The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The mo...

This course offers an overview of the role of information, flows, topology, and synchronization in complex networks. In the first part of the course the students will learn the theory behind these concepts in general terms. In particular, it will be introduced concepts from Ergodic theory, Dynamical Systems, Complex Networks, Theory of information, Networks Flow Theory, and Synchronization Theo...

for each x, y ∈ C. For a mapping T of C into itself, we denote by F(T) the set of fixed points of T . We also denote by N and R+ the set of positive integers and nonnegative real numbers, respectively. Baillon [1] proved the first nonlinear ergodic theorem. Let C be a nonempty bounded convex closed subset of a Hilbert spaceH and let T be a nonexpansive mapping of C into itself. Then, for an arb...

The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most rest...

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-...

Abstract The nonlinear filter associated with the discrete time signal-observation model (Xk,Yk) is known to forget its initial condition as k →∞ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condit...

Linear-quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed. Typically, such systems exhibit “complicated dynamics,” i.e., have nontrivial recurrence. The controllers, which use small perturbations of the nominal dynamics as input actuators, are based on modeling the tracking error as a linear dynamically ...

This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals Fε stored in the deformation of an ε-scaling of a stochastic lattice Γ-converge to a continuous energy functional when ε goes to zero. In particular, the limiting energy functional is of integral type, an...

Abstract. The nonlinear filter associated with the discrete time signal-observation model (Xk, Yk) is known to forget its initial condition as k → ∞ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial con...

Nonlinear Analysis B: in press, 2004. The Swift-Hohenberg fluid convection system with both local and nonlocal nonlinearities under the influence of white noise is studied. The objective is to understand the difference in the dynamical behavior in both local and nonlocal cases. It is proved that when sufficiently many of its Fourier modes are forced, the system has a unique invariant measure, o...