Fredholm Transform and Local Rapid Stabilization for a 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

Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation

We analyse the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretisations modeling its dynamics on coarse grids. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretisations excellently reproduce the steady states and the dynamics of the Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as an example to show how holistic discretisation may be successfully applied to fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.Comment: Without figures. See http://www.sci.usq.edu.au/staff/aroberts/ksdoc.pdf to download a version with the figure

Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.Comment: 31 page

Control theory for infinite dimensional dynamical systems and applications to falling liquid film flows

In this thesis, we study the problem of controlling the solutions of various nonlinear PDE models that describe the evolution of the free interface in thin liquid films flowing down inclined planes. We propose a control methodology based on linear feedback controls, which are proportional to the deviation between the current state of the system and a prescribed desired state. We first derive the controls for weakly nonlinear models such as the Kuramoto-Sivashinsky equation and some of its generalisations, and then use the insight that the analytical results obtained there provide us to derive suitable generalisations of the controls for reduced-order long-wave models. We use two long-wave models to test our controls: the first order Benney equation and the first order weighted-residual model, and compare some linear stability results with the full 2-D Navier-Stokes equations. We find that using point actuated controls it is possible to stabilise the full range of solutions to the generalised Kuramoto-Sivashinsky equation, and that distributed controls have a similar effect on both long-wave models. Furthermore, point-actuated controls are efficient when stabilising the flat solution of both long-wave models. We extend our results to systems of coupled Kuramoto-Sivashinsky equations and to stochastic partial differential equations that arise by adding noise to the weakly nonlinear models.Open Acces

Additive noise effects in active nonlinear spatially extended systems

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multiscale theory for such systems and apply it to the Kuramoto-Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset, so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case

Stabilizing non-trivial solutions of the generalized Kuramoto-Sivashinsky equation using feedback and optimal control

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