Linearized Kuramoto Sivashinsky Research Articles

Sampled-data observer design for Linear Kuramoto–Sivashinsky systems with non-local output

The aim of this paper is to provide a novel systematic methodology for the design of sampled-data observers for Linear Kuramoto–Sivashinsky systems (LKS) with non-local outputs. More precisely, we extend the systematic sampled-data observer design approach which is based on the use of an Inter-Sample output predictor to the class of LKS systems. By using a small-gain methodology we provide sufficient conditions ensuring the Input-to-Output Stability property of the estimation errors in the presence of measurement noise. Our Inter-Sample output predictor contains a tuning term which can enlarge significantly the Maximum Allowable Sampling Period.

Temporal fractal nature of linearized Kuramoto–Sivashinsky SPDEs and their gradient in one-to-three dimensions

Temporal fractal nature of linearized Kuramoto–Sivashinsky SPDEs and their gradient in one-to-three dimensions

Optimal convergence for time-dependent linearized Kuramoto–Sivashinsky type problems: A new approach

We continue the convergence analysis approach initiated in Fishelov and Croisille (2021). Here we consider the case of linear versions of Kuramoto–Sivashinsky type problems of the form ut+uxxxx+duxx+au=f in one dimension and ut+Δ2u+dΔu+au=f in two dimensions. We suggest compact schemes for their approximations. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence of the discrete in space, continuous in time schemes is actually four, thus the error tends to zero as O(h4), where h is the size of the mesh. Numerical results shown for the Kuramoto–Sivashinsky equations in 1D and in 2D assess the fourth-order accuracy of the scheme.

Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed $L^2$-Gain

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the one dimensional (1D) heat equation, under the assumption that one of the observation or control operators is bounded. This article suggests a constructive method for such controllers for 1D parabolic partial differential equations (PDEs) with both observation and control operators being unbounded. We consider the Kuramoto–Sivashinsky equation under either boundary or in-domain point measurement and boundary actuation in the presence of disturbances in the PDE and measurement. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm–Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> may be larger. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in a linear matrix inequality (LMI), for input-to-state stabilization (ISS) and guaranteed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> -gain, whose elements and dimension depend on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . The value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> implies its feasibility for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N+1$</tex-math></inline-formula> . Numerical examples demonstrate the efficiency of the method.

Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients

Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients

Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable.

Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.

Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation

The Kuramoto-Sivashinsky equation is a fourth-order partial differential equation used as a model for physical phenomena such as plane flame propagation and phase of turbulence. The inverse problem of recovering the second-order coefficient from the knowledge of the solution in final time, for the linear version of the equation, is studied in this article. The inverse problem is formulated as a regularized nonlinear optimization problem, from which the local uniqueness and the stability are proved. Finally, an algorithm for the reconstruction of the coefficient is proposed and several numerical simulations are presented.

Actuator and sensor placement for closed-loop control of convective instabilities

This work deals with the characterization of the closed-loop control performance aiming at the delay of transition. We focus on convective wavepackets, typical of the initial stages of transition to turbulence, starting with the linearized Kuramoto–Sivashinsky equation as a model problem representative of the transitional 2D boundary layer; its simplified structure and reduced order provide a manageable framework for the study of fundamental concepts involving the control of linear wavepackets. The characterization is then extended to the 2D Blasius boundary layer. The objective of this study is to explore how the sensor–actuator placement affects the optimal control problem, formulated using linear quadratic Gaussian (LQG) regulators. This is carried out by evaluating errors of the optimal estimator at positions where control gains are significant, through a proposed metric, labelled as $$\gamma $$ . Results show, in quantitative manner, why some choices of sensor–actuator placement are more effective than others for flow control: good (respectively, bad) closed-loop performance is obtained when estimation errors are low (respectively, high) in the regions with significant gains in the full-state-feedback problem. Unsatisfactory performance is further understood as dominant estimation error modes that overlap spatially with control gains, which shows directions for improvement of a given set-up by moving sensors or actuators. The proposed metric and analysis explain most trends in closed-loop performance as a function of sensor and actuator position, obtained for the model problem and for the 2D Blasius boundary layer. The spatial characterization of the $$\gamma $$ -metric provides thus a valuable and intuitive tool for the problem of sensor–actuator placement, targeting here transition delay but possibly extending to other amplifier-type flows.

Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control

Abstract 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 use of a Lyapunov function, we prove that the same feedback law exponential stabilize the original unstable infinite-dimensional system.

Simulations of Co-GISAXS during kinetic roughening of growth surfaces.

The recent development of surface growth studies using X-ray photon correlation spectroscopy in a grazing-incidence small-angle X-ray scattering (Co-GISAXS) geometry enables the investigation of dynamical processes during kinetic roughening in greater detail than was previously possible. In order to investigate the Co-GISAXS behavior expected from existing growth models, calculations and (2+1)-dimension simulations of linear Kuramoto-Sivashinsky and non-linear Kardar-Parisi-Zhang surface growth equations are presented which analyze the temporal correlation functions of the height-height structure factor. Calculations of the GISAXS intensity auto-correlation functions are also performed within the Born/distorted-wave Born approximation for comparison with the scaling behavior of the height-height structure factor and its correlation functions.

