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Stabilization of the Kawahara equation with saturated internal or boundary feedback controls
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In this work, we are interested in a detailed qualitative analysis of the Kawahara equation, which models numerous physical motivations such as magneto-acoustic waves in a cold plasma and gravity waves on the surface of a heavy liquid. First, we design a feedback control law, which combines a damping component and another one of finite memory type. Then, we are capable of proving that the problem is well-posed under a condition involving the feedback gains of the boundary control and the memory kernel. Afterward, it is shown that the energy associated with this system exponentially decays by employing two different methods: the first one utilizes the Lyapunov function and the second one uses a compactness-uniqueness argument which reduces the problem to prove an observability inequality.
Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to 0 exponentially as t → ∞. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.
This article deals with the stability problem for a higher-order dispersive model governed by the so-called Kawahara equation. To do so, a damping mechanism is introduced, which contains a distributed memory term, and then proves that the solutions of the system are exponentially stable, provided that specific assumptions on the memory kernel are fulfilled. This is possible thanks to the energy method that permits us to obtain a decay rate estimate of the energy of the problem.
The stability analysis of the linear and nonlinear Korteweg-de Vries equations in presence of a saturated feedback actuator is studied. The well-posedness is derived by using nonlinear semigroup results, Schauder's and Banach fixed point theorems. The exponential stability is shown thanks to an observability inequality, which is obtained via contradiction and compactness arguments. Finally, an alternative proof of the asymptotic stability of the linear Korteweg-de Vries equation is presented.
In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy and using the Lyapunov approach, to ensure that the unique solution of the Kawahara system exponentially decays. The second result is achieved by means of a compactness-uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the main novelty of this work is to characterize the critical set phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the Möbius transformations.
This work is devoted to presenting Massera-type theorems for the Kawahara system, a higher-order dispersive equation, posed in a bounded domain. Precisely, thanks to some properties of the semigroup and the decay of the solutions of this equation, we can prove its solutions are periodic, quasi-periodic, and almost periodic.
The aim of this article is to investigate the well-posedness and stability problems of the so-called Kawahara equation under the presence of an interior localized delayed damping. The system is shown to be well-posed. Furthermore, we prove that the trivial solution is exponentially stable in spite of the delay effect. Specifically, local and semi-global stability results are established according to the properties of the spatial distribution of the delay term.
In this work, we deal with the exponential stability of the nonlinear Korteweg–de Vries equation on a finite star-shaped network in the presence of delayed internal feedback. We start by proving the well-posedness of the system and some regularity results. Then, we state an exponential stabilization result using a Lyapunov function by imposing small initial data and a restriction over the lengths. In this part also, we are able to obtain an explicit expression for the decay rate. Then, we prove a semi-global exponential stability result, which is based on an observability inequality working directly on the nonlinear system. Next, we study the case where it may happen that a control domain with delay is outside the control domain without delay. In that case, we obtain also a local exponential stabilization result. Finally, we present some numerical simulations to illustrate the stabilization.
In this manuscript, we consider the internal control problem for the fifth-order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we can prove that there exists an internal control input such that solutions of the Kawahara equation satisfies an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half-line, and right half-line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the ``mass" of the Kawahara equation posed in the half-line.
In recent years, controllability problems for dispersive systems have been extensively studied. This work is dedicated to proving a new type of controllability for a dispersive fifth order equation that models water waves, what we will now call the overdetermination control problem. Precisely, we are able to find a control acting at the boundary that guarantees that the solutions of the problem under consideration satisfy an overdetermination integral condition. In addition, when we make the control act internally in the system, instead of the boundary, we are also able to prove that this condition is satisfied. These problems give answers that were left open in [6] and present a new way to prove boundary and internal controllability results for a fifth order KdV type equation.
We consider the Kawahara equation, a fifth order Korteweg-de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax-Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on L 2 Sobolev space, the so-called regional controllability, that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region.
This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability, ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.
In this study, we characterize the lengths of intervals for
which the linear Kawahara equation has a non-trivial solution,
whose energy is stationary. This gives rise to a family of complex
functions. Characterizing the lengths amounts to deciding which members
of this family are entire functions. Our approach is essentially
based on determining the existence of certain Mobius transformation.
This article deals with the design of saturated controls in the context of
partial differential equations. It is focused on a linear Korteweg-de Vries
equation, which is a mathematical model of waves on shallow water surfaces. In
this article, we close the loop with a saturating input that renders the
equation nonlinear. The well-posedness is proven thanks to the nonlinear
semigroup theory. The proof of the asymptotic stability of the closed-loop
system uses a Lyapunov function.
The exact boundary controllability of linear and nonlinear Korteweg-de
Vries equation on bounded domains with various boundary conditions is
studied. When boundary conditions bear on spatial derivatives up to
order 2, the exact controllability result by Russell-Zhang is directly
proved by means of Hilbert Uniqueness Method. When only the first
spatial derivative at the right endpoint is assumed to be controlled,
a quite different analysis shows that exact controllability holds too.
From this last result we derive the exact boundary controllability for
nonlinear KdV equation on bounded domains, for sufficiently small initial
and final states.
We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.
