Finite dimensional attractor for a composite system of wave/plate equations with localised damping

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.Comment: 40 pages, 1 figure; v2: added references for Section 1, submitte

Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $\mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically \cite {Chu2013-comp}---the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup $\left\{ e^{ \mathcal{A}t}\right\} _{t\geq 0}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator $\mathcal{A}$ which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $\mathbf{U}\in \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.Comment: 1 figur

Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system

This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k → ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well

Boundary local null-controllability of the Kuramoto-Sivashinsky equation

International audienceWe prove that the Kuramoto-Sivashinsky equation is locally controllable in 1D and in 2D with one boundary control. Our method consists in combining several general results in order to reduce the null-controllability of this nonlinear parabolic equation to the exact controllability of a linear beam or plate system. This improves known results on the controllability of Kuramoto-Sivashinsky equation and gives a general strategy to handle the null-controllability of nonlinear parabolic systems

Observation and control of a ball on a tilting

The ball and plate system is a nonlinear MIMO system that has interesting characteristics which are also present in aerospace and industrial systems, such as: instability, subactuation, nonlinearities such as friction, backlash, and delays in the measurements. In this work, the modeling of the system is based on the Lagrange approach. Then it is represented in the state-space form with plate accelerations as inputs to the system. These have a similar effect as applying torques. In addition, the use of an internal loop of the servo system is considered. From the obtained model, we proceed to carry out the analysis of controllability and observability resulting in that the system is globally weak observable and locally controllable in the operating range. Then, the Jacobi linearization is performed to use the linearized model in the design of linear controllers for stabilization. On the other hand, analyzing the internal dynamics of the ball and plate system turns out to be a non-minimum phase system, which makes it difficult to design the tracking control using the exact model. This is the reason why we proceed to make approximations. Using the approximate model, nonlinear controllers are designed for tracking using different approaches as: feedback linearization for tracking with and without integral action, backstepping and sliding mode. In addition, linear and nonlinear observers are designed to provide full state information to the controller. Simulation tests are performed comparing the different control and observation approaches. Moreover, the effect of the delay in the measurement is analyzed, where it is seen that the greater the frequency of the reference signal the more the error is increased. Then, adding the Smith predictor compensates the delay and reduces the tracking error. Finally, tests performed with the real system. The system was successfully controlled for stabilization and tracking using the designed controllers. However, it is noticed that the effect of the friction, the spring oscillation and other non-modeled characteristics significantly affect the performance of the control.Tesi

Modal Analysis of Fluid Flows: An Overview

Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades. For a non-expert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis in a presentation that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis