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On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results
Abstract
This article is concerned with a Kuramoto–Sivashinsky-Korteweg-de Vries equation in a bounded interval. The equation as well as one of the boundary conditions are supposed to be the subject of the presence of a parameter . Moreover, this specific boundary condition has a time-delay effect. As tends to zero, we show that one can obtain the findings of [4,58] concerning two Korteweg–de Vries equations. Indeed, we are able to retrieve the well-posedness and stability results for the Korteweg–de Vries problem without delay [58] and with delay [4] under the same conditions, as a singular limit of the Kuramoto–Sivashinsky equation with delay. The proof is based in the well-known Galerkin method together with the multiplier technique.
... However, when delays become time-varying, the complexity of analyzing system stability significantly rises. Numerous studies have delved into the stability analysis of partial differential equations (PDEs) with time-delay or time-varying delays, with notable references, such as [12,15,16,26,30,37]. ...
... Thanks to (12), the choice of ξ is feasible. ...
... Proposition 3.3. Let (u 0 , z 0 (·, −θ(0)·)) ∈ H and assume that the nonnegative functions a and b belonging to L ∞ (0, L) satisfy (11) and (12). We also suppose that (8), (9) and (10) holds and f ∈ L 1 (0, T, L 2 (0, L)). ...
The current paper investigates the question of stabilizability of the Kawahara equation with time-varying delay in the internal feedback. Under specific assumptions regarding the time-varying delay, we establish that the studied system is well-posed. Moreover, using an appropriate Lyapunov functional, we prove the exponential stability result. Finally, we present some conclusions.
The main concern of the present paper is to study the well-posedness and stability problem of two different dispersive systems subject to the effect of a distributed infinite memory term. The two systems are respectively governed by the one-dimensional Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations in a bounded domain [0,1]. In order to deal with the presence of the memory term, we adopt the history approach. First, we show that both problems are well-posed in appropriate functional spaces by means of the Fixed-Point Theorem provided that the initial condition is sufficiently small. Then, the energy method enables us to provide a decay estimate of the systems’ energy according to the assumptions satisfied by the physical parameters and the memory kernel.
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The aim of this paper is to investigate the stability problem for a nonlinear dispersive equation with memory. More precisely, the equation under consideration combines the well-known Korteweg-de Vries and Kuramoto–Sivashinsky equations, subject to the presence of a boundary memory term. The problem is shown to be well-posed for small initial data and provided that reasonable conditions hold for the parameters of the system and the memory kernel. In addition, we prove that the trivial solution is exponentially stable in spite of the memory effect. It is noteworthy that these outcomes are obtained under numerous instances of the physical parameters of the system.
The main concern of this article is to deal with the presence of a boundary delay in the nonlinear Kuramoto-Sivashinsky equation. First, we prove that the whole system is well-posed under a smallness assumption on the initial data and a condition on the parameter involved in the delay term. Then, the solution is shown to decay exponentially despite the presence of the delay. These findings are proved by considering several situations depending on the physical parameters of the system.
This paper addresses the impact of the presence of a boundary memory term in the third-order Korteweg–de Vries equation in a bounded interval [0,ℓ]. First, an overall literature review is provided. Indeed, a comprehensive discussion on the literature constitutes a survey part of the current paper. Thereafter, it is shown that the system under consideration possesses a unique solution under a smallness assumption on the initial data and an appropriate condition on the parameters and the kernel involved in the memory term. Last but not least, we demonstrate that the zero solution is exponentially stable as long as the length ℓ is small enough by means of Lyapunov method, which permits to provide an estimate of the exponential decay rate. These findings improve and complement those of Zhang (in: Desch W, Kappel F, Kunisch K (eds) Proceedings of international conference on control and estimation of distributed parameter systems: nonlinear phenomena, International Series of Numerical Mathematics, vol 118, Birkhauser, Basel, pp 371–389, 1994) (resp. Baudouin et al. in IEEE Trans Autom Control 64:1403–1414, 2019), where no memory term is present (resp. a delay occurs instead of memory).
Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is [0,1] [0,1]. First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the L2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.
In the last 40 years the study of initial boundary value problem for the Korteweg–de Vries equation has had the attention of researchers from various research fields. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view.
In this paper we stabilize the linear Kuramoto-Sivashinsky equation by means of a delayed boundary control. From the spectral decomposition of the spatial operator associated to the equation, we find that there is a finite number of unstable eigenvalues. After applying the Artstein transform to deal with the delay phenomenon, we design a feedback law based on the pole-shifting theorem to exponential stabilize the finite-dimensional system associated to the unstable eigenvalues. Then, thanks to the inversion of the Artstein transform and the use of a Lyapunov function, we obtain a delayed feedback law that exponential stabilize the original unstable infinite-dimensional system.
Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto–Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter ν. It is shown that if ν > 1, then the set of constant equilibrium solutions is globally asymptotically stable. If ν < 1, then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto–Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.
In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval u_t+u_x+u_{xxx}+uu_x=0, u(x,0)=\phi(x), 0<x<L, t>0, subject to the nonhomogeneous boundary conditions,
B_1u=h_1(t), B_2 u= h_2 (t), B_3 u= h_3 (t), t>0 where B_i u =\sum _{j=0}^2 \left(a_{ij} \partial ^j_x u(0,t) + b_{ij}
\partial ^j_x u(L,t)\right), i=1,2,3 and are real constants. Under some general assumptions imposed on the coefficients , , the IBVPs is shown to be locally well-posed in the space for any with and boundary values belonging to some appropriate spaces with optimal regularity.
In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one.
In this paper we study the null controllability property of the linear Kuramoto-Sivashinsky equation by means of either boundary or internal controls. In the Dirichlet boundary case, we use the moment theory to prove that the null controllability property holds with only one boundary control if and only if the anti-diffusion parameter of the equation does not belong to a critical set of parameters. Regarding the Neumann boundary case, we prove that the null controllability property does not hold with only one boundary control. However, it does always hold when either two boundary controls or an internal control are considered. The proof of the latter is based on the controllability-observability duality and a suitable Carleman estimate.
This paper presents a control problem for a one-dimensional nonlinear parabolic system, which consists of a Kuramoto-Sivashinsky-Korteweg de Vries equation coupled to a heat equation. We address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only. The local null-controllability of the system is proved. The proof is based on a Carleman estimate for the linearized system around the origin. A local inversion theorem is applied to get the result for the nonlinear system.
The problem of controlling and stabilizing solutions to the Kuramoto–Sivashinsky (KS) equation is studied in this paper. We
consider a generalized form of the equation in which the effects of an electric field and dispersion are included. Both the
feedback and optimal control problems are studied. We prove that we can control arbitrary non-trivial steady states of the
KS equation, including travelling wave solutions, using a finite number of point actuators. The number of point actuators
needed is related to the number of unstable modes of the equation. Furthermore, the proposed control methodology is shown
to be robust with respect to changing the parameters in the equation, e.g. the viscosity coefficient or the intensity of the
electric field. We also study the problem of controlling solutions of coupled systems of KS equations. Possible applications
to controlling thin film flows are discussed. Our rigorous results are supported by extensive numerical simulations.
The Korteweg-de Vries (KdV) and the Kuramoto-Sivashinsky (KS) partial differential equations are used to model nonlinear propagation of one-dimensional phenomena. The KdV equation is used in fluid mechanics to describe waves propagation in shallow water surfaces, while the KS equation models front propagation in reaction-diffusion systems. In this article, the boundary control of these equations is considered when they are posed on a bounded interval. Different choices of controls are studied for each equation.
These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in L 2 -norm.
The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s. Both
exact control and stabilization problems have been intensively studied since then and significant progresses have been made
due to many people's hard work and contributions. In this article, the authors intend to give an overall review of the results
obtained so far in the study but with an emphasis on its recent progresses. A list of open problems is also provided for further
investigation.
We are concerned with the boundary controllability to the trajec-tories of the Kuramoto–Sivashinsky equation. By using a Carleman estimate, we obtain the null controllability of the linearized equa-tion around a given solution. From a local inversion theorem we get the local controllability to the trajectories of the nonlinear sys-tem.
In this article, we study the boundary controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval. The control acts on the first spatial derivative at the left endpoint. First, we prove that this control system is null controllable. It is done using a spectral analysis and the method of moments. Then, we introduce a boundary feedback law stabilizing to zero the solution of the closed-loop system.
In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.
In this article, we present an inverse problem for the nonlinear 1D Kuramoto-Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhgem-Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.
The paper is devoted to distributed sampled-data control of nonlinear PDE system governed by 1-D Kuramoto–Sivashinsky equation. It is assumed that N sensors provide sampled in time spatially distributed (either point or averaged) measurements of the state over N sampling spatial intervals. Locally stabilizing sampled-data controllers are designed that are applied through distributed in space shape functions and zero-order hold devices. Given upper bounds on the sampling intervals in time and in space, sufficient conditions ensuring regional exponential stability of the closed-loop system are established in terms of Linear Matrix Inequalities (LMIs) by using the time-delay approach to sampled-data control and Lyapunov–Krasovskii method. As it happened in the case of diffusion equation, the descriptor method appeared to be an efficient tool for the stability analysis of the sampled-data Kuramoto–Sivashinsky equation. An estimate on the domain of attraction is also given. A numerical example demonstrates the efficiency of the results.
