Second-order Problems Research Articles

On duality in convex optimization of second-order differential inclusions with periodic boundary conditions

The present paper is devoted to the duality theory for the convex optimal control problem of second-order differential inclusions with periodic boundary conditions. First, we use an auxiliary problem with second-order discrete-approximate inclusions and focus on formulating sufficient conditions of optimality for the differential problem. Then, we concentrate on the duality that exists in periodic boundary conditions to establish a dual problem for the differential problem and prove that Euler-Lagrange inclusions are duality relations for both primal and dual problems. Finally, we consider an example of the duality for the second-order linear optimal control problem.

Fifth Order Improved Runge-Kutta Nystrom Method Using Trigonometrically-Fitting for Solving Oscillatory Problems

In this paper, the Trigonometrically Fitted Improved Runge-KuttaNystrom method is proposed as a novel method with four stages and fifth order for solving oscillatory problems. This method is intended to integrate second-order initial value problems using the trigonometrically fitting approach. To increase the method'saccuracy, the principal frequency of the problem푤∈ℝ, is used. It is discovered that the new method is more precise when compared with the other existing Runge-Kutta Nystrom and IRKN5 methods. To show how well the TFIRKN5 method works, test problems for second-order ordinary differential equations (ODEs) are solved. The numerical outcomes show that the novel approach outperforms methods that have already been published.

Second-Order Consensus for Multi-Agent Systems With Various Intelligent Levels via Intermittent Sampled-Data Control

This brief is concerned with the second-order consensus problem of multi-agent systems (MASs) over directed communication topologies. Due to engineering applications, agents in the network may have nonidentical intelligent degrees in practical scenarios. A novel intermittent sampled-data control protocol with different intelligent levels is proposed. Necessary and sufficient conditions are derived to guarantee the desired consensus under the protocol. Nevertheless, since the velocity states of real agents may not be measured, a novel protocol only utilizing past sampled position data with distinct intelligent levels is presented, under which the conditions of realizing consensus can be attained via similar analysis. Finally, the asymptotic consensus of MASs can be verified by some numerical simulations. The states of agents with the same intelligent level can be validated to converge to the common value as well.

Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses

Abstract We are concerned with Dirichlet problems of impulsive differential equations − u ″ ( x ) − λ u ( x ) + g ( x , u ( x ) ) + ∑ j = 1 p I j ( u ( x ) ) δ ( x − y j ) = f ( x ) for a.e. x ∈ ( 0 , π ) , u ( 0 ) = u ( π ) = 0 , \left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(x)-\lambda u\left(x)+g\left(x,u\left(x))+\mathop{\displaystyle \sum }\limits_{j=1}^{p}{I}_{j}\left(u\left(x))\delta \left(x-{y}_{j})=f\left(x)\hspace{1em}\hspace{0.1em}\text{for a.e.}\hspace{0.1em}\hspace{0.33em}x\in \left(0,\pi ),\\ u\left(0)=u\left(\pi )=0,\\ \end{array}\right. where λ \lambda is a parameter and runs near 1, f ∈ L 2 ( 0 , π ) f\in {L}^{2}\left(0,\pi ) , I j ∈ C ( R , R ) {I}_{j}\in C\left({\mathbb{R}},{\mathbb{R}}) , j = 1 , 2 , … , p j=1,2,\ldots ,p , p ∈ N p\in {\mathbb{N}} , the nonlinearity g : [ 0 , π ] × R → R g:\left[0,\pi ]\times {\mathbb{R}}\to {\mathbb{R}} satisfies the Carathéodory condition, δ = δ ( x ) \delta =\delta \left(x) denote the Dirac delta impulses concentrated at 0, which are applied at given points 0 < y 1 < y 2 < ⋯ < y p < π 0\lt {y}_{1}\lt {y}_{2}\hspace{0.33em}\lt \cdots \lt {y}_{p}\lt \pi . We show the existence and multiplicity of solutions to the aforementioned problem for λ \lambda in a neighborhood of 1 by using degree theory and bifurcation theory.

