Second-order Problems Research Articles

Second-order problems involving time-dependent subdifferential operators and application to control

<p style='text-indent:20px;'>The paper provides a new result concerning the existence of solutions for second-order evolution problems associated with time-dependent subdifferential operators involving both single-valued and mixed semi-continuous set-valued perturbations. Optimal control problems corresponding to such differential inclusions using relaxation theorems with Young measures are investigated. The existence of solutions for a coupled system governed by a second-order differential equation with an evolution problem is also addressed.</p>

Nonlinear Low-Frequency Response of a Floating Offshore Wind Turbine Integrated With a Steel Fish Farming Cage

In this article, the effects of Morison drag force and the second-order hydrodynamics on a state-of-the-art floating system integrating a floating offshore wind turbine with a steel fish farming cage (FOWT-SFFC) are studied. To numerically solve the second-order hydrodynamic problem with ease, a simplified structure is adopted. Convergence study is carried out on the full difference-frequency quadratic transfer function (QTF). The middle-field method is recommended to be used in the calculation of QTF. For the aero-hydro-servo-elastic simulations of FOWT-SFFC, the time-domain solver OrcaFlex is used. The effects of the Morison drag force and second-order hydrodynamics on the dynamic responses under a variety of conditions are discussed. The comparison between the results with and without Morison drag force shows that Morison drag force has twofold effects on the low-frequency responses, making the excitation and damping level both larger. The results obtained by including first-order wave loads only and involving both first-order and second-order wave loads are also compared. The comparison reveals that the low-frequency contents of the dynamic responses, particularly heave and pitch, can be boosted when the second-order hydrodynamic forces are included. In contrast to the full QTF method, the known Newman's approximation leads to underprediction of the low-frequency responses of FOWT-SFFC.

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations

The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations

Controllability results for second-order integro-differential equations with state-dependent delay

The purpose of this study is to use resolvent operators to investigate the existence and the controllability of a mild solution to a second-order semilinear integro-differential problem. To construct our criterion, we use a fixed point theorem in conjunction with measures of noncompactness. A practical example is used to illustrate the obtained results.

Real-Time Optimal Control for Eco-Driving and Powertrain Energy Management

This paper presents a real-time receding-horizon control solution for the optimal control problem on combined eco-driving and powertrain energy management for a series-hybrid electric vehicle. By defining the optimal control problem in the spatial domain, discretizing the dynamics and applying several relaxations to the constraints, the problem is formulated as a second-order cone problem. The online real-time solution is achieved through the reformulation into a receding horizon control problem. To reduce computational complexity, several move-blocking strategies are applied. Combining powertrain control of the hybrid powertrain with eco-driving, leads to a reduction of 33.7%, compared to only applying energy management.

On periodic Ambrosetti-Prodi-type problems

<abstract><p>This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.</p></abstract>

The second-order problem of other minds.

The target article proposes that people perceive social robots as depictions rather than as genuine social agents. We suggest that people might instead view social robots as social agents, albeit agents with more restricted capacities and moral rights than humans. We discuss why social robots, unlike other kinds of depictions, present a special challenge for testing the depiction hypothesis.

On the integral representations of solutions of some matrix pencils of the Sturm-Liouville equation

In the present work, we obtain some integral representations for special solutions, which play an important role in solving direct and inverse problems for secondorder and fourth-order matrix Sturm-Liouville pencils. We also investigate some useful properties of the special solutions.

COMPLEMENTARY ENERGY PRINCIPLE FOR CABLE NETWORKS IN TERMS OF FORCE VECTOR VARIABLE

An alternative approach to the variation formulation of static equilibrium problem for cable networks is proposed. The dual minimization principle is derived in terms of force variables. The full vector force in the cable member is chosen as the dual variable. Unlike the previously proposed treatments of the problem using the scalar axial force variable the elastic energy retains the most general form until the closing steps. It is only assumed to be a convex function of the member end-to-end vector. As a result the nonsmooth asymmetric response of the cable does not appear in the form of inequality in the very beginning. Instead it shows up in the final form of the complementary energy functional, which turn out to be non-differentiable for zero member forces. The kinematic relations between member elongations and nodal displacements convert into the force equilibrium conditions in terms of nodal reactions and member force vectors. The Dirichlet boundary conditions appear in the dual form as a linear term in the complementary energy functional. For the linear elastic cables this approach lead to a second-order cone optimization problem with two sets conical constraints. Such formulation provides flexibility and thus can be potentially extended to account for phenomena such as inter-fiber friction and adhesion. Keywords: complementary energy, variational principles, cable networks

Modified Filon-type methods for second-order highly oscillatory systems with a time-dependent frequency matrix

We propose a modified Filon-type method for solving second-order highly oscillatory systems with a time dependent matrix. We first transform the second-order problem into a first-order one and then integrate the transformed one by the Filon-type method combined with the waveform relaxation method. The fundamental matrix in the formula of variation of constants can be calculated directly such that our proposed modified Filon-type method do not need to compute the nested integral commutators in the Magnus expansion. The global error of the modified Filon-type method is presented. Numerical examples show that our proposed numerical solver is efficient and accurate for highly oscillatory problems.

