Free Initial Conditions Research Articles

Global-in-time probabilistically strong solutions to stochastic power-law equations: Existence and non-uniqueness

We are concerned with the power-law fluids driven by an additive stochastic forcing in dimension d⩾3. For the power index r∈(1,3d+2d+2), we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in Llocp([0,∞);L2)∩C([0,∞);W1,max{1,r−1}),p⩾1 for every divergence free initial condition in L2∩W1,max{1,r−1}. This result in particular implies non-uniqueness in law. Our result is sharp in the three dimensional case in the sense that the solution is unique if r⩾3d+2d+2.

Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.

Novel Nonlinear Control Method for First‐Order Systems and Its Implementation in a Temperature System

In this work, a novel nonlinear control theory design for first‐order systems is developed, contributing to the improvement of the existing theory. The theory will allow a design of the open loop and closed‐loop controllers that ensure the tracking of any reference, constant, or variant in time with a free initial condition where the Laplace transform was used to find all the analytical solutions, avoiding the transfer function theory. Moreover, the closed‐loop control will be the best option to speed up or slow down the reference convergence rate in the desired finite time. Then, an algorithm indicating the steps for designing a closed‐loop controller and achieving proper tuning for a real‐time application is shown. Finally, this manuscript presents the results and discussions of the theory implemented in a prototype tank of a liquid temperature control system, where the effectiveness of the applied temperature control can be seen.

On inflationary parameters in scalar–tensor theories

The equivalence between [Formula: see text] and scalar–tensor theories is revisited, we consequently explored different [Formula: see text] models. After consideration of specific definition of the scalar field, we derived the potentials [Formula: see text] for each [Formula: see text] model focusing on the early Universe, mostly the inflation epoch. For a given potential, we applied the slow-roll approximation approach to each [Formula: see text] model and obtained the expressions for the spectral index [Formula: see text] and tensor-to-scalar ratio [Formula: see text]. We determined the corresponding numerical values associated with each of the [Formula: see text] models. Our results showed that for certain choice of parameter space, the values of [Formula: see text] and [Formula: see text] are consistent with the Planck survey results and others produce numerical values that are in the same range as suggested by Planck data. We further constructed the Klein–Gordon equations (KGEs) of each [Formula: see text] model. We found numerical solutions to each KGE considering different values of free parameters and initial conditions of each [Formula: see text] model. All models showed that the scalar field decreases as time increases, an indication that there is less content of the scalar field in the late Universe.

Value Iteration-Based H∞ Controller Design for Continuous-Time Nonlinear Systems Subject to Input Constraints

In this paper, a novel integral reinforcement learning method is proposed based on value iteration (VI) to design the $H_{\infty }$ controller for continuous-time nonlinear systems subject to input constraints. To confront the control constraints, a nonquadratic function is introduced to reconstruct the ${L_{2}}$ -gain condition for the $H_{\infty }$ control problem. Then, the VI method is proposed to solve the corresponding Hamilton–Jacobi–Isaacs equation initialized with an arbitrary positive semi-definite value function. Compared with most existing works developed based on policy iteration, the initial admissible control policy is no longer required which results in a more free initial condition. The iterative process of the proposed VI method is analyzed and the convergence to the saddle point solution is proved in a general way. For the implementation of the proposed method, only one neural network is introduced to approximate the iterative value function, which results in a simpler architecture with less computational load compared with utilizing three neural networks. To verify the effectiveness of the VI-based method, two nonlinear cases are presented, respectively.

