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We give a convergence theorem for the difference approximation for the evolution equation ( d / d t ) u ( t ) ∈ A u ( t ) (d/dt)u(t) \in Au(t) and a generation theorem of nonlinear semigroups for "directed" dissipative operators A A in a real Hilbert space.

Problems of the dynamic simulation of controls in parabolic systems involving dissipative operators are considered. Algorithms, stable with respect to information noise and computation errors, which reconstruct the unknown controls for a fairly general class of systems, are described. Examples are presented. A method was proposed previously /1, 2/ for investigating problems of reconstructing the characteristics of dynamic systems, based on ideas of the theory of positional control /3/ and the theory of ill-posed problems /4/ and valid for systems with a finite number of degrees of freedom. This method will now be extended to new classes of systems with distributed parameters.

A distributed-port-Hamiltonian system is a generalized model for passivity-based controls. The system representation has been extended to an infinite-dimensional conservative system derived from variational calculus, which is called a field-port-Lagrangian system. A lot of practical systems for control engineering include dissipative elements; however such a non-conservative structure usually cannot be defined by a variational problem. This paper shows that a system with the dissipative structure can be defined as a time-varying fieldport-Lagrangian system by a gauge transformation. First, we show that the gauge transformation generates a time-dependent Lagrangian density functional that introduces the time-varying port-representation. Next, we present that a class of dissipative systems can be identified with a conservative system possessing an internal irreversible energy flow. Finally, we illustrate an equation of elastic films with viscosity damping with the time-varying port-representation.

We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role of a dissipative element. The coupling between the two -- quantum and classical -- parts of the compound system is treated in the spirit of the mean-field approximation and justification of the validity of such an approach is given. The equations of motion of the classical subsystem are solved explicitly and an effective dissipative Schrodinger equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its application to two basic problems: the decay of the quasi-stationary state and the calculation of the nonlinear resonance line shape., Comment: 7 pages, uses RevTeX style, 7 eps figures embedded

Given a right-continuous Markov process (Xt)t ≥ 0 on a second countable metrizable space E with transition semigroup (pt)t ≥ 0, we prove that there exists a σ-finite Borel measure μ with full support on E, and a closed and densely defined linear operator [Formula: see text] generating (pt)t ≥ 0 on Lp (E; μ). In particular, we solve the corresponding Cauchy problem in Lp (E; μ) for any initial condition [Formula: see text]. Furthermore, for any real β > 0 we show that there exists a generalized Dirichlet form which is associated to (e-βt pt)t ≥ 0. If the β-subprocess of (Xt)t ≥ 0 corresponding to (e-βt pt)t ≥ 0, β > 0, is μ-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for [Formula: see text]. If (Xt)t ≥ 0 is transient, then β can be chosen to be zero.

If X is a non-degenerate derivation on R and H = −X 2 we examine conditions for the closure of H to generate a weakly* continuous semigroup on L ∞ which extends to the L p -spaces. We give an example which cannot be extended and an example which extends but for which the real part of the generator on L 2 is not lower semibounded.

The von Neumann entropy of various quantum dissipative models is calculated in order to discuss the entanglement properties of these systems. First, integrable quantum dissipative models are discussed, i.e., the quantum Brownian motion and the quantum harmonic oscillator. In case of the free particle, the related entanglement of formation shows no non-analyticity. In case of the dissipative harmonic oscillator, there is a non-analyticity at the transition of underdamped to overdamped oscillations. We argue that this might be a general property of dissipative systems. We show that similar features arise in the dissipative two level system and study different regimes using sub-Ohmic, Ohmic and and super-Ohmic baths, within a scaling approach., Comment: 7 pages, 3 figures

Dissipative particle dynamics is a model of multi-phase fluid flows described by a system of stochastic differential equations. We consider the problem of N particles evolving on the one-dimensional periodic domain of length L and, if the density of particles is large, prove geometric convergence to a unique invariant measure. The proof uses minorization and drift arguments, but allows elements of the drift and diffusion matrix to have compact support, in which case hypoellipticity arguments are not directly available.