Multi-scale computational method for nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties

For most of materials, the properties of the materials are usually dependent on the temperature, which causes the nonlinear physical and mechanical material behaviors. In this paper, an innovative multi-scale computational method is developed for efficiently simulating nonlinear dynamic thermo-mechanical problems of composite materials. The nonlinearities of these multi-scale problems were caused by the temperature-dependent properties of each component material in the composites and the heterogeneities of composite materials are taken into account by periodic distributions of representative unit cells on the micro-scale. Firstly, a second-order two-scale (SOTS) computational model for accurately analyzing nonlinear dynamic thermo-mechanical behaviors of composite materials is established based on multi-scale asymptotic analysis and Taylor series method. The proposed SOTS computational model includes the first-order and second-order auxiliary cell problems on micro-scale, homogenized problem on macro-scale and macro–micro coupled SOTS asymptotic solutions. Then we theoretically explain the crucial importance of developing the SOTS solutions by the error analysis in the point-wise sense. Next, a rigorous error analysis with an explicit rate of the SOTS approximate solutions is given under some simplifications and assumptions in the integral sense. In addition, a multi-scale numerical algorithm is proposed in detail to effectively simulate these nonlinear multi-scale problems. Finally, several typical numerical examples are carried out to confirm the effectiveness and correctness of the proposed multi-scale numerical algorithm, especially for large-scale engineering structures. This study offers an efficient and high-accuracy multi-scale computational scheme that enables the effective simulation and analysis of nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties.

Spectrum and global bifurcation results for nonlinear second-order problem on all of $${\mathbb {R}}$$

Spectrum and global bifurcation results for nonlinear second-order problem on all of $${\mathbb {R}}$$

On second-order [formula omitted]-sub-step explicit algorithms with controllable dissipation and adjustable bifurcation point for second-order hyperbolic problems

This paper proposes a self-starting, second-order accurate, composite s-sub-step explicit method, within which the first five explicit members are developed, analyzed, and compared. Each member attains maximal stability bound, reaching 2×s, where s denotes the number of sub-steps. Identical diagonal elements in the amplification matrix are required in the undamped case, and algorithmic dissipation is controlled and measured at the bifurcation point. Further, if the bifurcation point is regarded as a free parameter during the dissipation analysis, the methods can achieve simultaneously controllable algorithmic dissipation and adjustable bifurcation point. Apart from these, the optimization of low-frequency dissipation is performed in the four- and five-sub-step members. As the number of sub-steps increases, the proposed explicit members can embed better numerical characteristics. This paper provides two approaches to constructing the updating velocity schemes in all sub-steps. The first approach enlarges the stability in the damped case, while the second one enhances the solution accuracy for damping problems. Numerical examples are solved to validate the proposed explicit members’ numerical performance. The novel algorithms are often more accurate and efficient than the existing methods.

A superconvergent ultra-weak discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

A superconvergent ultra-weak discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

Implementation of a projection and rescaling algorithm for second-order conic feasibility problems

This paper documents a computational implementation of a projection and rescaling algorithm for solving one of the alternative feasibility problems where L is a linear subspace in , is its orthogonal complement, and is the interior of a direct product of second order cones. The gist of the projection and rescaling algorithm is to enhance a low-cost first-order method (a basic procedure) with an adaptive reconditioning transformation (a rescaling step). We give a full description of a Python implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our Python code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.

Adaptive multi-step Runge–Kutta–Nyström methods for general second-order ordinary differential equations

In this study, a variable step size formulation of multi-step general Runge–Kutta–Nyström (MSGN) methods to directly integrate general second-order initial value problems (IVPs) is considered. This formula is carried out using an embedded explicit pair where: the higher-order formula is an accurate and the lower-order formula uses to estimate the local error. Numerical results show that the new methods perform much better in terms of functions evaluations, implementation cost and execution time compared to other existing high quality embedded Runge–Kutta (ERK) methods in the literature.

Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form [Formula: see text]. Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order [Formula: see text] in the [Formula: see text]-norm for the solution, when piecewise polynomials of degree at most [Formula: see text] are used. We use these results to prove that the UWDG solution is superconvergent with order [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] towards a special projection of the exact solution. We further prove that the [Formula: see text]-degree UWDG solution and its derivative are [Formula: see text] superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree [Formula: see text]. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

Estimating Time of Urbanization with Moderate-Resolution Sensors

The continuous recording of images of the Earth’s surface through Earth observation satellites has enabled the study of large-scale dynamical process, such as urbanization. Mapping the urban land cover at a given time is a first-order problem that has been largely addressed, and for which several acceptable solutions exist. A second-order problem is the analysis of change through time, which has been generally addressed through change vector or change matrix analyses, but they tend to neglect the spatial dimensions in favor of the temporal one. In this study, we propose the estimation of the time of urbanization across space as a useful indicator of the spatiotemporal structure of the urbanization process. A map of the time of urbanization can be considered a third-order product, as it compresses time series of urban land cover in one image that can effectively help identify areas of accelerated development and the time when they occurred. We apply linear spectral unmixing to quarterly composites of Landsat 8 imagery over the metropolitan corridor in central Mexico, from which a time series of urban percentage coverage is generated for the period between 2013 and 2021. Time filters were then applied to the time series and the time of urbanization was detected as the time when an increase of 25 percent of urbanization or larger first occurred. The accuracy assessment of the urbanization time showed a good correspondence with reference points that were randomly selected in the area of study.

Distributionally robust portfolio optimization with second-order stochastic dominance based on wasserstein metric

In portfolio optimization, we may be dealing with misspecification of a known distribution, that stock returns follow it. The unknown true distribution is considered in terms of a Wasserstein-neighborhood of P to examine the tractable formulations of the portfolio selection problem. This study considers a distributionally robust portfolio optimization problem with an ambiguous stochastic dominance constraint by assuming the unknown distribution of asset returns. The objective is to maximize the worst-case expected return and subject to an ambiguous second-order stochastic dominance constraint. The expected return robustly stochastically dominates the benchmark in the second order over all possible distributions within an ambiguity set. It is also shown that the Wasserstein-moment ambiguity set-based distributionally robust portfolio optimization can be reduced to a semidefinite program and second-order conic programming. We use a cutting plane to solve our second-order stochastic dominance constraint portfolio optimization problem with ambiguity sets based on the Wasserstein metric. Then we decompose this class of distributionally robust portfolio optimization into semi-infinite programming and apply the cutting surface method to solve it. The captured optimization programs are applied to real-life data for more efficient comparison. The problems are examined in depth using the optimal solutions of the optimization programs based on the different setups.

A Novel Damping Control of Grid-Connected Converter Based on Optimal Split-Inductor Concept.

In this paper, a grid-connected converter is investigated. Since the AC side of the grid-connected converter is the LC filter, there is a second-order system resonance problem, and the conventional passive damping control has an inherent limitation of excessive power loss. Based on the mathematical model, a new damping control method is proposed in this paper. It is compared with the traditional solution in terms of damping effect, power loss and system stability. The optimal inductor split ratio is also discussed. The theoretical analysis demonstrates that the proposed method can not only achieve almost the same damping effect as the conventional solution, but also reduce the power loss of the damping resistor. The experimental tests are carried out and the experimental results verify the effectiveness of the proposed method.

Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method

Rapid and accurate rendezvous and proximity operations for spacecraft are crucial to the success of most space missions. In this paper, a sequential convex programming method, combined with the first-order and second-order Birkhoff pseudospectral methods, is proposed for the autonomous rendezvous and proximity operations of spacecraft. The original nonlinear and nonconvex close-range rendezvous problem with thrust constraints and no-fly zone constraints is converted into its convex version by using the sequential convexification techniques; then, the Birkhoff pseudospectral method is used to transcribe the dynamic constraints into a series of linear algebraic equality constraints, in other words, a convex second-order conic programming problem with a relatively small condition number. Thus, the resulting problem can be accurately and efficiently solved by a convex solver. The simulation results indicate that the proposed methods, especially the second-order Birkhoff pseudospectral method, have obvious advantages over other methods in computational efficiency and sensitivity.