Comparison of binary black hole initial data sets

We present improvements to construction of binary black hole initial data used in SpEC (the Spectral Einstein Code). We introduce new boundary conditions for the extended conformal thin sandwich elliptic equations that enforce the excision surfaces to be slightly inside rather than on the apparent horizons, thus avoiding extrapolation into the black holes at the last stage of initial data construction. We find that this improves initial data constraint violations near and inside the apparent horizons by about 3 orders of magnitude. We construct several initial data sets that are intended to be astrophysically equivalent but use different free data, boundary conditions, and initial gauge conditions. These include free data chosen as a superposition of two black holes in time-independent horizon-penetrating harmonic and damped harmonic coordinates. We also implement initial data for which the initial gauge satisfies the harmonic and damped harmonic gauge conditions; this can be done independently of the free data, since this amounts to a choice of the time derivatives of the lapse and shift. We compare these initial data sets by evolving them. We show that the gravitational waveforms extracted during the evolution of these different initial data sets agree very well after excluding initial transients. However, we do find small differences between these waveforms, which we attribute to small differences in initial orbital eccentricity, and in initial BH masses and spins, resulting from the different choices of free data. Among the cases considered, we find that superposed harmonic initial data leads to significantly smaller transients, smaller variation in BH spins and masses during these transients, smaller constraint violations, and more computationally efficient evolutions. Finally, we study the impact of initial data choices on the construction of zero-eccentricity initial data.

Gravity with free initial conditions: A solution to the cosmological constant problem testable by CMB B -mode polarization

In standard general relativity the universe cannot be started with arbitrary initial conditions, because four of the ten components of the Einstein's field equations (EFE) are constraints on initial conditions. In the previous work it was proposed to extend the gravity theory to allow free initial conditions, with a motivation to solve the cosmological constant problem. This was done by setting four constraints on metric variations in the action principle, which is reasonable because the gravity's physical degrees of freedom are at most six. However, there are two problems about this theory; the three constraints in addition to the unimodular condition were introduced without clear physical meanings, and the flat Minkowski spacetime is unstable against perturbations. Here a new set of gravitational field equations is derived by replacing the three constraints with new ones requiring that geodesic paths remain geodesic against metric variations. The instability problem is then naturally solved. Implications for the cosmological constant $\Lambda$ are unchanged; the theory converges into EFE with nonzero $\Lambda$ by inflation, but $\Lambda$ varies on scales much larger than the present Hubble horizon. Then galaxies are formed only in small $\Lambda$ regions, and the cosmological constant problem is solved by the anthropic argument. Because of the increased degrees of freedom in metric dynamics, the theory predicts new non-oscillatory modes of metric anisotropy generated by quantum fluctuation during inflation, and CMB B-mode polarization would be observed differently from the standard predictions by general relativity.

Tachyon inflation in an AdS braneworld with backreaction

We analyze the inflationary scenario based on the tachyon field coupled with the radion of the second Randall–Sundrum model (RSII). The tachyon Lagrangian is derived from the dynamics of a 3-brane moving in the five-dimensional bulk. The AdS5 geometry of the bulk is extended to include the radion. Using the Hamiltonian formalism we find four nonlinear field equations supplemented by the modified Friedmann equations of the RSII braneworld cosmology. After a suitable rescaling we reduce the parameters of our model to only one free parameter related to the brane tension and the AdS5 curvature. We solve the equations numerically assuming a reasonably wide range of initial conditions determined by physical considerations. Varying the free parameter and initial conditions we confront our results with the Planck 2015 data.

Absorption kinetics and swelling stresses in hydrothermally aged epoxies investigated by photoelastic image analysis

The present work proposes an experimental optical methodology able to measure the transient swelling stresses induced by the water uptake ageing of polymers. In particular, the work describes the implementation of a Photoelastic technique to quantify internal stresses arising during the hydrothermal ageing of cast epoxy samples. The material investigated is a model DGEBA/DDS epoxy system. Curing and post-curing cycles have been optimised in order to obtain a fully cured, high Tg, and completely stress free initial condition. Rectangular beam samples were then left in a hydrothermal bath at 90 °C, and regularly monitored by gravimetric and photoelastic analyses. The quantitative evolution of stresses induced by water ingress was obtained by a Photoelastic Stress Analysis technique applied during water absorption, and successive desorption in controlled conditions. The evolution of stresses is correlated with gravimetric data, allowing to gain a new insight to investigate the complex swelling and diffusion kinetics of water ingress into thermoset polymer networks.

