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Runge–Kutta methods are used to solve stiff systems of ordinary differential equations and differential-algebraic equations. The solution of such problems often exhibits order reduction, when, for prescribed accuracy, the actual order of a method is lower than its classical order, which inevitably increases computational costs. To avoid order reduction, the method has to have a sufficiently high stage order. However, methods with the most convenient and efficient implementation have a low stage order. Accordingly, a task of importance is to construct methods of low stage order that have properties of methods with a higher stage order. The construction of methods of this type is addressed in the present paper. Singly diagonally implicit and explicit methods and methods inverse to explicit ones are considered. Results of solving test problems are presented. [ABSTRACT FROM AUTHOR]

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems, we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics. [ABSTRACT FROM AUTHOR]

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices. [ABSTRACT FROM AUTHOR]

This paper investigates the production of nanoparticles via detonation. To extract valuable knowledge regarding this route, a phenomenological model of the process is developed and simulated. This framework integrates the mathematical description of the detonation with a model representing the particulate phenomena. The detonation process is simulated using a combination of a thermochemical code to determine the Chapman–Jouguet (C-J) conditions, coupled with an approximate spatially homogeneous model that describes the radial expansion of the detonation matrix. The conditions at the C-J point serve as initial conditions for the detonation dynamic model. The Mie–Grüneisen Equation of State (EoS) is used, with the "cold curve" represented by the Jones–Wilkins–Lee Equation of State. The particulate phenomena, representing the formation of metallic oxide nanoparticles from liquid droplets, are described by a Population Balance Equation (PBE) that accounts for the coalescence and coagulation mechanisms. The variables associated with detonation dynamics interact with the kernels of both phenomena. The numerical approach employed to handle the PBE relies on spatial discretization based on a fixed-pivot scheme. The dynamic solution of the models representing both processes is evolved with time using a Differential-Algebraic Equation (DAE) implicit solver. The strategy is applied to simulate the production of alumina nanoparticles from Ammonium Nitrate Fuel Oil aluminized emulsions. The results show good agreement with the literature and experience-based knowledge, demonstrating the tool's potential in advancing understanding of the detonation route. [ABSTRACT FROM AUTHOR]

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned. [ABSTRACT FROM AUTHOR]

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave's Advantage™ quantum annealer, Fujitsu's digital annealer V3, D-Wave's quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms. [ABSTRACT FROM AUTHOR]

During the fault period, a phase angle jump may occur at the stator or the point of common coupling, which will deteriorate the low-voltage ride-through (LVRT) characteristics of a doubly fed induction generator (DFIG). The existing LVRT studies focus on the impact of a voltage drop on DFIGs but often ignore that of a phase angle jump. The time-domain simulation is accurate in describing the response of a DFIG during the LVRT process, but it is time-consuming for a DFIG with the full-order model. In this paper, by using the voltage magnitude and phase angle of the stator or the point of common coupling as the inputs, and the state variables as the outputs, the transfer function of a DFIG is derived to analyze its response and find the LVRT measures against the voltage drop and, especially, the phase angle jump. Firstly, the differential-algebraic equations of the DFIG are linearized to propose their transfer function model. Secondly, considering its high-order characteristic, a model reduction method for the transfer function of the DFIG using the Schur decomposition is proposed, and the analytical expression of the output variables of the DFIG with the phase angle jump is derived by the inverse Laplace transformation to judge the necessity of the LVRT measures. Finally, the simulation results of the DFIG are provided to verify the accuracy of the transfer function model and its reduced-order form and validate the feasibility of the LVRT against the phase angle jump with the proposed models. [ABSTRACT FROM AUTHOR]

Due to their several attractive properties, BDF-type multistep methods are usually the method-of-choice for solving stiff initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). Recently, a class of BDF-type methods based on linear barycentric rational interpolants (LBRIs), referred to as RBDF methods, was introduced for solving ODEs. In the present paper, we are going to introduce a new family of LBRIs-based BDF-type formulas for the numerical solution of ODEs and DAEs with desirable stability properties and smaller error constants than those of the RBDF methods. Numerical experiments of the proposed methods on some well-known IVPs for ODEs and DAEs with index ≤ 3 illustrate the efficiency and capability of the methods in solving such problems. [ABSTRACT FROM AUTHOR]

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied. [ABSTRACT FROM AUTHOR]

