Linear Differential-algebraic Equations Research Articles

Solvability Analysis of high-order Linear Differential-Algebraic Equations with time-varying coefficients

In this paper, we study the solvability analysis of arbitrarily high-order linear differential-algebraic equations (DAEs) with time-varying coefficients, using the algebraic-behavior approach. We propose a concept of strangeness-index and construct condensed forms for high-order linear DAEs. We also discuss other structural properties like the existence and uniqueness of a solution, consistency and smoothness requirements for an initial vector and for an inhomogeneity. Our work extends the algebraic approach for DAEs and combines this approach with the behavior approach to establish a reformulation algorithm that reveals an underlying ordinary differential equation (ODE) and all hidden constraints in the DAE. This direct treatment of the system addresses the limitations of the classical approach to transforming the system into a first-order DAE. We illustrate our theoretical results with applications in mechanical systems and electrical circuits. This comprehensive study into general high-order systems is a natural progression from the extensive study of first and second-order DAEs.

Application of Adomian Decomposition Method for Solving Helmholtz Equation

The Adomian decomposition method (ADM) is effective for solving linear and nonlinear ordinary and partial differential equations, integral equations, algebraic equations and integro-differential equations and it leads to an analytical solution in the form of an infinite power series. We will solve the Helmholtz equation in two dimensional by the Adomian decomposition method.
 Key words: Application of Adomian Decomposition Method, Helmholtz Equation, Adomian Decomposition Method. 

Impulsive Stability for Positive Linear Differential-Algebraic Equations With Time-Varying Delay

This paper deals with the impulsive stability problem for positive linear differential-algebraic equations (DAEs) with a time-varying delay. First, a sufficient condition for the positivity of the considered system is addressed. Then, a new exponential stability criterion for positive linear DAEs with a time-varying delay is obtained based on a suitable change of variables. According to our results, a stable impulse-free system can keep its original stability property under certain destabilizing impulsive perturbations. It should be noted that it is the first time the exponential stability results for impulsive positive linear DAEs are given. For this purpose, delay-range dependent sufficient conditions for such systems are stated in the paper. An additional numerical example is provided to demonstrate the results' efficacy.

On the spectral theory of linear differential-algebraic equations with periodic coefficients

On the spectral theory of linear differential-algebraic equations with periodic coefficients

On impulse-free solutions and stability of switched nonlinear differential–algebraic equations

In this paper, we investigate solutions and stability properties of switched nonlinear differential–algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs.

Canonical Subspaces of Linear Time-Varying Differential-Algebraic Equations and Their Usefulness for Formulating Accurate Initial Conditions

Accurate initial conditions have the task of precisely capturing and fixing the free integration constants of the flow considered. This is trivial for regular ordinary differential equations, but a complex problem for differential-algebraic equations (DAEs) because, for the latter, these free constants are hidden in the flow. We deal with linear time-varying DAEs and obtain an accurate initial condition by means of applying both a reduction technique and a projector based analysis. The highlighting of two canonical subspaces, the flow-subspace and its canonical complement, plays a special role. In order to be able to apply different DAE concepts simultaneously, we first show that the very different looking rank conditions on which the regularity notions of the different concepts (elimination of unknowns, reduction, dissection, strangeness, and tractability) are based are de facto consistent. This allows an understandingof regularity independent of the methods.

Linear port-Hamiltonian DAE systems revisited

Port-Hamiltonian systems theory provides a systematic methodology for the modeling, simulation and control of multi-physics systems. The incorporation of algebraic constraints has led to a multitude of definitions of port-Hamiltonian differential–algebraic equations (DAE) systems in the literature. This paper presents extensions of results obtained in Gernandt et al. (2021); Mehrmann and van der Schaft (2023) in the context of maximally monotone structures, and shows that any such structure can be written as the composition of a Dirac and a resistive structure. This yields an alternative, but equivalent, definition of linear port-Hamiltonian DAE systems with certain advantages. In particular, it leads to simpler coordinate representations, as well as to explicit expressions for the associated transfer functions.

Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

On Singular Points of Linear Differential-Algebraic Equations with Perturbations in the Form of Integral Operators

On Singular Points of Linear Differential-Algebraic Equations with Perturbations in the Form of Integral Operators

Model Reduction of Linear Differential–Algebraic Equation Systems Using Constructed Spectral Projectors

This contribution introduces linear differential-algebraic equation (DAE) systems and provides the explicit construction steps of spectral projectors for DAEs with index 1 or 2. Further, an interpolatory projection-based model order reduction technique using spectral projectors is applied to an eddy current problem as well as an electrical circuit to compute a reduced order model.