Introducing a novel algorithm for minimum-time low-thrust orbital transfers with free initial condition

In this work, a new algorithm is presented with regard to the free initial condition for solving optimal control problems. The reason for presentation of such an algorithm is to develop a suitable method that simplifies difficulties of the optimal control problems that researchers face in common methods of optimal control theory. Also, initial condition as true anomaly is considered to be free in this optimal control problem. To do so, issues such as optimal control theory, orthogonal functions in the Hilbert space, and evolutionary optimizations such as genetic algorithm-particle swarm optimization and imperial competition algorithm are utilized. The algorithm was solved for low-thrust orbital transfer problems which included nonlinear dynamic equations. To validate the algorithm, simplified Edelbaum low-thrust equations are compared with the proposed analytical solutions. Next, the algorithm is investigated for the low-thrust orbital transfers with respect to the equinoctial orbital equations of the minimum-time problem. Results are achieved for two evolutionary optimization methods genetic algorithm-particle swarm optimization and imperial competition algorithm and three orthogonal functions such as Fourier, Chebyshev, and Legendre. Two optimization methods and three orthogonal functions are covered and compared precisely. With respect to the results, this algorithm has the capability to overcome difficulties of the optimal control problems and can be considered as a novelty in this field for the free initial condition problems.

Painlevé IV: A numerical study of the fundamental domain and beyond

The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast ‘pole field solver’ (Fornberg and Weideman, 2011). The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for PIV, the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation and also explore solution regimes which have not been described in the previous literature.

Comparing and analyzing min-time and min-effort criteria for free true anomaly of low-thrust orbital maneuvers with new optimal control algorithm

This research presents a new algorithm for solving optimal control problems with regard to path planning with a free initial condition. To conduct research in this subject, issues such as optimal control theory, orthogonal functions in the Hilbert space, and the evolutionary optimizations such as Genetic algorithm (GA), Particle swarm optimization (PSO), Genetic algorithm-Particle swarm optimization (GA-PSO) and Imperial competition algorithm (ICA) are utilized. The algorithm is proposed for low-thrust orbital transfer problems, which include nonlinear dynamic equations. To validate the algorithm, Edelbaum low-thrust equations are compared with proposed analytical solutions. Afterwards, the algorithm is investigated for low-thrust orbital transfers concerning equinoctial equations of the min-time and min-effort problems and compared with pseudo-spectral method. Also two performance indices are compared from the viewpoint of space mission analysis and design. In addition, to obtain the best point for starting maneuvers, initial condition of true anomaly is considered to be free. Once the orbital transfer problem is solved, the fuzzy decision maker has to choose the best configuration among the free true anomaly solutions and performance indices to upgrade the proposed algorithm. This configuration balanced two performance indices by considering free true anomaly. As a result, this novel algorithm is able to overcome difficulties of optimal control problems using the global optimization and multi combination performance indices. Another innovation of this new method in the field of optimal control theory is to set initial conditions with more than one performance criterion free.

Global solutions to planar magnetohydrodynamic equations with radiation and large initial data

The existence and uniqueness of the global classical solution for the planar magnetohydrodynamic equations are proved for large initial data. The model equations are coupled with the thermal radiation and are supplemented with free boundary and initial conditions. The existence proof relies on the classical Leray–Schauder fixed theorem together with some new a priori estimates in Lagrangian coordinates. The result holds for more general heat conductivity, which is plausible and interesting in physics. Particularly, it is also valid for the case of constant transport coefficients.

Painlevé IV with both Parameters Zero: A Numerical Study

The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast “pole field solver” [1]. The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. The present study applies this new computational tool to the special case when both of its parameters are zero. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature.

