Dynamics of Quasi-Stable Dissipative Systems

Čhuešhov, Igor

doi:10.1007/978-3-319-22903-4

Cited by 171 publications

(236 citation statements)

References 158 publications

(526 reference statements)

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“…The existence and properties of the attractor for panel configurations are shown in [25] (though earlier proofs have certainly appeared for these configurations). This work also discusses the application of a recent tool: the theory of quasi-stability [9,12]. The theory provides the existence of so called exponential attractors [9,12].…”

Section: Long-time Behavior Of Trajectoriesmentioning

confidence: 99%

“…This work also discusses the application of a recent tool: the theory of quasi-stability [9,12]. The theory provides the existence of so called exponential attractors [9,12]. A generalized fractal exponential attractor for the dynamics (S(t), H) is a forward invariant compact set A exp ⊂ H, with finite fractal dimension (possibly in a weaker topology), that attracts bounded sets with uniform exponential rate.…”

Section: Long-time Behavior Of Trajectoriesmentioning

confidence: 99%

“…However, to study the qualitative properties of the dynamics in the post-flutter regime, analysis will require some physical nonlinear restoring force to keep trajectories bounded in time [16,14,10]. Indeed, for the post-flutter dynamics of a given system, the theory of nonlinear dynamical systems is often used, e.g., compact global attractors [12,10] as well as exponential attractors [25,9,12].…”

Section: Introductionmentioning

confidence: 99%

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A thorough look at the (in)stability of piston-theoretic beams

Howell¹,

Huneycutt²,

Webster³

et al. 2019

Mathematics in Engineering

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We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated dynamical system for solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses are provided, focusing both at the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.

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Section: Long-time Behavior Of Trajectoriesmentioning

confidence: 99%

Section: Long-time Behavior Of Trajectoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A thorough look at the (in)stability of piston-theoretic beams

Howell¹,

Huneycutt²,

Webster³

et al. 2019

Mathematics in Engineering

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“…The theory of quasi-stable dynamical systems has been developed rather thoroughly in recent years [8,10]. This includes more general definitions of quasi-stable dynamical systems [8] than what we present below. For ease of exposition and application in our analysis we focus on this more narrow definition.…”

Section: Quasi-stability and Practical Applicationsmentioning

confidence: 99%

Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping

Howell¹,

Lasiecka²,

Webster³

2016

EECT

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We consider a nonlinear (Berger or Von Karman) clamped plate model with a piston-theoretic right hand side-which include non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of strong tools such as backward-in-time smallness of velocities and finiteness of the dissipation integral. Modern quasi-stability techniques are utilized to show the existence of compact global attractors and generalized fractal exponential attractors. Specific results depending on the size of the damping parameter and the nonlinearity in force. For the Berger plate, in the presence of large damping, the existence of a proper global attractor (whose fractal dimension is finite in the state space) is shown via a decomposition of the nonlinear dynamics. This leads to the construction of a compact set upon which quasi-stability theory can be implemented. Numerical investigations for appropriate 1-D models are presented which explore and support the abstract results presented herein.2 these two terms is the key driver of interest in this plate problem (and in the general flutter phenomenon). Such a non-dissipative, piston-theoretic model has been intensively studied in for different boundary conditions and resistive damping forces, see [5,9,10].In line with the aeroelasticity literature [16,17] (and references therein), we primarily focus on the theory of large deflections for the plate. Taking the boundary to be clamped (as we do) or hinged, von Karman theory is considered accurate [15,16,24]. Moreover, as we consider thin plates, we do not consider rotational inertia effects [24], nor in plane accelerations [25]. In addition, by making an ad hoc simplification to von Karman's scalar plate model, one arrives at a "friendlier" plate model, known as the Berger equation [3, 33, 39] 2 . Other work on flutter dynamics have considered Berger-like beam and plate models (see [9] and references therein), as well as the scalar von Karman equations [10,13,14,38]. Physically, when the plate has no portion of its boundary free, the Berger approximation of von Karman's dynamics is taken to be valid [20,34,37].Remark 1.1. In 1-D, the appropriate model (reflecting an appropriate singular limit of the full von Karman dynamics [25] as in-plane accelerations vanish) is the 1-D analog of the Berger model presented here (sometimes also referred to as a Timoshenko beam [32], or Krieger beam [30]). In Section 7 we consider some numerical results for this 1-D Berger beam model.Many of the results herein are simultaneously valid for both the von Karman and Berger plate dynamics. However, the principal results presented below in Section 1.4.1 on the existence and ...

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“…In existence theory they provide bounds on the transients of solutions and so in some situations conditions for forward completeness of trajectories, [1]. A criterion for the existence of a bounded absorbing ball is formulated in [8,Theorem 2.1.2]. In the finite-dimensional case they have also been used to investigate the geometric structure of the general solution of differential equations [41] and to analyze coordinate-free notions of growth rates [15].…”

Section: Introductionmentioning

confidence: 99%

Non-coercive Lyapunov functions for infinite-dimensional systems

Mironchenko

Wirth

2019

Journal of Differential Equations

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We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.

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Dynamics of Quasi-Stable Dissipative Systems

Cited by 171 publications

References 158 publications

A thorough look at the (in)stability of piston-theoretic beams

A thorough look at the (in)stability of piston-theoretic beams

Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping

Non-coercive Lyapunov functions for infinite-dimensional systems

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