Related papers: Bounds on mean energy in the Kuramoto-Sivashinsky …
We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be…
We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion…
We investigate a fully discrete finite element approximation for the stochastic Kuramoto-Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler-Maruyama scheme in time. Rigorous…
We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees…
We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under the assumption that one of the observation or control operators is bounded.…
This paper investigates the approximation of stochastic delay differential equations (SDDEs) via the backward Euler-Maruyama (BEM) method under generalized monotonicity and Khasminskii-type conditions in the infinite horizon. First, by…
The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $\nu=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits…
We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be…
Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on the Lyapunov dimension, among all trajectories of a dynamical system governed by ordinary differential equations. The first approach…
Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized…
In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming (LP) techniques of…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…
This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations…
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled…
A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…