Studied here is the eventual dissipation of solutions to initial–boundary value problems for the modified Kawahara equation with and without a localized damping term included. It is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical. In contrast, the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length.
A characterization of compact sets in Lp (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L
loc
1
(0, T; X) and if {fn/t} is bounded in L
loc
1
(0, T; Y) then {fn} is relatively compact in Lp(0,T; B), p
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 paper, we consider the Kawahara equation posed on a bounded interval with a distributed control. First, we establish a Carleman estimate for the Kawahara equation with internal observation. Then, applying this Carleman estimate, we can show that the Kawahara equation is null controllable. Furthermore, we prove the null controllability with constraints on the state for the Kawahara equation.
In this paper, we establish a global Carleman estimate for the Kawahara equation. Based on this estimate, we obtain the Unique Continuation Property (UCP) for this equation and the global exponential stability for the Kawahara equation with a very weak localized dissipation. © 2018 American Institute of Mathematical Sciences. All rights reserved.
Introduction.- Part I: Generalities.- Description of Systems Considered: Problem Statement.- Robust Stabilization under Control Constraints: An Overview.- Part II: Stability Analysis and Stabilization.- Analysis via the Use of Polytopic Models.- Synthesis via the Polytopic Model.- Analysis via the Use of Sector Nonlinearities Model.- Analysis via the Saturation Regions Model.- Part III: Anti-windup.- An Overview on Anti-windup Techniques.- Anti-windup Compensators Synthesis.- Appendices: Fundamental Properties on Stability Theory.- Fundamental Properties on Robust Control.- Mathematical Tools.
This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. Two kinds of control are considered: a distributed action and a boundary control. It is supposed that the control signal is subject to a cone-bounded nonlinearity. This kind of feedback laws includes (but is not restricted to) saturating inputs. By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups techniques. Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closed-loop system is proven by Lyapunov techniques. Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations.
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.
Steady solutions of a generalized Korteweg-de Vries equation which has an additional fifth order derivative term are investigated on the basis of a numerical calculation. It is found that either compressive or rarefactive solitary waves are possible to exist according as the dispersion is negative or positive and that the solitary waves take the oscillatory structure when the coefficient of the fifth order derivative term dominates over that of the third order one.
The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justification of this modeling and show that the solution of the Kawahara equation approximates that of the full problem of capillary-gravity waves in an appropriate sense for a long time interval. We also consider the case where the bot-tom is not flat and derive coupled Kawahara type equations whose solution approximates that of the full problem in that case.
In this paper we study solvability of the Cauchy problem of the Kawahara equation \(
\partial _{t} u + au\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0
\) with L
2 initial data. By working on the Bourgain space X
r,s
(R
2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H
r
(R) and −1 < r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L
2(R).
We investigate the stability of a solitary wave solution of the Korteweg-de Vries equation when a fifth order spatial derivative term is added. We show that the solution ceases to be strictly localized but develops an infinite oscillating tail and we compute the amplitude of the latter.
Hunter and Scheurle have shown that capillary-gravity water waves in the vicinity of Bond number (Bo)≈ are consistently modelled by the Korteweg-de Vries equation with the addition of a fifth derivative term. This wave equation does not have strict soliton solutions for Bo < because the near-solitons have oscillatory “wings” that extend indefinitely from the core of the wave. However, these solutions are “arbitrarily small perturbations of solitary waves” because the amplitude of the “wings” is exponentially small in the amplitude ϵ of the “core”. Pomeau, Ramani, and Grammaticos have calculated the amplitude of the “wings” by applying matched asymptotics in the complex plane in the limit ϵ → 0.In this article, we describe a mixed Chebyshev/radiation function pseudospectral method which is able to calculate the “weakly non-local solitons” for all ϵ. We show that for fixed phase speed, the solitons form a three-parameter family because the linearized wave equation has three eigensolutions. We also show that one may repeat the soliton with even spacing to create a three-parameter of periodic solutions, which we also compute.Because the amplitude of the “wings” is exponentially small, these non-local capillary gravity solitons are as interesting as the classical, localized solitons that solve the Korteweg-de Vries equation.
Necessary and sufficient conditions for globally stabilizing linear systems with bounded controls are known. However, it has been shown in [5] that, for single-input systems, no saturation of a linear feedback can globally stabilize a chain of integrators of order n, with n ≥ 3. In this paper, we propose a nonlinear combination of saturation functions of linear feedbacks that globally stabilizes a chain of integrators of arbitrary order. The appealing feature of the proposed control is that it is fairly easy to construct. It is linear near the origin and can also be used to achieve trajectory tracking for a class of trajectories restricted by the absolute on the input.
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 known that a linear system
x ˙= Ax + Bu can be stabilized by means of a
smooth bounded control if and only if it has no eigenvalues with
positive real part, and all the uncontrollable modes have a negative
real part. The authors investigate, for single-input systems, the
question of whether such systems can be stabilized by means of a
feedback u =σ( h ( x )), where h is
linear and σ( s ) is a saturation function such as
sign( s ) min(| s |,1). A stabilizing feedback of this
particular form exists if A has no multiple eigenvalues, and
also in some other special cases such as the double integrator. It is
shown that for the multiple integrator of order n , with n
⩾3, no saturation of a linear feedback can be globally
stabilizing
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