Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

Funnel control for a moving water tank

We study tracking control for a moving water tank system, which is modelled using the Saint-Venant equations. The output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is to achieve that the tracking error evolves within a prespecified performance funnel. Exploiting recent results in funnel control we show that it suffices to show that the operator associated with the internal dynamics of the system is causal, locally Lipschitz continuous and maps bounded functions to bounded functions. To show these properties we consider the linearized Saint-Venant equations in an abstract framework and show that it corresponds to a regular well-posed linear system, where the inverse Laplace transform of the transfer function defines a measure with bounded total variation.Comment: 11 page

Rapid Convergence of Solution for Hybrid System with Causal Operators

We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of order k≥2

The Cauchy Problem for the Einstein Equations

Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced systems of equations which are hyperbolic is examined in detail and some new results on that subject are presented. Relevant background from the theory of partial differential equations is also explained at some lengthComment: 98 page

Strict and non-strict inequalities for implicit first order causal differential equations

In this paper, some fundamental differential inequalities for the implicit perturbations of nonlinear first order ordinary causal differential equations have been established

Quantum noise in optical fibers I: stochastic equations

We analyze the quantum dynamics of radiation propagating in a single mode optical fiber with dispersion, nonlinearity, and Raman coupling to thermal phonons. We start from a fundamental Hamiltonian that includes the principal known nonlinear effects and quantum noise sources, including linear gain and loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are treated. This allows quantum Langevin equations to be calculated, which have a classical form except for additional quantum noise terms. In practical calculations, it is more useful to transform to Wigner or +$P$ quasi-probability operator representations. These result in stochastic equations that can be analyzed using perturbation theory or exact numerical techniques. The results have applications to fiber optics communications, networking, and sensor technology.Comment: 1 figur

Field-theoretic methods

Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that facilitates the computation of such correlation functions and moments, and furthermore allows a systematic development of perturbation expansions and other useful approximative schemes. It is explained how nonlinear stochastic processes may be mapped onto exponential probability distributions, whose weights are determined by continuum field theory actions. Such mappings are madeexplicit for (1) stochastic interacting particle systems whose kinetics is defined through a microscopic master equation; and (2) nonlinear Langevin stochastic differential equations which provide a mesoscopic description wherein a separation of time scales between the relevant degrees of freedom and background statistical noise is assumed. Several well-studied examples are introduced to illustrate the general methodology.Comment: Article for the Encyclopedia of Complexity and System Science, B. Meyers (Ed.), Springer-Verlag Berlin, 200