Quadratic Dissipation Research Articles

Minimum principles for the trajectories of systems governed by rate problems

Recently, Mielke and Ortiz [2007. A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var., in press] have proposed a variational reformulation of evolutionary problems that characterizes entire trajectories of a system as minimizers of certain energy–dissipation functionals. In this paper we present two examples of energy–dissipation functionals for which relaxations and optimal scalings can be rigorously derived. The first example concerns a one-dimensional bar characterized by a quadratic dissipation function and a bistable energy density; the second example concerns the coarsening kinetics of island growth in thin films exhibiting a preferred slope. In both cases, we present closed-form relaxations in the local limit of the problem and optimal scaling relations for the nonlocal problems. The relaxations rigorously characterize macroscopic properties of complex microstructural evolution by means of well-posed effective problems. The scaling relations rigorously characterize some average properties of the coarsening kinetics of the systems and lead to predictions on the growth exponents.

Tidal interactions of close-in extrasolar planets: The OGLE cases

Close-in extrasolar planets experience extreme tidal interactions with their host stars. This may lead to a reduction of the planetary radius and a spin-up of stellar rotation. Tidal interactions have been computed for a number of extrasolar planets in circular orbits within 0.06 AU, namely for OGLE-TR-56 b. We compare our range of the tidal dissipation value with two dissipation models from Sasselov (2003) and conclude that our choices are equivalent to these models. However, applied to the planet OGLE-TR-56 b, we find in contrast to Sasselov (2003) that this planet will spiral-in toward the host star in a few billion years. We show that the average and maximum value of our range of dissipation are equivalent to the linear and quadratic dissipation models of Sasselov (2003). Due to limitations in the observational techniques, we do not see a possibility to distinguish between the two dissipation models as outlined by Sasselov (2003). OGLE-TR-56 b may therefore not be well suited to serve as a test case for dissipation models. The probable existence of OGLE-TR-3 b at 0.02 AU and the discovery of OGLE-TR-113 b at 0.023 AU and OGLE-TR-132 b at 0.03 AU may also counter Sasselovs (2003) assumption of a pile-up stopping boundary at 0.04 AU.

Thermodynamics of rate-independent plasticity

We show that the singular dissipative potential of the phenomenological rate-independent plasticity can be obtained by homogenization of a micro-model with quadratic dissipation. The essential ingredient making this reduction possible is a rugged energy landscape at the micro-scale, generating under external loading a regular cascade of subcritical bifurcations. Such landscape may appear as a result of a sufficiently strong pinning or jamming of defects, leading to elastic micro-metastability. The rate-independent plastic deformation emerges in this description as a continuous succession of infinitesimal viscous events; the limiting procedure presumes the elimination of small time and length scales. We present an explicit example of a simple viscoelastic mass-spring system whose macroscopic dissipative behavior is plastic, rate independent.

Quantum Bouncer with Dissipation

Effects on the spectra of the quantum bouncer due to dissipation are given when a linear of quadratic dissipation is taken into account. Classical constants of motions and Hamiltonians are deduced for these systems and their quantized eigenvalues are estimated through perturbation theory. Differences were found comparing the eigenvalues of these two quantities.

Global stability and quadratic Hamiltonian structure in Lotka–Volterra and quasi-polynomial systems

We show that the global stability of quasi-polynomial (QP) and Lotka–Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

Discrete-Time Q-Dissipative Control with Application to Stability Margin Guarantee

This paper studies discrete-time dissipative control and proposes a design method for a controller ensuring a given stability margin. The paper begins by generalizing an existing continuous-time dissipative control method into the discrete-time case, with emphasis on a quadratic dissipativity, measured by Q, of the loop transfer system that is not necessarily stable. Then, by taking suitable Q, the feedback system is designed so that it is robustly stable against a pre-specified range of perturbations inserted in the feedback loop. This robustness includes the classical gain and phase margins as a special case.

Nonlinear equations for a slow magnetogasdynamic quasi-monochromatic wave in a medium with dispersion and dissipation

Evolutionary equations are obtained in the vicinity of a singularity for a slow wave with allowance for the quadratic nonlinearity, dispersion, and dissipation. It is demonstrated that the propagation of quasi-monochromatic waves in a current-conducting gas—liquid mixture is described by ordinary differential equations for the amplitudes in the aforementioned region.