The article deals with the problem of a dynamic model of the truck crane considering the elasticity of the rope and the ground. The dynamic model of a truck crane in the vertical plane has been established. When considering the oscillation of a truck crane, it is a dynamic system with a large number of degrees of freedom. This model will allow us to evaluate the dynamic response of the machine, analyze the vibration of the heavy object being lifted and lowered, and subsequently study stability and control problems. In this paper, the dynamic models of the cranes in the working plane have been established, built by Lagrange's multiplier Equations, which are systems of differential-algebraic Equations with the generalized coordinates of the machine's motions and the elastic coordinates. The Lagrange Equation provides a simple method for solving dynamic problems. The advantage of this Equation is that the form and number of Equations do not depend on the number of objects in the investigated system, nor do they depend on the way the objects move. The number of Lagrange Equations depends only on the number of degrees of freedom of the system. [ABSTRACT FROM AUTHOR]

This article emphasizes the significance of dynamic analysis in mechanical engineering, particularly in the context of multibody systems. By accurately modeling the interactions between interconnected mechanical components, engineers can optimize system design and reduce the need for physical prototyping. The article also highlights several research papers published in the journal Machines, covering topics such as cutting forces, Formula SAE cars, two-wheeled vehicles, and cable-driven parallel robots. These contributions demonstrate the wide-ranging applications of dynamic analysis in designing efficient and reliable mechanical systems in various industries. [Extracted from the article]

The paper examines entire solutions of differential equations of the Briot–Bouquet type. It is shown that (under some conditions satisfied by the polynomials and ) all entire transcendental solutions of such equations are quasipolynomials. [ABSTRACT FROM AUTHOR]

The presence of impermeant molecules within a cell can lead to an increase in cell volume through the influx of water driven by osmosis. This phenomenon is known as the Donnan (or Gibbs-Donnan) effect. Animal cells actively transport ions to counteract the Donnan effect and regulate their volume, actively pumping Na+ out and K+ into their cytosol using the Na+/K+ ATPase (NKA) pump. The pump-leak equations (PLEs) are a system of algebraic-differential equations to model the membrane potential, ion (Na+, K+, and Cl-), and water flux across the cell membrane, which provide insight into how the combination of passive ions fluxes and active transport contribute to stabilizing cell volume. Our broad objective is to provide analytical insight into the PLEs through three lines of investigation: (1) we show that the provision of impermeant extracellular molecules can stabilize the volume of a passive cell; (2) we demonstrate that the mathematical form of the NKA pump is not as important as the stoichiometry for cell stabilization; and (3) we investigate the interaction between the NKA pump and cation-chloride co-transporters (CCCs) on cell stabilization, showing that NCC can destabilize a cell while NKCC and KCC can stabilize it. We incorporate extracellular impermeant molecules, NKA pump, and CCCs into the PLEs and derive the exact formula for the steady states in terms of all the parameters. This analytical expression enables us to easily explore the effect of each of the system parameters on the existence and stability of the steady states. [ABSTRACT FROM AUTHOR]

In pursuing quick and precise progressive flooding simulations for decision-making support, the linearised method has emerged and undergone refinement in recent years, becoming a reliable tool, especially for onboard decision support. This study consolidates and enhances the modelling approach based on a system of differential-algebraic equations capable of accommodating compartments filled with floodwater. The system can be linearised to permit analytical solutions, facilitating the utilization of larger time increments compared to conventional solvers for differential equations. Performance enhancements are achieved through the implementation of an adaptive time-step mechanism during the integration process. Furthermore, here, a correction coefficient for opening areas is introduced to enable the accurate modelling of free outflow scenarios, thereby mitigating issues associated with the assumption of deeply submerged openings used in governing equations. Experimental validation is conducted to compare the method's efficacy against recent model-scale tests, specifically emphasising the improvements stemming from the correction for free outflow. [ABSTRACT FROM AUTHOR]

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs), in which the embedded optimization problem is solved by global optimization. This leads to differential inclusions for cases in which there are multiple global optimizers at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and reduce the order of convergence of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behaviour of the integrator. This allows the computation of discrete tangent and adjoint sensitivities for optimal control problems. [ABSTRACT FROM AUTHOR]

