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Concept# Stability theory

Summary

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalue

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In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical mode

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability

Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or a

Related courses (21)

Related units (2)

ME-523: Nonlinear Control Systems

Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Divers exemples illustrent la théorie (exercices papier crayon et simulations à l'ordinateur).

MATH-301: Ordinary differential equations

Le cours donne une introduction à la théorie des EDO, y compris existence de solutions locales/globales, comportement asymptotique, étude de la stabilité de points stationnaires et applications, en particulier aux systèmes dynamiques et en biologie.

MATH-459: Numerical methods for conservation laws

Introduction to the development, analysis, and application of computational methods for solving conservation laws with an emphasis on finite volume, limiter based schemes, high-order essentially non-oscillatory schemes, and discontinuous Galerkin methods.

This thesis is a study of wave maps from a curved background to the standard two dimensional sphere S2. The target is always assumed to be embedded in R3 in the standard way. The do- main manifold (the "curved background") will be diffeomorphic to S2 × R, but the Lorentzian metric will not necessarily be the standard one. The current work is based on the following two results by Shatah, Tahvildar-Zadeh [47] and Krieger, Schlag, Tataru [25]. In [47] Shatah and Tahvildar-Zadeh established the existence and stability in the energy norm of a compact family of stationary (satisfying a certain periodicity condition in time) and equiv- ariant (satisfying a certain spatial symmetry) wave maps from S2 × R to S2 equipped with the standard metrics. Stability was proved under perturbations in the same spatial equivariance class. In [25], the authors construct a one parameter family of blow up solutions to the co-rotational wave maps equation from R2+1 to S2 with the standard metrics. Co-rotational means that the wave maps are assumed to have rotation number equal to one. These solutions are obtained as perturbations of the rescaled standard rotational harmonic map 2arctanr from R2 to S2. Here, we prove the conclusions of these papers for different metrics on the domains of wave maps. More specifically, our results can be grouped into two parts: stationary maps and blow ups. In the first part following the method of [25], we construct a one parameter family of co-rotational blow up wave maps from S2 ×R to S2. These solutions can be parameterized by ν ∈ ( 1 , 1]. The metric on the domain sphere is taken to be the standard one. 2 In the second part, we consider wave maps from S2 × R to S2 where the metric on the do- main S2 is now taken to be a general SO(1)−symmetric metric of the form dr2 + f 2(r)dθ2. The special case f (r ) = sin r corresponds to the standard metric considered by Shatah and Tahvildar-Zadeh. We prove the same stability result as in [47] in this general setting.

We exhibit non-equivariant perturbations of the blowup solutions constructed in [18] for energy critical wave maps into $\mathbb{S}^2$. Our admissible class of perturbations is an open set in some sufficiently smooth topology and vanishes near the light cone. We show that the blowup solutions from [18] are rigid under such perturbations, including the space-time location of blowup. As blowup is approached, the dynamics agrees with the classification obtained in [7], and all six symmetry parameters converge to limiting values. Compared to the previous work [16] in which the rigidity of the blowup solutions from [18] under equivariant perturbations was proved, the class of perturbations considered in the present work does not impose any symmetry restrictions. Separation of variables and decomposing into angular Fourier modes leads to an infinite system of coupled nonlinear equations, which we solve for small admissible data. The nonlinear analysis is based on the distorted Fourier transform, associated with an infinite family of Bessel type Schrödinger operators on the half-line indexed by the angular momentum $n$. A semi-classical WKB-type spectral analysis relative the parameter $\hbar=\frac{1}{n+1}$ for large $|n|$ allows us to effectively determine the distorted Fourier basis for the entire infinite family. Our linear analysis is based on the global Liouville-Green transform as in the earlier work [4, 5].

2020Related lectures (39)

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Discretization methods such as finite differences or finite elements were usually employed to provide high fidelity solution approximations for reduced order modeling of parameterized partial differential equations. In this paper, a novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a new fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the effectiveness and accuracy of IGA-based POD techniques for the model order reduction of the acoustic wave equation.