Odd Order Integrator with Two Complex Functions Control Parameters for Solving Systems of Initial Value Problems

In this study, a numerical integrator that is based on a nonlinear interpolant, for the local representation of the theoretical solution is presented. The resulting integrator aims to solve second and higher-order initial value problems as systems of first-order initial value problems. The method is designed to have two complex functions as control parameters. The control parameters may become real, depending on the nature of the second-order initial value problems to be solved. The generalization and properties of the scheme are also presented.

Distributed optimal dispatching method for smart distribution network considering effective interaction of source-network-load-storage flexible resources

Smart distribution networks (SDNs) can integrate the flexible resources from source-network-load-storage (SNLS) to cope with the fluctuation due to a high proportion of distributed generation (DG). However, such SNLS resources are characterized by complex coupling relationships; their control authority may belong to different stakeholders. Challenged by the above, the laminar flow structure from the communication field is introduced for distributed optimal dispatching in SDNs. A day-ahead laminar dispatching method considering the effective interaction of SNLS resources is proposed. First, the applicability of the laminar flow structure is analyzed. An upper-layer dispatching model for the SDN and a lower-layer dispatching model for users with DG are established. Then, by introducing intermediate variables, the lower dispatching model is transformed into a quadratic programming problem and the upper dispatching model is transformed into a second-order cone relaxation programming problem. Selecting the tie-line power flow as the exchanged information in the boundary, the upper- and lower-layer models are alternately solved until the convergence criterion is met. Finally, an improved IEEE 33-bus system is experimentally analyzed. We find that the SNLS flexible resource dispatching scheme can be obtained with only a few iterations, and the DG consumption can be significantly improved.

A Single-Step Modified Block Hybrid Method for General Second-Order Ordinary Differential Equations

A multistep collocation approach is used to derive a single-step modified block hybrid method (MBHM) of order five for solving general second-order initial-value problems (IVPs) of ordinary differential equations (ODEs). The new method's basic convergence property is established, and its numerical accuracy is demonstrated using numerical examples from the literature. The new method outperforms similar methods in terms of accuracy, earning it a recommendation as a likely candidate for solving general second-order ODEs.

Bipartite Consensus for Second-Order Multiagent Systems With Matrix-Weighted Signed Network

The second-order scalar-weighted consensus problem of multiagent systems has been well explored. However, in some practical antagonistic interaction networks, the interdependencies of multidimensional states of the agents must be described by matrix coupling. In order to highlight the influence of matrix coupling in the antagonistic interaction network, we investigate the second-order matrix-weighted bipartite consensus problem on undirected structurally balanced signed networks. Under the proposed bipartite consensus protocol, an algebraic condition is obtained for achieving second-order bipartite consensus via utilizing matrix-valued Gauge transformation and stability theory. Then, using the obtained criteria, a more direct algebraic graph condition is given for reaching bipartite consensus. Besides, because of the existence of negative (positive) semidefinite connections, the matrix-weighted network may have clustering phenomena, which means that matrix weights play a critical role in achieving consensus. An algebraic graph condition for admitting cluster bipartite consensus is provided. By designing matrix weights in practical scenarios, the required number of clusters can be obtained. Finally, the theoretical results are verified by five simulation examples.

Exploring the Digital Revolution in Education in India during the COVID-19 Pandemic

Abstract One important response to COVID-19 was the intensification of the use of digital media to deliver education. However, the results are paradoxical, since the digital revolution did not lead to improvement of the social quality of teachers’ working circumstances. We analyze “internal” or subjective oriented constitutional and “external” or objective orientated conditional factors related to teachers that determine the adaptation of digitalization, taking a social quality perspective. Through a case study in the most advanced educational hub of India—Delhi—we find that the digital revolution helped India to address the first-order problems in digital transformation, namely concerning objective infrastructural facilities. The second-order problems, particularly changing the subjective belief structures of teachers related to the integration of technologies, appear to remain a challenge. As India has recently adopted a new education policy (2020), the findings of our study have significant relevance to improving the accessibility and utilization of digital technology in educational spaces.

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.