A system of equations with nonlinearity in the electric field potential and temperature is proposed for describing the heating of semiconductor elements on an electrical board with thermal and electrical breakdowns possibly arising over time. A method for numerical diagnostics of solution blow-up is considered. In the numerical analysis of the problem, the original system of partial differential equations is reduced to a differential-algebraic system, which is solved using a single-stage Rosenbrock scheme with complex coefficients. The blow-up of the exact solution is detected using an asymptotically sharp a posteriori error estimate obtained by computing approximate solutions on sequentially refined grids. The blow-up time is numerically estimated in the case of various initial conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study solvability and stability of stochastic differential-algebraic equations (SDAEs) with delay. It is difficult to investigate these properties for SDAEs with delay because of the singularity of the leading coefficient matrix. An index-ν concept is derived for solvability of these equations. Stability of SDAEs with delay is studied by using the method of Lyapunov functions and comparison principle. An example is given to illustrate the results. To the best of our knowledge, these results are novel for SDAEs with delay. [ABSTRACT FROM AUTHOR]

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology. [ABSTRACT FROM AUTHOR]

The article considers and discusses the comparison of interpolation and extrapolation methods of estimation of the boundary trajectory of electron beams propagated in ionized gas. All estimations have been computed using root-polynomial functions to numerically solve a differential-algebraic system of equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that in the case of solving a self-connected interpolation-extrapolation task, the average error of the beam radius estimation is generally smaller. This approach was especially effective in estimating the focal beam radius. An algorithm for solving self-connected interpolation-extrapolation tasks is given, and its efficiency is explained. Corresponding graphic dependencies are also given and analyzed. [ABSTRACT FROM AUTHOR]

Proton Exchange Membrane Fuel Cells are a favorite technology for decarbonizing the transportation sector. However, their large-scale democratization is hampered by their high cost compounded by their unsatisfactory lifespan. To anticipate potential degradation while keeping improving performance, it is essential to maintain an acceptable humidity range inside the cells, especially at the membrane level. However, membrane humidity level is not directly measurable, alternative techniques must be considered to recover this key variable. Here, we develop a real-time software sensor of the membrane water content at the fuel cell's heart. We build a model describing the membrane water balance, electrochemical behavior, and species mass balance. We then reduce the model and perform an Adaptive Extended Kalman Filter. We perform sensitivity analyses in both steady-state and transient conditions. We validate the filter on a "Worldwide Harmonized Light Vehicles Test Cycles" test procedure. Finally, we obtain a fast and accurate model-based software sensor. • Real-time capable software sensor for PEMFC membrane water content. • Observer based on membrane model accounting for most of major transport phenomena. • Adaptive Extended Kalman filter applied to differential-algebraic equations. • Sensitivity analysis of membrane water content shows a 5% coefficient of variation.

This paper proposes an improved expanding interior algorithm (EIA) to estimate the region of attraction (ROA) of power systems with wind power generation based on sum of squares (SOS) programming. An ordinary differential equation (ODE) model is derived for the doubly‐fed induction generator‐based wind turbine (DFIGWT), which is named as an enhanced synchronous‐generator‐mimicking (ESGM) model. The ESGM model bridges the gap between the requirement of an ODE model in ROA estimation and the conventional differential‐algebraic equation (DAE) model of the DFIGWT system. The ESGM model is able to accurately reflect the low frequency dynamics of the DFIGWT. Moreover, an improved EIA is designed to estimate the ROA based on SOS programming, which has higher efficiency than the existing ROA estimation algorithms based on SOS programming. It is able to adaptively search for the Lyapunov function and obtain an optimal estimation of the ROA in an iterative process. The accuracy and efficiency of this algorithm are verified in three test systems composed of DFIGWTs and synchronous generators (SGs). The morphological changes in the ROA of the test systems caused by the penetration of DFIGWT are examined. [ABSTRACT FROM AUTHOR]

In a finite-dimensional space, we consider a linear stochastic differential equation in Itô form with a singular constant matrix on the left-hand side. Taking into account various economic applications of such equations, they are classified as Leontief type equations, since under some additional assumptions, a deterministic analog of the equation in question describes the famous Leontief input–output balance model taking into account reserves. In the literature, these systems are more often called differential–algebraic or descriptor systems. In general, to study this type of equations, one needs higher-order derivatives of the right-hand side. This means that one must consider derivatives of the Wiener process, which exist in the generalized sense. In the previous papers, these equations were studied using the technique of Nelson mean derivatives of random processes, whose description does not require generalized functions. It is well known that mean derivatives depend on the -algebra used to find them. In the present paper, the study of this equation is carried out using mean derivatives with respect to a new -algebra that was not considered in the previous papers. [ABSTRACT FROM AUTHOR]