The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.
This article concerns the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non critical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results.
This paper is concerned with the optimal boundary control of a non-dimensional non-linear parabolic system consisting of the Kuramoto–Sivashinsky–Korteweg–de Vries equation and a heat equation. By the Dubovitskii and Milyutin functional analytical approach, first in the fixed final horizon case we prove the Pontryagin maximum principle of the optimal control problem of this coupled system. Then under weaker additional conditions, we study the controlled system in the free final horizon case and present further investigational results of current interests. The necessary optimality conditions are established for optimal control problems in these two cases. Finally, a remark on how to utilize the obtained results is also made for illustration.
This paper is addressed to study the null controllability with constraints on the state for the Kuramoto-Sivashinsky equation. We first consider the linearized problem. Then, by Kakutani fixed point theorem, we show that the same result holds for the Kuramoto-Sivashinsky equation. © 2015, American Institute of Mathematical Sciences. All rights reserved.
In this paper, we establish a new global Carleman estimate for the linearized Kuramoto–Sivashinsky equation which is a one-parameter Carleman estimate. Based on this estimate, we can obtain the local exact controllability to the trajectories and Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation.
This paper is devoted to the study of the local rapid exponential
stabilization problem for a controlled Kuramoto-Sivashinsky equation on a
bounded interval. We build a feedback control law to force the solution of the
closed-loop system to decay exponentially to zero with arbitrarily prescribed
decay rates, provided that the initial datum is small enough. Our approach uses
a method we introduced for the rapid stabilization of a Korteweg-de Vries
equation. It relies on the construction of a suitable integral transform and
can be applied to many other equations.
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.
Robust control theory, a generalization of optimal control theory, has been proposed as an effective technique when control algorithms are sensitive to a broad class of external disturbances. In [T. R. Bewley, R. Teman and M. Ziane, Physica D 138, No. 3-4, 360-392 (2000; Zbl 0981.76026)], a general framework for the robust control of the Navier-Stokes equations in finite time horizon was developed. In this article the robust boundary control for the Kuramoto-Sivashinsky equation is considered in the same spirit: a robust boundary control problem is formulated, and the existence and uniqueness for the robust control problem are proved. A data assimilation problem corresponding to the Kuramoto-Sivashinsky equation is considered, existence and uniqueness of solution are derived. This approach is also applicable as well to other equations with a structure similar to that of the Kuramoto-Sivashinsky equation.
This paper considers the problem of robust global stabilization of the Kuramoto–Sivashinsky equation subject to Neumann and
Dirichlet boundary conditions. The aim is to derive non-linear robust boundary control laws which make the system robustly
globally asymptotically stable in spite of uncertainty in both the instability parameter and the anti-diffusion parameter.
A unique approach this paper introduces for achieving the required robustness is spatially dependent scaling of uncertain
elements in Lyapunov-based stabilization. An important advantage of this approach is flexibility to obtain robust control
laws with small control effort. The control laws can be implemented as Dirichlet-like boundary control as well as Neumann-like
boundary control. Furthermore, it is shown that they guarantee the stability and boundedness in terms of both L2 and L∞.
We consider a coupled system of Kuramoto-Sivashinsky equations depending on a
suitable parameter and study its asymptotic behavior for t large, as
. Introducing appropriate boundary conditions we show that the
energy of the solutions decays exponentially uniformly with respect to the
parameter . In the limit, as , we obtain a coupled system
of Korteweg-de Vries equations known to describe strong interactions of two long
internal gravity waves in a stratified fluid for which the energy tends to zero
exponentially as well. The decay fails when the length of the space interval L
lies in a set of critical lengths.
Considered herein is the Korteweg-de Vries equation with a Kuramoto-Sivashinsky
dissipative term appended. This evolution equation, which arises as a model for a number
of interesting physical phenomena, has been extensively investigated in a recent paper of
Ercolani, McLaughlin and Roitner. The numerical simulations of the initial-value problem
reported in the just-mentioned study showed solutions to possess a more complex range of
behavior than the unadorned Korteweg-de Vries equation. The present work contributes some
basic analytical facts relevant to the initial-value problem and to some of the
conclusions drawn by Ercolanet al. In addition to showing the initial-value problem
is well posed, we determine the limiting behavior of solutions as the dissipative or the
dispersive parameter tends to zero.
