where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the -norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.
... The price to be paid is the lack of any Kato smoothing effect (the system being conservative), which makes the extension of the control results to the nonlinear Boussinesq system more delicate than for KdV. We refer the reader to [7,8,9,10,11,12,13,16,18,19,20,21,27,29,30,31,32,33] for the control and stabilization of KdV, and [14,15,17] for the critical lengths concerning some other dispersive equations. ...
We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley- Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
... The price to be paid is the lack of any Kato smoothing effect (the system being conservative), which makes the extension of the control results to the nonlinear Boussinesq system more delicate than for KdV. We refer the reader to [7,8,9,10,11,12,13,16,18,19,20,21,27,29,30,31,32,33] for the control and stabilization of KdV, and [14,15,17] for the critical lengths concerning some other dispersive equations. ...
We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
... The price to be paid is the lack of any Kato smoothing effect (the system being conservative), which makes the extension of the control results to the nonlinear Boussinesq system more delicate than for KdV. We refer the reader to [7,8,9,10,11,12,13,16,18,19,20,21,27,29,30,31,32,33] for the control and stabilization of KdV, and [14,15,17] for the critical lengths concerning some other dispersive equations. ...
We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
... For the stabilization and controllabilty for a linear model arising in pulse propagation in optical fiber see Silva-Vasconcellos [23]. ...
We analyze the stabilization and internal exact control for the Kuramoto-Sivashinsky equation (KS) in a bounded interval. That is, we consider the following equation: u t +αu xx +βu xxx +ηu xxxx +uu x =0· The above equation is a model for small amplitude long waves. Using multiplier techniques we prove the exponential decay of the total energy associated with the KS system and in the linear case, using HUM method, we study its internal exact controllability.
The purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.
We study the boundary controllability of a nonlinear Korteweg–de Vries equation with
the Dirichlet boundary condition on an interval with a critical length for which it has been shown
by Rosier that the linearized control system around the origin is not controllable. We prove that the
nonlinear term gives the local controllability around the origin.
This paper is concerned with the the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a lo-calized damping term. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived.
It is known that the linear Korteweg-de Vries (KdV) equation with homogeneous Dirichlet boundary conditions and Neumann boundary control is not controllable for some critical spatial domains. In this paper, we prove in these critical cases, that the nonlinear KdV equation is locally controllable around the origin provided that the time of control is large enough. It is done by performing a power series expansion of the solution and studying the cascade system resulting of this expansion. (c) 2008 Elsevier Masson SAS. All rights reserved.
This paper is concerned with the internal stabilization of the generalized Korteweg--de Vries equation on a bounded domain. The global well-posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval [1,4). The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua [Quart. Appl. Math., 60 (2002), pp. 111-129].
In this work we are interested in the study of controllability and
stabilization of the linearized Benjamin-Ono equation with
periodic boundary conditions, which is a generic model for the
study of weakly nonlinear waves with nonlocal dispersion. It is
well known that the Benjamin-Ono equation has infinite number of
conserved quantities, thus we consider only controls acting in the
equation such that the volume of the solution is conserved. We
study also the stabilization with a feedback law which gives us an
exponential decay of the solutions.
We study the stabilization of global solutions of the linear Kawahara (K) equation in a bounded interval under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. We also prove that the decay of solutions, in absence of damping, fails for some critical values of the length L and we define precisely this countable set. Finally, we include some remarks about nonlinear problem and we analyze the exact boundary control for linear Kawahara system.
A modified nonlinear Schrödinger equation (MNLS) which is obtained by adding a third-order dispersion term to the nonlinear Schrödinger equation (NLS) is investigated numerically under the periodic boundary condition. It is found that the third-order dispersion term can give rise to chaotic behaviour. It is also found that when the coefficient of the third-order dispersion term becomes small the solutions of the MNLS equation come to have resemblance in the overall aspect to those of the NLS equation.
Consider a body of water of finite depth under the influence of gravity, bounded below by a flat, impermeable surface. If viscous and surface tension effects are ignored, and assuming that the flow is incompressible and irrotational, the fluid motion is governed by the Euler equations together with suitable boundary conditions on the rigid surfaces and on the air-water interface. In special regimes, the Euler equations admit of simpler, approximate models that describe pretty well the fluid response to a disturbance. In situations where the wavelength is long and the amplitude is small relative to the undisturbed depth, and if the Stokes number is of order one, then various model equations have been derived. Two of the most standard are the KdV-equation (0.1)ut+ux+uux+uxxx=0and the RLW-equation (0.2)ut+ux+uux−uxxt=0.Bona, Pritchard and Scott showed that solutions of these two evolution equations agree to the neglected order of approximation over a long time scale, if the initial disturbance in question is genuinely of small-amplitude and long-wavelength. The same formal argument that allows one to infer (0.2) from (0.1) in small-amplitude, long-wavelength regimes also produces a third equation, namelyut+ux+uux+uxtt=0.Kruskal, in a wide-ranging discussion of modelling considerations, pointed to this equation as an example that might not accurately describe water waves. Its status has remained unresolved. It is our purpose here to show that the initial-value problem for the latter equation is indeed well posed. Moreover, we show that for small-amplitude, long waves, solutions of this model also agree to the neglected order with solutions of either (0.1) or (0.2) provided the initial data is properly imposed.11AMS(MOS) subject classifications (1985 revision). 35B05, 35E15, 35Q20, 76D33, 76E30
We establish global well-posedness for the initial value problem (IVP) associated to the so-called Benney–Lin equation. This model is a Korteweg–de Vries equation perturbed by dissipative and dispersive terms which appears in fluid dynamics. We also study the limiting behaviour of solutions to this IVP when the parameters of the perturbed terms approach 0.
The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H−1-norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q. Appl. Math. 2002; LX(1):111–129; ESAIM Control Optim. Calculus Variations 2005; 11(3):473–486) and gives a satisfactory answer to a problem suggested in (Q. Appl. Math. 2002; LX(1):111–129). Copyright © 2007 John Wiley & Sons, Ltd.
We study the stabilization of global solutions of the
Kawahara (K) equation in a bounded interval, under the effect of
a localized damping mechanism. The Kawahara equation is a model
for small amplitude long waves. Using multiplier techniques and
compactness arguments we prove the
exponential decay of the solutions of the (K) model. The proof
requires of a unique continuation theorem and the smoothing effect
of the (K) equation on the real line, which are proved in this work.
It is shown that the third-order dispersion of an optical fibre can be used to reduce the mutual interaction between solitons when the fibre is operated nonlinearly. This provides a means to improve the bandwidth of nonlinear fibre.