A problem of optimal control with free initial state

We are studying an optimal control problem with free initial condition. The initial state of the optimized system is not known exactly, information on initial state is exhausted by inclusions x0 2 X0. Accessible controls for optimization of continuous dynamic system are discrete controls defined on quantized axes. The method presented is based on the concepts and operations of the adaptive method (9) of linear programming. The results are illustrated by a fourth order problem, efficiency estimates of proposed methods are given.

Direct Method for Resolution of Optimal Control Problem with Free Initial Condition

The theory of control analyzes the proprieties of commanded systems. Problems of optimal control (OC) have been intensively investigated in the world literature for over forty years. During this period, series of fundamental results have been obtained, among which should be noted the maximum principle (Pontryagin et al., 1962) and dynamic programming (Bellman, 1963). For many of the problems of the optimal control theory (OCT), adequate solutions are found (Bryson and Yu-chi, 1969, Lee and Markus, 1967, Gabasov and Kirillova, 1977, 1978, 1980). Results of the theory were taken up in various fields of science, engineering, and economics. The present paper aims at extending the constructive methods of Balashevich et al., (2000) that were developed for the problems of optimal control with the bounded initial state is not fixed are considered.

Stable inversion of SISO nonminimum phase linear systems through output planning: An experimental application to the one-link flexible manipulator

In this paper, we introduce a new technique that allows for causal stable inversion of linear nonminimum phase systems. This approach is presented through its application to a flexible arm robot. In contrast with the available techniques, which amounts to finding a particular initial value for the inverse system to have bounded state trajectories, the presented scheme starts from any free initial conditions of the inverse dynamics, and searches for proper output trajectories with a polynomial form. Those output trajectories are computed such that the effect of the unstable zeros are completely cancelled. Furthermore, the scheme deals equally with hyperbolic as well as nonhyperbolic systems. The stable inverse is then incorporated into an output tracking controller, using a classical static state feedback. This global controller permits an exact tracking of the planned output. Experiments for a one-link flexible arm show good end-effector tracking results.

On solving optimal control problems with free initial condition using iterative dynamic programming

AbstractFor solving optimal control problems where the initial conditions of some of the state variables are not specified, a procedure based on iterative dynamic programming (IDP) is presented. In this procedure, the free initial conditions are taken to be additional control variables for the first time stage only; then the search for the optimal initial conditions and also the optimal control policy is carried out simultaneously using IDP. The procedure is straightforward, and as illustrated with two nonlinear optimal control problems, for each case the optimum performance index is readily obtained.

ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS

This note contains three main results. Firstly, a complete solution of the Linear Non-Homogeneous Matrix Differential Equations (LNHMDEs) is presented that takes into account both the non-zero initial conditions of the pseudo state and the non-zero initial conditions of the input. Secondly, in order to characterise the dynamics of the LNHMDEs correctly, some important concepts such as the state, slow state (smooth state) and fast state (impulsive state) are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth (slow) response and the fast (implusive) response. As a third result, a new characterization of the impulsive free initial conditions of the LNHMDEs is given.

Higher order moments of the matter distribution in scale-free cosmological simulations with large dynamic range

We calculate reduced moments $\overline \xi_q$ of the matter density fluctuations, up to order $q=5$, from counts in cells produced by Particle--Mesh numerical simulations with scale--free Gaussian initial conditions. We use power--law spectra $P(k) \propto k^n$ with indices $n=-3,~-2,~-1,~0,~1$. Due to the supposed absence of characteristic times or scales in our models, all quantities are expected to depend on a single scaling variable. For each model, the moments at all times can be expressed in terms of the variance $\overline \xi_2$, alone. We look for agreement with the hierarchical scaling ansatz, according to which $\overline \xi_q \propto \overline \xi_2^{~q-1}$. For $n\leq -2$ models we find strong deviations from the hierarchy, which are mostly due to the presence of boundary problems in the simulations. A small, residual signal of deviation from the hierarchical scaling is however also found in $n \geq -1$ models. For the first time, due to our large dynamic range and careful checks of scaling and shot--noise effects, we are able to detect evolution away from the perturbation theory result.