Restricted Constant of Motion for the One-Dimensional Harmonic Oscillator with Quadratic Dissipation and Some Consequences in Statistic and Quantum Mechanics

A restricted constant of motion, Lagrangian and Hamiltonian, for the harmonic oscillator with quadratic dissipation is deduced. The restriction comes from the consideration of the constant of motion for the velocity of the particle either for v ≥ 0 or for v < 0. A study is done about the implications that these restricted variables have on the specific heat of a thermodynamical system of oscillators with this dissipation, and on the quantization of this dissipative system.

Dissipative control of linear discrete-time systems with dissipative uncertainty

This paper focuses on the dissipative control of uncertain linear discrete-time systems. The uncertainty under consideration is characterized by a dissipative system, which contains commonly used uncertainty structures, such as normbounded and positive real uncertainties, as special cases. We consider the design of a feedback controller which can achieve asymptotic stability and strict quadratic dissipativeness for all admissible uncertainties. Both the linear static state feedback and the dynamic output feedback controllers are considered. It is shown that the robust dissipative control problem can be solved in terms of a scaled quadratic dissipative control problem without uncertainty. Linear matrix inequality (LMI) based methods for designing robust controllers are derived. The result of this paper unifies existing results on discrete-time H and positive real control and it provides a more flexible and less conservative control design as it al ows for a bet er trade-off between phase and gain performances.

Dissipative control for linear discrete-time systems

This paper focuses on the problem of quadratic dissipative control for linear discrete-time systems. We consider the design of feedback controllers to achieve asymptotic stability and strict quadratic dissipativeness. Both the linear static state feedback and the dynamic output feedback controllers are considered. Necessary and sufficient conditions for the solutions of the quadratic dissipative control problems are obtained using the linear matrix inequality (LMI) approach. The main result of this paper unifies existing results on H ∞ and positive real control and it provides a more flexible and less conservative control design as it allows for a better trade-off between phase and gain performances.

Robust dissipative control for linear systems with dissipative uncertainty

This paper focuses on the problem of quadratic dissipative control for linear systems with or without uncertainty. We consider the design of feedback controllers to achieve (robust) asymptotic stability and strict quadratic dissipativeness. Both linear static state feedback and dynamic output feedback controllers are considered. First, the equivalence between strict quadratic dissipativeness of linear systems and a H performance is established. Necessary and sufficient conditions for the solution of the quadratic dissipative control problem are then obtained using a linear matrix inequality (LMI) approach. As for uncertain systems, we consider structured uncertainty characterized by a dissipative system. This uncertainty description is quite general and contains commonly used types of uncertainty, such as norm-bounded and positive real uncertainties, as special cases. It is shown that the robust dissipative control problem can be solved in terms of a scaled quadratic dissipative control problem without uncertainty. LMIbased methods for designing robust dissipative controllers are also derived. The results of this paper unify existing results on H and positive real control and provide a more flexible and less conservative robust control design as it allows for a better trade-off between phase and gain performances.

Non-Gaussian solution for random rocking of slender rigid block

This paper adopts a random vibration approach to study the response of the slender rigid block to seismic action. The problem is strongly non-linear because of (i) the restoring term and (ii) the quadratic dissipation of energy due to the inelastic impacts, modeled as an impulsive process. The excitation process is firstly assumed to be a Gaussian white noise; secondly, a non-stationary filtered Gaussian white noise is assumed to simulate seismic shaking more accurately. The solution of the associated Fokker-Planck equation in terms of moments of the response is obtained by means of a non-Gaussian closure technique, that enables the complete statistical definition of the approximated transient response process to be achieved. The mean upcrossing rates and the response spectra in terms of displacement are evaluated. The reliability of the solutions derived is assessed by comparing them with Monte Carlo simulations.

On the reduction of certain frequency-shaped linear quadratic dissipative design problems to an H/sub ∞/ formulation

This paper concerns a concept of frequency-based quadratic dissipativity and its reduction to an equivalent H/sub /spl infin// (or /spl mu/) condition. The paper is motivated by the observation that a strictly proper system cannot be strictly passive; hence, results on the synthesis of positive real systems using the Cayley transform directly cannot be applied in most physical situations.