Continuous-time Equations Research Articles

Linear Stieltjes equation with generalized riemann integral and existence of regulated solutions

In this work we establish an existence theorem of regulated solutions for a class of Stieltjes equations which involve generalized Riemann kind of integrals. The general method applied consists in considering the continuous-time Stieltjes equation as limit of discrete processes. This approach will prove fruitful in the study of the controllability of Stieltjes systems, because it will be possible to get properties on the continuous time equation by transferring properties of the discrete ones.

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form.

A continuous time model of the bandwagon effect in collective action

The paper offers a complex and systematic model of the bandwagon effect in collective action using continuous time equations. The model treats the bandwagon effect as a process influenced by ratio between the mobilization efforts of social activists and the resources invested by the government to counteract this activity. The complex modeling approach makes it possible to identify the conditions for specific types of the bandwagon effect, and determines the scope of that effect. Relying on certain behavioral assumptions, these conditions are only indirectly connected to individual beliefs, so that a given bandwagon effect can be explained and planned without knowing the exact preferences of the players' preferences.

Singular perturbation analysis of cheap control problem for sampled data systems

This paper studies the discrete-time cheap control problem for sampled data systems using the theory of singular perturbations. It is shown, by using the two time-scale property of singularly perturbed systems, that the problem can be solved in terms of two reduced-order subproblems for which computations can be done in parallel, thus increasing the computational speed. Similarly to the continuous-time case, the singular perturbation approach enables the decomposition of the algebraic Riccati equation into two reduced-order pure-slow and pure-fast continuous-time algebraic equations.

Coarsening by Ginzburg-Landau Dynamics

We study slowly moving solutions of the real Ginzburg–Landau equation on the infinite line, by a method inspired by the work of J. Carr and R. L. Pego. These solutions are functions taking alternately positive or negative values on large intervals. We give a formula for the speed of the motion of the zeros of these functions and present a description of the collapse of two nearby zeros. Since we work on an infinite domain, we can find initial conditions which lead to infinite sequences of collapses of kinks. These results justify in part the approach to the coarsening problem by a simplified model proposed by A. J. Bray, B. Derrida, and C. Godreche. Their model starts from a random distribution of domain lengths and evolves by successive elimination of the smallest domain. Our model is on one hand more realistic, since it controls the original Ginzburg–Landau equation in continuous time. On the other hand, we fail to gain sufficient control over the vanishing of pairs of kinks “embedded” within pairs of kinks at a greater distance.

Continuous panel models with time dependent parameters

It is shown how the likelihood function can be computed in this general linear setting by using a Kaiman filter algorithm. The estimation method is tested in a simulation study using a bivariate growth model and applied to the Brownian bridge. Panel data are modeled as dynamic structural equations in continuous time t (stochastic differential equations). The continuously moving latent state vector y(t) is mapped to an observable discrete time series (or panel) zni = zn (t i,) with the help of a measurement equation including errors of measurement (continuous‐discrete state space model). Therefore the approach is able to handle data with irregularly observed waves, missing values and arbitrarily interpolated exogenous influences (control variables). In order to model development and growth models, the system parameter matrices are assumed to be time dependent.

Delay-differential equation versus 1D-map: Application to chaos control

Theoretical and experimental results on the control of chaotic motion in an electronic circuit modeled by a delay-differential equation are presented. The model is derived from a nonlinear optical system with time-delay feedback. Our control method is based on Pyragas approach and applied to the infinite-dimensional system. Generalized mean Floquet multipliers and mean Lyapunov exponents related to delay-differential equations are introduced in order to compare a discrete-time approximation with the continuous-time equation. This control method is also used to find the so-called isomers related to harmonic periodic solutions predicted by Ikeda and Matsumoto.

