State Parametrization Research Articles

A numerical method for solving optimal control problems using state parametrization

A numerical method for solving a special class of optimal control problems is given. The solution is based on state parametrization as a polynomial with unknown coefficients. This converts the problem to a non-linear optimization problem. To facilitate the computation of optimal coefficients, an improved iterative method is suggested. Convergence of this iterative method and its implementation for numerical examples are also given.

The Feasibility of Constraining Dark Energy Using LAMOST Redshift Survey

We consider using future redshift surveys with the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) to constrain the equation of state of dark energy ω. We analyze the Alcock & Paczynski (AP) effect imprinted on the two-point correlation function of galaxies in redshift space. The Fisher matrix analysis is applied to estimate the expected error bounds of ω0 and ωa from galaxy redshift surveys, ω0 and ωa being the two parameters in the equation of state parametrization ω(z) = ω0+ωaz/(1+z). Strong degeneracies between ω0 and ωa are found. The direction of the degeneracy in ω0−ωa plane, however, rotates counter-clockwise as the redshift increases. LAMOST can potentially contribute in the redshift range up to 0.5. In combination with other high redshift surveys, such as the proposed Kilo-Aperture Optical Spectrograph project (KAOS), the joint constraint derived from galaxy surveys at different redshift ranges is likely to efficiently break the degeneracy of ω0 and ωa. We do not anticipate that the nature of dark energy can be well constrained with LAMOST alone, but it may help to reduce the error bounds expected from other observations, such as the Supernova/Acceleration Probe (SNAP).

Direct reconstruction of the quintessence potential

We describe an algorithm which directly determines the quintessence potential from observational data, without using an equation of state parametrization. The strategy is to numerically determine observational quantities as a function of the expansion coefficients of the quintessence potential, which are then constrained using a likelihood approach. We further impose a model selection criterion, the Bayesian Information Criterion, to determine the appropriate level of the potential expansion. In addition to the potential parameters, the present day quintessence field velocity is kept as a free parameter. Our investigation contains unusual model types, including a scalar field moving on a flat potential, or in an uphill direction, and is general enough to permit oscillating quintessence field models. We apply our method to the ``gold`` Type Ia supernovae sample of Riess et al. [A. G. Riess et al. (Supernova Search Team Collaboration), Astrophys. J. 607, 665 (2004)] confirming the pure cosmological constant model as the best description of current supernovae luminosity-redshift data. Our method is optimal for extracting quintessence parameters from future data.

Stopping cross sections forN4+→Hat low projectile velocity

We study the time-dependent dynamics of N 4 + ions colliding with atomic hydrogen for projectile energies ranging from a fraction of an eV/amu up to 25 keV/amu using the electron-nuclear dynamics (END) formalism. The END theory obtains the electron-nuclear coupled equations of motion from the time-dependent variational principle employing a coherent state parametrization of the wave function. This approach leads to a simultaneous nonadiabatic dynamics of all the electrons and nuclei. We calculate and discuss dynamical trajectories, deflection functions, final charge states, differential cross sections, and energy loss. Quantum effects of the forward peak scattering are emphasized. Due to the strong interaction between the heavy and the hydrogen atom, a diffuse ion (vide infra) is formed, leading to acceleration or energy gain of the projectile. For the case of the electron transfer cross section, we found that it does not follow the Langevin-type cross section at low projectile energies as reported by other methods. Present results show good agreement with available experimental data.

Toward an ab Initio Treatment of the Time-Dependent Schrödinger Equation of Molecular Systems

The time-dependent variational principle (TDVP) is employed to produce equations of motion that approximate the time-dependent Schrodinger equation. Choices of wave function and basis sets are discussed. The use of electron translation factors and the electronic and nuclear parts of the molecular wavefunction are put in the context of the electron nuclear dynamics (END) theory. The role of wave function parameters as dynamical variables is discussed, and the use of coherent state parametrization is explored.

Computational method based on state parametrization for solving constrained nonlinear optimal control problems

In this paper, we present a method for solving nonlinear optimal control problems subject to terminal state constraints and control saturation constraints. This method is based on using the quasilinearization and the state variables parametrization by using Chebyshev polynomials. In this method, there is no need to integrate the system state equations or the costate equations. By virtue of the quasilinearization and the state parametrization, the difficult constrained nonlinear optimal control problem is approximated by a sequence of small quadratic programming problems which can be solved easily. To show the effectiveness of the proposed method, the simulation results of a numerical example are shown.

H++H, He, and H2 scattering using a new time-dependent method for electron nuclear dynamics

In this paper we apply the recently proposed and implemented electron nuclear dynamics (END) theory [J. Chem. Phys. 96, 6820 (1992)] to the study of prototypical ion–atom and ion–molecule collisions. The END theory obtains the equations of motion from the time-dependent variational principle (TDVP) employing a group theoretical coherent state (CS) parametrization of the wave function. The approach leads to a fully dynamical treatment of electrons and nuclei without invoking potential energy surfaces. The present implementation of the END theory constitutes the simplest ab initio model with the electrons described by a single determinantal wave function and the nuclei treated classically (or equivalently, with frozen Gaussian wave packets in the limit of a narrow widths). The method is applied to the H++H, He, and H2 collision processes in the energy range of 200–5000 eV. Results for the elastic and charge transfer differential cross sections, the differential probabilities, and the rainbow angles are presented and compared with experimental data. Also, the dynamical trajectories, deflection functions, and differential vibrational excitation for the H2 target are calculated and discussed. Effects of initial state molecular orientations, in the case of the H2 target, are considered. In general, the results provided by this model implementation of the END theory are in good agreement with experimental data.

Optimal control of linearly constrained linear systems via state parametrization

This paper presents a general computational tool for determining the near-optimal trajectories of linear, lumped parameter, dynamic systems subjected to linear constraints. In the proposed approach each state variable is approximated by the sum of a third-order polynomial and a finite term Fourier-type series. This enables a linearly constrained optimal control problem to be converted into a linearly constrained mathematical programming problem. Simulation results demonstrate that the approach accurately predicts the optimal performance index and the optimal state and control trajectories.

A simple phenomenology for 2 γ+ states

Based on the microscopic structure of the first gamma vibrational state, a simple but efficient parametrization for the energies of the 2 γ + states in all (non-magic) nuclei with Z>20 is introduced. A simple analytic formula, similar to that existing for the 3 1 − -states, with mass and total active nucleon number dependence, fits the 2 γ + energies rather well.

Discrete and continuous models for computation of optimal vertical highway alignment

This paper describes a dynamic programming model and a state parametrization model formulated to solve two-dimensional highway location problems. A numerical example is presented to compare the solutions obtained using the two models. Features of the two approaches, with respect to their applicability in solving three-dimensional highway location problems in particular, are highlighted. This study forms the basis of a three-dimensional model that computes optimal horizontal and vertical alignments simultaneously.