We consider a coupled system of Kuramoto--Sivashinsky (KS) equations in a bounded
interval depending on a suitable parameter . As tends to zero, we obtain a
coupled system of Korteweg--de Vries (KdV) equations known to describe strong interactions
of two long internal gravity waves in a stratified fluid. Existence and uniqueness of
global solutions of the KS model is established as well.
We consider second-order evolution equations in an abstract setting with
damping and time delay and give sufficient conditions ensuring exponential
stability. Our abstract framework is then applied to the wave equation, the
elasticity system and the Petrovsky system.
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.
This work addresses the problem of global exponential stabilization of the Kuramoto–Sivashinsky equation (KSE) subject to periodic boundary conditions via distributed static output feedback control. Under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the KSE and a necessary and sufficient stability condition is satisfied, linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the KSE. The controllers are designed on the basis of finite-dimensional approximations of the KSE which are obtained through Galerkin's method. The theoretical results are confirmed by computer simulations of the closed-loop system.
A nonlinear analysis is presented of flame front stability assuming stiochiometric composition of the combustible mixture. The constant-density model of a premixed flame is considered. An asymptotic nonlinear partial differential equation, which describes the evolution of the disturbed flame front, is derived. It is shown that plane flame front is stable if the mean Lewis number of the reactants exceeds some critical value close to unity. If this condition does not hold the plane flame front is unstable, and numerical studies demonstrate that the flame assumes the form of a nonstationary cellular surface in a state of continual chaotic division and recombination of cells. A description is given of the effect of local violation of stoichiometry in the vicinity of the disturbed reaction zone, as a result of which the locally-excess component penetrates into the combustion product region.
The Kuramoto-Sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films. They are characterized by the coexistence of coherent spatial structures with temporal chaos. We investigate some global dynamical properties, including nonlinear stability. We demonstrate their low modal behavior, in terms of determining modes; and that the fractal dimension of all attractors is bounded by a universal constant times ~L13/8, where ~L is a dimensionless pattern cell size (in the one-dimensional case). Such equations are indeed a paradigm of low-dimensional behavior for infinite-dimensional systems.
A turbulent state in a distributed chemical reaction system is studied
theoretically. The chaotic behavior here is basically due to the
unstable growth of a spatial inhomogeneity taking place in an
oscillating medium. It is argued that textit{phase turbulence} and
textit{amplitude turbulence} have to be discriminated from each other
according to their distinct origins. Two prototype equations describing
respective types of turbulence are derived by means of some asymptotic
methods, and their solutions turn out toe exhibit successive
bifurcations. It is found that the phase turbulence arises essentially
from the interaction among a few unstable phase modes, while the
amplitude turbulence may well appear in the presence of only one
unstable mode with is a mixed mode of the phase and amplitude. In
particular, some similarity of the amplitude turbulence to the Lorenz
chaos is pointed out. Throughout the present paper, the chaotic behavior
is discucssed in connection with spatial pattern, so that no discrete
approximations, such as a box-model or mode-truncation, are employed.
The paper is concerned with adaptive stabilization of a reaction–diffusion system governed by the Kuramoto–Sivashinsky equation (a non-linear partial differential equation). Under the existence of bounded deterministic disturbances, the adaptive stabilizer is constructed by the concept of high-gain non-linear output feedback and the estimation mechanism of the unknown parameters. In the control system, the global asymptotic stability and convergence of the system state to zero will be guaranteed. The problem of set point regulation is also considered.
An asymptotic nonlinear integrodifferential equation is derived for spontaneous instability of the plane front of a laminar flame.If the combustible mixture is deficient in the light component, spontaneous instability will lead to self-turbulization of the flame, and the flame front assumes a strongly nonstationary cellular structure.If there is an excess of the light component, spontaneous instability produces stationary, irregular wrinkles on the flame front, and the flame continues to propagate in a laminar regime.It is shown that in all cases spontaneous instability of the flame implies an increase in its propagation velocity.
This work focuses on linear finite-dimensional output feedback control of the Kuramoto–Sivashinsky equation (KSE) with periodic boundary conditions. Under the assumption that the linearization of the KSE around the zero solution is controllable and observable, linear finite-dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution, for any value of the instability parameter. The controllers are synthesized on the basis of finite-dimensional approximations of the KSE which are obtained through Galerkin’s method. The performance of the controllers is successfully tested through computer simulations.