An Adaptive MIMO VSC Algorithm, and its Application to the Control of an Underwater Vehicle

Discrete-time Variable Structure Control (VSC) for MIMO systems is addressed in this note. Two control techniques, consisting in VSC laws cascaded to a Generalized Minimum Variance (GMV) controller, are presented. The former algorithm refers to a completely known system, while the latter has been designed to deal with parameter uncertainties in the plant. The presented algorithms have been applied to the problem of position and orientation control of an underwater Remotely Operated Vehicle (ROV). Resulting performances have been tested by vsimulation, modelling the ROV with a nonlinear continuous-time equations system. Parameter variations have been applied to the plant parameters, and the presence of measurement noise has been considered.

Vulnerability to noise of a phase-locked oscillator array

The effect of noise on the in-phase attractor of a set of N globally coupled oscillators is studied; these discrete-time maps are associated with the continuous-time equations of motion for a series array of Josephson junction oscillators. We investigate both geometrical properties of the basin of attraction in the large N limit, and the implications of this geometry on the average time for the system to escape from the phase-locked mode. Our main results are that the attractor basin maintains a box-shaped “core” of finite radius even as N → ∞, and that the in-phase attractor of a large N array is much less vulnerable to noise than are the out-of-phase attractors.

Vulnerability to noise of a phase-locked oscillator array

Abstract The effect of noise on the in-phase attractor of a set of N globally coupled oscillators is studied; these discrete-time maps are associated with the continuous-time equations of motion for a series array of Josephson junction oscillators. We investigate both geometrical properties of the basin of attraction in the large N limit, and the implications of this geometry on the average time for the system to escape from the phase-locked mode. Our main results are that the attractor basin maintains a box-shaped “core” of finite radius even as N → ∞, and that the in-phase attractor of a large N array is much less vulnerable to noise than are the out-of-phase attractors.

Particle paths in a Schwarzschild spacetime via the Regge calculus

As a test of the Regge calculus Williams and Ellis (1984) computed, among other quantities, the precession of the perihelia of Mercury. Unfortunately they did not obtain anywhere near the correct values. Their results varied between -2.74 and 0.42 radians per orbit. Their best result was 9.1*10-4 radians per orbit whereas the correct analytic value is 5.0*10-7 radians per orbit. A continuous-time version of their equations is presented. It is shown numerically that, for a sufficiently fine discretization, the global error between the Regge and Schwarzschild geodesics varies linearly with the typical length scale for the Regge simplices. Some simple modifications to the continuous-time equations are then presented. It is shown both numerically and analytically that the modified equations yield paths that converge quadratically to the Schwarzschild geodesics. The modified Regge equations are then applied to the problem of computing the precession of the perihelia of Mercury. The result is a precession of 5.0*10-7 radians per orbit.

Tsirel'son's equation in discrete time

Motivated by Tsirel'son's equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solution. In general, for any time,n, the σ-field generated by the past of a solution up to timen is shown to be equal, up to negligible sets, to the σ-field generated by the 3 following components: the infinitely remote past of the solution, the past to the noise up to timen, together with an adequate independent complement.

Stochastic Estimation Approach for the Evaluation of Thermal-Hydraulic Parameters in Pressurized Water Reactors

A method based on the extended Kalman filter is developed for the estimation of the core coolant mass flow rate in pressurized water reactors. The need for flow calibration can be avoided by a direct estimation of this parameter. A reduced-order neutronic and thermal-hydraulic model is developed for the Loss-of-Fluid Test (LOFT) reactor. The neutron detector and core-exit coolant temperature signals from the LOFT reactor are used as measurements in the parameter estimation algorithm. The estimation sensitivity to model uncertainties was evaluated using the ambiguity function analysis. This also provides a lower bound on the measurement sample size necessary to achieve a certain estimation accuracy. A sequential technique was developed to minimize the computational effort needed to discretize the continuous time equations, and thus achieve faster convergence to the true parameter value. The performance of the stochastic approximation method was first evaluated using simulated random data, and then applied to the estimation of coolant flow rate using the operational data from the LOFT reactor at 100 and 65% flow rate conditions.

On Wald's equations in continuous time

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).

On Wald's equations in continuous time

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).