The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity and KYP inequalities is investigated. [ABSTRACT FROM AUTHOR]

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods. [ABSTRACT FROM AUTHOR]

A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed, and conditions are determined under which the absolute stability of the system can be proved with the use of such a functional. In the case of power-law nonlinearities, estimates for the rate of convergence of the solution to the origin are obtained. The stability of the corresponding system with parameter switching is analyzed. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained. [ABSTRACT FROM AUTHOR]

We present a straightforward way to solve a model predictive control problem for a power network system given as a nonlinear differential‐algebraic equation (DAE) in a distributed way using the consensus alternating directions method of multipliers (consensus ADMM) algorithm. While no convergence‐ or stability results are available for fully nonlinear DAE models, this gives unprecedented experimental evidence that power network systems of the presented structure allow to be controlled in this way, unlocking the numerous combined advantages of distributed and predictive control schemes in the context of energy distribution networks, as well as broadening the field of use for the consensus ADMM algorithm to nonlinear DAE models. [ABSTRACT FROM AUTHOR]

We investigate a stochastic algebraic-differential equation, modelling dynamically distorted signals in an electronic device with the presence of noise in the ingoing signal, on the basis of the theory of mean derivatives of stochastic processes. A new point here is that it is a second-order equation with mean derivatives where the second-order derivative is taken from the so-called Newton-Nelson equation of Nelson's stochastic mechanics (a version of quantum mechanics). This is important since it allows us to take into account not only the noise in the ingoing signal, but also the noise generated by quantum effects in the device. The equation is ill-defined at time t = 0 . Fixing an arbitrary time instant t 0 > 0 , we prove the existence of the process that satisfies the equation under consideration for t ≥ t 0 . [ABSTRACT FROM AUTHOR]

Transient stability preventive control (TSPC), a method to efficiently withstand the severe contingencies in a power system, is mathematically a transient stability constrained optimal power flow (TSC-OPF) issue, attempting to maintain the economical and secure dispatch of a power system via generation rescheduling. The traditional TSC-OPF issue incorporated with differential-algebraic equations (DAE) is time consumption and difficult to solve. Therefore, this paper proposes a new TSPC method driven by a naturally inspired optimization algorithm integrated with transient stability assessment. To avoid solving complex DAE, the stacking ensemble multilayer perceptron (SEMLP) is used in this research as a transient stability assessment (TSA) model and integrated into the optimization algorithm to replace transient stability constraints. Therefore, less time is spent on challenging calculations. Simultaneously, sensitivity analysis (SA) based on this TSA model determines the adjustment direction of the controllable generators set. The results of this SA can be utilized as prior knowledge for subsequent optimization algorithms, thus further reducing the time consumption process. In addition, a naturally inspired algorithm, Aptenodytes Forsteri Optimization (AFO), is introduced to find the best operating point with a near-optimal operational cost while ensuring power system stability. The accuracy and effectiveness of the method are verified on the IEEE 39-bus system and the IEEE 300-bus system. After the implementation of the proposed TSPC method, both systems can ensure transient stability under a given contingency. The test experiment using AFO driven by SEMLP and SA on the IEEE 39-bus system is completed in about 35 s, which is one-tenth of the time required by the time domain simulation method. [ABSTRACT FROM AUTHOR]

The energy transition relies on the integration of non-conventional renewable energy sources. These disruptive technological developments alter the functioning and operation of the electric power system. This paper examines the impacts of wind power on transient stability indicators of the power system, using an implicit formulation and the nine-bus test system. The research findings indicate that transient stability indicators are sensitive to the location and duration of faults. Furthermore, there is an observed trend of increasing maximum rotor speed deviation and oscillation duration, indicating reduced stability margins. [ABSTRACT FROM AUTHOR]

Challenges in the operation of power systems arise from several factors such as the interconnection of large power systems, integration of new energy sources and the increase in electrical energy demand. These challenges have required the development of fast and reliable tools for evaluating the operation of power systems. The load margin (LM) is an important index in evaluating the stability of power systems, but traditional methods for determining the LM consist of solving a set of differential-algebraic equations whose information may not always be available. Data-Driven techniques such as Artificial Neural Networks were developed to calculate and monitor LM, but may present unsatisfactory performance due to difficulty in generalization. Therefore, this article proposes a design method for Physics-Informed Neural Networks whose parameters will be tuned by bio-inspired algorithms in an optimization model. Physical knowledge regarding the operation of power systems is incorporated into the PINN training process. Case studies were carried out and discussed in the IEEE 68-bus system considering the N-1 criterion for disconnection of transmission lines. The PINN load margin results obtained by the proposed method showed lower error values for the Root Mean Square Error (RMSE), Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) indices than the traditional training Levenberg-Marquard method. [ABSTRACT FROM AUTHOR]