Effect of Social Influence on a Two-Party Election: A Markovian Multiagent Model

In digital social networks, the filtering algorithms employed by the platform management to sieve the contents shared among the users can alter the social influence intensity. In this article, a Markov multiagent model of opinion dynamics is used to analyze possible opinion manipulation under apparently neutral interventions on the influence intensity. We consider a two-party election, whose voters, modeled as heterogeneous agents, are connected in a social network with an arbitrary topology. The equations describing the variance of the vote share, both in transient and steady state, are derived. The key is the solution of the second-order marginalization problem under the form of a numerically tractable characterization of pairwise joint probabilities of the voters’ opinions. In particular, these probabilities are computed by means of a Lyapunov-like matrix differential equation driven by first-order moments. This result is used to answer some important questions, like the possible nonmonotonic effect of the influence intensity on the vote volatility and the interplay of topology and individuals’ stubbornness to determine the electoral balance between two parties.

Power constraint waveform design based on ADMM algorithm for MIMO radar transmit beampattern

To address the problems of high sidelobe and large fluctuation amplitude of the mainside on matching the radar beampattern of multiple-input multiple-output (MIMO) radar, we proposed a transmission beampattern design method based on alternating direction method of multipliers (ADMM) under power limitation and peak-to-average ratio (PAR) constraint. The method used the Least-Most-Rule for the transmitting beamform design, i.e., the ratio of the maximum value of sidelobe to the minimum value of mainlobe as objective function. Meanwhile, the sum of power over the mainside and sidelobe is constrained to match the desired pattern. Then, the log-sum-exp function and Taylor's formula are used to approximate the optimization problem to produce a fourth-order polynomial on the beamform vector; finally, we introduce a dual variable and equation constraints and apply ADMM method to convert the fourth-order polynomial problem into the second-order polynomial problem for solution. In addition, the computational complexity of the proposed method is also analyzed. Two sets of experiments verify the effectiveness of the method in the simulation part.

Second-order cone and semidefinite methods for the bisymmetric matrix approximation problem

Approximating the closest positive semi-definite bisymmetric matrix using the Frobenius norm to a data matrix is important in many engineering applications, communication theory and quantum physics. In this paper, we will use the interior point method to solve this problem. The problem will be reformulated into various forms, in the beginning as a semi-definite programming problem and later, into the form of a mixed semidefintie and second-order cone optimization problem. Numerical results comparing the efficiency of these methods with the alternating projection algorithm will be reported.

Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a mth-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order problem with the boundary conditions given at two ends. These are coupled through a left-end boundary condition. Traditionally, the asymptotic solution within the boundary layer is carried out in the stretched coordinates by either analytic or numerical method. The present paper executes the mth-order asymptotic series solution in terms of the original coordinates. After introducing 2(m+1) new variables, the outer and inner problems are transformed together to a set of 3(m+1) first-order initial value problems with the given zero initial conditions; then, the Runge–Kutta method is applied to integrate the differential equations to determine the 2(m+1) unknown terminal values of the new variables until they are convergent. The asymptotic-numerical solution exactly satisfies the boundary conditions, which are different from the conventional asymptotic solution. Several examples demonstrated that the newly proposed method can achieve a better asymptotic solution. For all values of the perturbing parameter, the method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy of the solution in the entire domain. We derive the sufficient conditions, which terminate the series of asymptotic solutions for inner and outer problems of the SPP without having the spring term. For a specific case, we can derive a closed-form asymptotic solution, which is also the exact solution of the considered SPP.

Singular perturbation and initial layer for the abstract Moore-Gibson-Thompson equation

We investigate the singular limit of a third-order abstract equation in time, in relation to the complete second-order Cauchy problem on Banach spaces, where the principal operator is the generator of a strongly continuous cosine family. Assuming that an initial datum is ill prepared, the initial layer problem is studied. We show convergence, which is uniform on compact sets that stay away from zero, as long as initial data are sufficiently smooth. Our method employs suitable results from the theory of general resolvent families of operators. The abstract formulation of the third-order in time equation is inspired by the Moore-Gibson-Thompson equation, which is the linearization of a model that currently finds applications in the propagation of ultrasound waves, displacement of certain viscoelastic materials, flexible structural systems that possess internal damping and the theory of thermoelasticity.