For a one-dimensional dissipative system with position depending coefficient, two constant of motion are deduce. These constants of motion bring about two Hamiltonians to describe the dynamics of same classical system. However, their quantization describe the dynamics of two completely different quantum systems., 8 pages, no figures, LaTeX format

A space of boundary values is constructed for minimal symmetric singular Sturm-Liouville operator acting in the Hilbert space L-w(2) [a, b), -infinity < a < b

We establish the existence of conservative dilations for various types of dissipative non-commutative N-D systems. As a corollary, the criteria for existence of conservative dilations of corresponding dissipative commutative N-D systems are obtained. We show that such a criterion is fulfilled for all dissipative commutative Givone-Roesser systems, however there are also types of commutative N-D systems for which this criterion is fulfilled not for all dissipative systems. New results announced in this short paper appear without proofs, which we intend to present elsewhere.

The classification of solutions for some dissipative systems by the infor- mation of the spectrum is established. Its generator is non self-adjoint Schrodinger operator with rank one singular perturbation. For the proof, a generalized Parseval formula is constructed.

We show that a dissipative current component is present in the dynamics generated by a Liouville-master equation, in addition to the usual component associated with Hamiltonian evolution. The dissipative component originates from coarse graining in time, implicit in a master equation, and needs to be included to preserve current continuity. We derive an explicit expression for the dissipative current in the context of the Markov approximation. Finally, we illustrate our approach with a simple numerical example, in which a quantum particle is coupled to a harmonic phonon bath and dissipation is described by the Pauli master equation., Comment: To appear in Phys. Rev. Lett

For hard spheres the interaction potential diverges, that is, an infinite force arises when two particles are brought into contact. This causes serious problems when the standard tools of kinetic theory are applied to hard-sphere fluids. The pseudo-Liouville operator was developed to avoid this problem. This chapter illustrates its application to dissipative gases.

The imaginary part of a dissipative operator L is weak if it is pre-sented by a positive operator T such that the square T 2 of it is a product of an operator with a finite trace and an operator from Macaev class. For a dissipative operator with a weak imaginary part the families of incoming and outgoing scattered waves form a non-orthogonal and often even over-complete system {Ψin, Ψout} of eigenfunctions of the corresponding self-adjoint dilation L. The rescription of L in the spectral representation associated with {Ψin, Ψout} gives the Symmetric Functional Model of L, and the characteristic function S of L coincides with the transmission coefficient of the outgoing waves. A general construction based on the self-adjoint delation and an example of the Lax-Phillips Semigroup for the 1-D wave equation on the infinite string with a bounded non-negative potential supported by semi-axis are considered.

Let H be separable Hilbert space with fundamental symmetry J=P+−P− where P± are mutually orthogonal projectors such that J2 is the identity operator. A short proof of the following result is presented in the paper. If A is a maximal dissipative operator in the Krein space K = { H , J } , the domain of A contains P + ( H ) and the operator P+AP− is compact then there is an A-invariant maximal nonnegative subspace ℒ such that the spectrum of the restriction A|ℒ lies in the closed upper-half plane. This theorem contains the well-known results on this subject of Pontrjagin, Langer, Krein and Azizov. As an application of this theorem the criterion for the stability of a rotaing top with a cavity filled with an ideal liquid is given.

The final self-similar state of decaying two-dimensional (2D) turbulence in 2D incompressible viscous flow is analytically and numerically investigated for the case with periodic boundaries. It is proved by theoretical analysis and simulations that the sinh-Poisson state $c\ensuremath{\omega}=\ensuremath{-}\mathrm{sinh}(\ensuremath{\beta}\ensuremath{\psi})$ is not realized in the dynamical system of interest. It is shown by an eigenfunction spectrum analysis that a sufficient explanation for the self-organization to the decaying self-similar state is the faster energy decay of higher eigenmodes and the energy accumulation to the lowest eigenmode for given boundary conditions due to simultaneous normal and inverse cascading by nonlinear mode couplings. The theoretical prediction is demonstrated to be correct by simulations leading to the lowest eigenmode of ${(1,0)+(0,1)}$ of the dissipative operator for the periodic boundaries. It is also clarified that an important process during nonlinear self-organization is an interchange between the dominant operators, which leads to the final decaying self-similar state.

Inventiones mathematicae, 163 (2), ISSN:0020-9910, ISSN:1432-1297

A space of boundary values is constructed for minimal symmetric discrete Dirac operators in the limit-circle case. A description of all maximal dissipative, maximal accretive and self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a self-adjoint dilation of a maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and its characteristic function. Finally, we prove the completeness of the system of eigenvectors and associated vectors of dissipative operators.