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As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10 − 6 , and the absolute errors for algebraic variables are maintained at 10 − 5 . Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems. [ABSTRACT FROM AUTHOR]

The object of research is the behavior of the load of the tower crane during the break of the sling. One of the most problematic areas is the safety of work and the prevention of emergency situations. Despite the presence of mandatory safety measures, during cargo transportation, one of the sling branches may be destroyed due to the presence of a dynamic component during the operation of the crane, or errors of the slinger when securing the cargo. Also, the presence of hidden internal or unnoticed defects in the sling construction itself cannot be ruled out. Also, one of the most problematic places is the chaotic fluctuations of the load, which negatively affect the stability of the crane and safety. The paper describes the case of the destruction of one of the branches of a two-rope sling during the transportation of a long product by a tower crane. The proposed method of cargo behavior analysis is based on the use of a dynamic description of cable system failure modes within the framework of setting and solving differential-algebraic equations. This makes it possible to more accurately describe the behavior of the cargo when the sling breaks. The obtained results show that the application of the proposed method makes it possible to bring the mathematical model of the two-link mathematical pendulum significantly closer to the actual mutual oscillations of the load during the sling break. This is due to the fact that the proposed method has a number of features, in particular, high sensitivity to changes in the behavior of the cargo and a quick reaction to a rope break. These results can be used in practice in the design and operation of tower cranes. Thanks to the application of the proposed method, it is possible to obtain accurate values of cargo behavior indicators and timely detection of a rope break. Compared to similar known methods, this method has such advantages as high efficiency, reliability and safety of operation. [ABSTRACT FROM AUTHOR]

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C 2 n . We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly. [ABSTRACT FROM AUTHOR]

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted. [ABSTRACT FROM AUTHOR]

In this paper, a stage structured predator-prey model with Holling type II functional response is formulated, considering the juvenile prey as the favorite food for the generalist predator. Further, the existence of predators is ensured by the sufficient amount of alternative food available in the habitat. The proportional harvesting of the adult prey is incorporated with the assumption that only the adult prey are of economic worth. The existence and local stability of the distinct equilibrium points of the system are investigated. The bifurcation from origin to predator free equilibrium state is obtained for the bifurcation parameter-effort on harvesting. The occurrence of Hopf bifurcation about the interior equilibrium state is established for arbitrary model parameters and the supercritical nature of this bifurcation is proved by fixing these parametric values. An algebraic equation is included to this modified model to analyze the economic benefits resulted from the harvesting of adult prey. The singularity-induced bifurcation (SIB) about the coexisting equilibrium state of differential-algebraic system is deduced along the parameter v at v = 0 , v being the profit/loss due to harvesting. The state feedback controller is recommended to eliminate the SIB about the coexisting equilibrium state for the differential algebraic system. Adopting an appropriate feedback control would ensure the stability of co-existence interior equilibrium state along with economic profit from harvesting. Numerical examples are used to elaborate the analytical results obtained. [ABSTRACT FROM AUTHOR]

A problem of solvability with parameter for a differential-algebraic equation is considered. For solving the problem is applied Weierstrass canonical form. Problem is reduced to an initial value problem with parameter for differential equations. Conditions for the existence and uniqueness of the problem with parameter for differential-algebraic equations are established. [ABSTRACT FROM AUTHOR]

In this article, the problem for a differential-algebraic equation with a significant loads is studied. Unlike previously studied problems for differential equations with a significant loads, in the considered equation, there is a matrix in the left part with a derivative that is not invertible. Therefore, the system of equations includes both differential and algebraic equations. To solve the problem, we propose a modification of the Dzhumabaev’s parametrization method. The considered problem is reduced to a parametric problem for the differential-algebraic equation with significant loads. We apply the Weierstrass canonical form to this problem. We obtain parametric initial value problem for a differential equations and an algebraic equations with a significant loads. The solvability conditions for the considered problem are established.

This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration. [ABSTRACT FROM AUTHOR]