Approximation of Lyapunov Functions from Noisy Data

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm

Stability and Optimal Control of Switching PDE-Dynamical Systems

Selected results for the stability and optimal control of abstract switched systems in Banach and Hilbert space are reviewed. The dynamics are typically given in a piecewise sense by a family of nonlinearly perturbed evolutions of strongly continuous semigroups. Stability refers to characterizations of asymptotic decay of solutions that holds uniformly for certain classes of switching signals for time going to infinity. Optimal control refers to the minimization of costs associated to solutions by appropriately selecting switching signals. Selected numerical results verify and visualize some of the available theory

Estimates for principal Lyapunov exponents: A survey

This is a survey of known results on estimating the principal Lyapunov exponent of a time-dependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further research are hinted.Comment: 33 page

Spatial discretization error in Kalman filtering for discrete-time infinite dimensional systems

We derive a reduced-order state estimator for discrete-time infinite dimensional linear systems with finite dimensional Gaussian input and output noise. This state estimator is the optimal one-step estimate that takes values in a fixed finite dimensional subspace of the system's state space --- consider, for example, a Finite Element space. We then derive a Riccati difference equation for the error covariance and use sensitivity analysis to obtain a bound for the error of the state estimate due to the state space discretization

On nonlinear stabilization of linearly unstable maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For G\^ateaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fr\'echet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fr\'echet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.Comment: New section 1.5 and several references added. 20 pages, no figur

Long time dynamics and coherent states in nonlinear wave equations

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to appear in Fields Institute Communications: Advances in Applied Mathematics, Modeling, and Computational Science 201

Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit

In this paper we consider a discrete Klein-Gordon (dKG) equation on \ZZ^d in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength \eps. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order \mathcal{O}(\exp(\eps^{-1})), is also obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.Comment: 20 page

Bregman Distances in Inverse Problems and Partial Differential Equation

The aim of this paper is to provide an overview of recent development related to Bregman distances outside its native areas of optimization and statistics. We discuss approaches in inverse problems and image processing based on Bregman distances, which have evolved to a standard tool in these fields in the last decade. Moreover, we discuss related issues in the analysis and numerical analysis of nonlinear partial differential equations with a variational structure. For such problems Bregman distances appear to be of similar importance, but are currently used only in a quite hidden fashion. We try to work out explicitely the aspects related to Bregman distances, which also lead to novel mathematical questions and may also stimulate further research in these areas

Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling

We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly.Comment: 41 page

Attractors for gradient flows of non convex functionals and applications

This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor.Comment: 46 page