The existence of a dissipative flux vector is known to be compatible with reversible processes, provided a timelike conformal Killing vector (CKV) $\chi^\alpha=\frac{V^\alpha}{T}$ (where $V^\alpha$ and $T$ denote the four-velocity and temperature respectively) is admitted by the space-time. Here we show that if a constitutive transport equation, either within the context of standard irreversible thermodynamics or the causal Israel--Stewart theory, is adopted, then such a compatibility also requires vanishing dissipative fluxes. Therefore, in this later case the vanishing of entropy production generated by the existence of such CKV is not actually associated to an imperfect fluid, but to a non-dissipative one. We discuss also about Landau damping., Comment: 3 pages Revtex. To appear in Phys. Lett. A

It is well known that classical Lagrangians do not exist for dissipative systems. The paper presents three formulations of a mathematical Lagrangian for a dissipative system. The first one depends on the generalized coordinate x, its velocity x/spl dot/, the image variable y and its velocity y/spl dot/. This Lagrangian can be obtained directly from differential equations of a dynamic system. For linear reciprocal dissipative systems, the image variable can be substituted by the generalize coordinate with reversed time scales. The second formulation which can resolve the problem for nonlinear dynamical systems depends on kinetic energy, potential energy, and the time integral of a dissipation function. The last formulation which is a time dependent function can be obtained for a special type of linear system.

In general, dissipative problems involve both frictional and fluctuation forces. As it turns out, solutions of many of these problems are encompassed by the generalized quantum Langevin equation. Here, we use this equation to obtain a general expression for the work done by the fluctuation force on a quantum particle. At equilibrium, this work is necessary to balance the energy lost by the particle due to the frictional force.

Based on the solution to the master equation of the density operator describing the amplitude dissipative channel, we derive the time evolution law of the coarse-graining-smoothed Wigner operator in this channel, which demonstrates how an initial pure state evolves into a mixed state, exhibiting decoherence.

The standard {\em system-plus-reservoir} approach used in the study of dissipative systems can be meaningfully generalized to a dissipative coupling involving the momentum, instead of the coordinate: the corresponding equation of motion differs from the Langevin equation, so this is called {\em anomalous} dissipation. It occurs for systems where such coupling can indeed be derived from the physical analysis of the degrees of freedom which can be treated as a dissipation bath. Starting from the influence functional corresponding to anomalous dissipation, it is shown how to derive the effective classical potential that gives the quantum thermal averages for the dissipative system in terms of classical-like calculations; the generalization to many degrees of freedom is given. The formalism is applied to a single particle in a double-well and to the discrete $\phi^4$ model. At variance with the standard case, the fluctuations of the coordinate are enhanced by anomalous dissipative coupling., Comment: 12 pages, 5 figures, to be published in Phys. Rev. E

The class of strongly regular J-inner matrix functions (m.f.’s) U sR (J) was introduced and investigated in the paper [ArD]. In this paper we investigate the Livsic—Brodskii nodes with strongly regular J-inner characteristic m.f.’s of Hardy class. In particular, we prove that the basic operator of such a node is similar to a dissipative operator.

The problem of separation of an invariant subspace of a dissipative operator corresponding to spectral singularities is studied by means of the functional model. The results obtained are applied to Boltzmann operator. The asymptotic behaviour of the semigroup generated by dissipative operator with spectral singularities is discussed.

It is proved that every dissipative-repulsive periodic system admits a periodic solution, which is comparable with some well-known results due to Yoshizawa, Hale and Lopes, and Burton and Zhang for dissipative systems.

In this paper a nonautonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and attracts from $-\infty$. A stochastic weak attractor is constructed in phase space with respect to two metrics and is compact in the lower one.

International audience; The third edition of the now standard Dissipative Systems Analysis and Control presents a revised and expanded treatment of dissipative systems theory, constituting a self-contained, advanced introduction for graduate students, researchers and practising engineers. It examines linear, nonlinear and nonsmooth systems, with many examples in each chapter; occasional infinite-dimensional examples are also included. Throughout, emphasis is placed on the use of the dissipative properties of a system for the design of stable and robust feedback control laws or state observers. The theory is substantiated by experimental results and by reference to its application in illustrative physical cases (Lagrangian systems, passivity-based and adaptive controllers are covered thoroughly).The third edition is substantially updated to accommodate new material within the existing chapter structure. The additions include:the introduction of negative imaginary transfer functions;the design of stable state observers that use passivity as a tool for their stability analysis;a new section on robust set-valued control of uncertain Lagrangian systems;extended section on discrete-time systems, especially the preservation of dissipativity after discretization;a thorough treatment of nonsmooth set-valued Lur'e systems well-posednees and stability;an extended chapter on the Kalman–Yakubovich–Popov Lemma; andover 1000 references.

Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.

In a Hilbert space H with indefinite J-metric, the uniform well-posedness of the Cauchy problem is studied for the equation dx/dt=Ax, where A is a maximal J-dissipative operator. In particular, in the case of a Pontryagin space an analog of Phillips's theorem on the relation between the maximal dissipative operator A and the uniform well-posedness of the Cauchy problem is obtained.

The present work is devoted to the study of travelling wave solutions of a convective dissipative system, the so-called (k, e) system modelling turbulent compressible flows. This system exhibits non conservative products for both the first order and second order operators. A particular attention is payed to the laminar and turbulent viscosities which ratio dictates the shape of the second order operator. In that aim, we introduce an equivalent system involving nonconservative products that are weighted by this ratio with the additional benefit of a second order operator in conservation form. Two limit cases are distinguished since they yield convective-dissipative systems in full conservation form.

We study trajectory attractors of autonomous dissipative hyperbolic equations depending on a small parameter e ≥ 0. We assume that the interaction function of perturbed equation (with e > 0) can be of arbitrary polynomial growth so that the corresponding Cauchy problem can have a nonunique solution. We prove that the trajectory attractors A e are upper semicontinuous as e → 0+ in an appropriate topological space. Two different perturbation problems are considered.

In a unitary approach generally used in quantum optics, we consider Lindblad's Markovian master equation and the non- Markovian master equation of Ford, Lewis and O'Connell. We show that the second-order master equation in the hierarchy obtained from a Krylov-Bogoliubov expansion corresponds to the Born approximation. By time averaging, and neglecting the rapidly varying terms, Lindblad's master equation is obtained. With these two equations, we calculate the decay spectrum. We find that for rather low dissipated energies, only the non-Markovian master equation provides correct results. Based on the independent oscillator model of the dissipative coupling, explicit expressions of the dissipative coefficients are obtained.

The Bak-Sneppen model of self-organized biological evolution of an infinite ecosystem of randomly interacting species is represented in terms of an infinite set of variables which can be considered as an analog to the set of integrals of motion of completely integrable system. Each of this variables remains to be constant but its influence on the evolution process is restricted in time and after definite moment its value is excluded from description of the system dynamics., Comment: LaTeX, 7 pages

We consider an initial and boundary value problem the one dimensional wave equation with damping concentrated at an interior point. Our main results assert that the decay rate is uniform for regular initial data and give lower estimates of the decay rate. An essential intermediate step is the description of the spectrum of the associate dissipative operator.

A method for the numerical solution of the dissipative Liouville–von Neumann equation is presented. The reduced density operator is expanded in a basis of time-dependent wave functions. This guarantees that the size of the basis required in the representation of the density operator is minimal. Equations of motion for the expansion wave functions and the density matrix elements are obtained from the Dirac–Frenkel variational principle. The numerical effort of the method scales proportional to N, where N is the dimension of the relevant Hilbert space. As a first example, the dynamics of a three-mode system with vibronic coupling (with N≈50 000) coupled to a dissipative bath has been studied, modeling the S1/S2 states of pyrazine. For the cw-absorption spectrum, fast convergence with respect to the number of expansion wave functions has been obtained.

The dissipative scalar field theory by means of the deformation quantization formalism is studied. Following the ideas presented by G. Dito and F. J. Turrubiates [Phys. Lett. A352, 309 (2006)] for quantum mechanics, a star product which contains the dissipative effect for the damped oscillation modes of the field is constructed. Employing this approach the expectation values of some observables in the quantum mechanical case as well as certain correlation functions for the field case are obtained under a particular dissipative process.

Dissipative systems do not have a natural space in the variational formulation of mechanics. We introduce a change of variables such that in the transformed frame the dissipative Newtonian equations mimic those for conservative systems. We present solutions of (a) a linearly damped harmonic oscillator, (b) the corresponding quadratically damped system, (c) a modified Emden-type equation and (d) a generalized Emden equation using their first integrals and deal with the corresponding inverse variational problem to derive Lagrangian and Hamiltonian representations. We confirm that, as opposed to the original damped equations, the reduced equations can be solved by the use of Hamilton–Jacobi theory.