Local controllability of the stabilized Kuramoto–Sivashinsky system by a single control acting on the heat equation

In this paper we consider the so called stabilized Kuramoto–Sivashinsky system which couples a fourth order and a second order parabolic equations. We prove that this system is locally controllable to the trajectories by a single distributed control acting only on the heat equation. The main novelty is a new Carleman inequality for the solutions of a linear Kuramoto–Sivashinsky equation with nonhomogeneous boundary conditions.

Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

A new global Carleman estimate for the one-dimensional Kuramoto–Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem

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.

Adaptive and Model-Based Control Theory Applied to Convectively Unstable Flows

Research on active control for the delay of laminar–turbulent transition in boundary layers has made a significant progress in the last two decades, but the employed strategies have been many and dispersed. Using one framework, we review model-based techniques, such as linear-quadratic regulators, and model-free adaptive methods, such as least-mean square filters. The former are supported by an elegant and powerful theoretical basis, whereas the latter may provide a more practical approach in the presence of complex disturbance environments that are difficult to model. We compare the methods with a particular focus on efficiency, practicability and robustness to uncertainties. Each step is exemplified on the one-dimensional linearized Kuramoto–Sivashinsky equation, which shows many similarities with the initial linear stages of the transition process of the flow over a flat plate. Also, the source code for the examples is provided.

From Brownian-Time Brownian Sheet to a Fourth Order and a Kuramoto–Sivashinsky-Variant Interacting PDEs Systems

We introduce n-parameter ℝ d -valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each “time” parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of n linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) connection to fourth order linear PDEs. Our BTBS does not belong to the classical theory of random fields; and to prove our new PDEs connections, we generalize our BTP approach in [4, 5] and we mix it with the Brownian sheet connection to a linear PDE system, which we also give along with its corresponding nonlinear second order PDE and a 2nth order linear PDE that we also connect to Brownian sheet. In addition, we introduce the n-parameter d-dimensional linear Kuramoto–Sivashinsky (KS) sheet kernel (or “transition density”); and we link it to an intimately connected system of new linear Kuramoto–Sivashinsky-variant interacting PDEs, generalizing our earlier one parameter imaginary-Brownian-time-Brownian-angle kernel and its connection to a linear KS PDE. The interactions here mean that our PDEs systems are to be solved for a family of functions, a feature shared with some well-known fluids dynamics models. The interacting PDEs connections established here open up another new fundamental front in the rapidly growing field of iterated-type processes and their connections to both new and important higher order PDEs and to some equivalent fractional Cauchy problems. We connect the BTBS fourth order interacting PDEs system given here with an interacting fractional PDE system and further study it in another article.

Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation

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.

A BROWNIAN-TIME EXCURSION INTO FOURTH-ORDER PDES, LINEARIZED KURAMOTO–SIVASHINSKY, AND BTP-SPDES ON ℝ+ × ℝd

In recent articles we have introduced the class of Brownian-time processes (BTPs) and the Linearized Kuramoto–Sivashinsky process (LKSP). Probabilistically, BTPs represent a unifying class for some different exciting processes like the iterated Brownian motion (IBM) of Burdzy (a process with fourth-order properties) and the Brownian–snake of Le Gall (a second-order process); they also include many additional new and quite interesting processes. The LKSP is closely connected to the Kuramoto–Sivashinsky PDEs, one of the most celebrated PDEs in modern applied mathematics. We start by surveying the fourth-order PDE connections to BTPs and the LKSP that we uncovered in two recent articles. In the second part of this paper we introduce BTP-SPDEs, these are SPDEs in which the PDE part is that solved by running a BTP. We consider a BTP-SPDE driven by an additive spacetime white noise on the time-space set ℝ+ × ℝd; and we prove the existence of a unique real-valued, Lp(Ω,ℙ) for all p ≥ 1, BTP solution to such BTP-SPDEs for 1 ≤ d ≤ 3. This contrasts sharply with the standard theory of reaction-diffusion type SPDEs driven by spacetime white noise, in which real-valued solutions are confined to one spatial dimension. Like the PDEs case, BTP-SPDEs also provide a valuable insight into other fourth-order SPDEs of applied science. We carry out such a program in forthcoming articles.

Absolute and Convective Instability for Evolution PDEs on the Half‐Line

We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let ω(k) be the associated symbol, i.e., let exp[ikx−ω(k)t] be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition q0(x), where q0(x) decays as |x| → ∞. By making use of a certain transformation in the complex k‐plane, which leaves ω(k) invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half‐line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial‐boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second‐order equation is a linearized Ginzburg–Landau equation arising in Rayleigh–Bénard convection and in the stability of plane Poiseuille flow, while the fourth‐order equation is a linearized Kuramoto–Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.