The application of fuzzy transform method to the initial value problems of linear differential–algebraic equations

The application of fuzzy transform method to the initial value problems of linear differential–algebraic equations

Upper central exponent of linear stochastic differential algebraic equations of index 1

The current study was carried out to assess the antioxidant expression of the seed’s black-testa peanut cultivar namely CNC1 in the artifical drought condition. The oxidative stress with a burst of O2 .- and H2O2 in peanut leaves resulted the obviously cellular membrane damage that was illustrated by an increase of lipid peroxidation. A high induced activity of enzymatic antioxidants such as SOD, CAT and POX would detoxify the excess of O2 .- and H2O2. An increase in content of amino acid proline protects the stressed cells by adjusting intercellular osmotic potential. The antioxidative expression of CNC1 plants was similar to that in L20 cutivar - a drought tolerant cutivar of peanut.

Relative genericity of controllablity and stabilizability for differential-algebraic systems

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E,A,B)in mathbb {R}^{ell times }times mathbb {R}^{ell times n}times mathbb {R}^{ell times m}, while for relative genericity we allow the restricted set Sigma _{ell ,n,m}^{le r} := {(E,A,B)in mathbb {R}^{ell times n}times mathbb {R}^{ell times n}times mathbb {R}^{ell times m} ,big vert ,textrm{rk},_{mathbb {R}} E le r}, where rin {mathbb {N}}. Our main results are characterizations of generic controllability and generic stabilizability in Sigma _{ell ,n,m}^{le r} in terms of the numbers ell , n, m, r.

An Ultraweak Variational Method for Parameterized Linear Differential-Algebraic Equations

We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations with respect to the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive a certified detailed discretization which provides an approximate solution in an ultraweak setting as well as for model reduction with respect to time in the spirit of the Reduced Basis Method. A computable sharp error bound is derived. Numerical experiments are presented that show that this method yields a significant reduction and can be combined with well-known system theoretic methods such as Balanced Truncation to reduce the size of the DAE.

Differential Equivalence for Linear Differential Algebraic Equations

Differential-algebraic equations (DAEs) are a widespread dynamical model that describes continuously evolving quantities defined with differential equations, subject to constraints expressed through algebraic relationships. As such, DAEs arise in many fields ranging from physics, chemistry, and engineering. In this article, we focus on linear DAEs, and develop a theory for their minimization up to an equivalence relation. We present differential equivalence, which relates DAE variables that have equal solutions at all time points (thus requiring them to start with equal initial conditions) and extends the line of research on bisimulations developed for Markov chains and ordinary differential equations. We apply our results to the electrical engineering domain, showing that differential equivalence can explain invariances in certain networks as well as analyze DAEs, which could not be originally treated due to their size.

On the numerical solution of second-order stiff linear differential-algebraic equations

This article addresses systems of linear ordinary differential equations with an identically degenerate matrix in the main part. Such formulations of problems in literature are usually called differential-algebraic equations. In this work, attention is paid to the problems of the second order. Basing on the theory of matrix pencils and polynomials, sufficient conditions for existence and uniqueness of the equations’ solution are given. To solve them numerically, authors investigate a multistep method and its version based on a reformulated notation of the original problem. This representation makes it possible to construct methods whose coefficient matrices can be calculated at previous points. This approach has delivered good results in numerical solution of first-order differential-algebraic equations that contain stiff and rapidly oscillating components and have singular matrix pencil. The stability of proposed numerical algorithm is investigated for the well-known test equation. It is shown that this difference scheme has the first order of convergence. Numerical calculations of the model problem are presented.

Functional Approach for Solving Reduced Order of Index-Four Hessenberg Differential-Algebraic Control System

This research investigates differential-algebraic equations with higher index (index four). Specifically, a functional analytic approach is proposed to find the solution of (index four) Hessenberg differential-algebraic equations (DAEs). The approximate solution of the proposed functional approach for a suitable separable Hilbert space is obtained with the help of the direct optimisation technique and Ritz basis functions method. Illustrations have been ranked from an 8 × 8 test system of index-four Hessenberg linear DAEs to a 4 × 4 DAE of rotating masses as well as an 8 × 8 differential-algebraic generator model, where it reformulated index-four linear Hessenberg DAEs. Their approximate solutions were obtained using the present approach with comparisons. The numerical results demonstrate the simplicity of the proposed approach and express suitable accuracy and efficiency.

Parameterization technique for solving variational formulation of index three linear time-varying singular control systems

This paper deals with finding an approximate solution to an index three time-varying linear differential algebraic control equations based on the theory of variational formulation. The critical points of the formulated variational functional subject to the differential- algebraic equation with the class of consistent initial conditions are shown to be the solution of the proposed index-three DAEs with control problem (singular control systems) and vise- versa. The given variational problem is also transformed from indirect method into direct method using generalized Ritz bases technique. From the numerical illustrations results, the efficiency and very good accuracy as well as the simplicity of the present approach are obtained.

Jacobi Neural Network Method for Solving Linear Differential-Algebraic Equations with Variable Coefficients

Jacobi Neural Network Method for Solving Linear Differential-Algebraic Equations with Variable Coefficients

Sensitivity analysis of random linear differential–algebraic equations using system norms

We consider linear dynamical systems composed of differential–algebraic equations (DAEs), where a quantity of interest (QoI) is assigned as output. Physical parameters of a system are modelled as random variables to quantify uncertainty, and we investigate a variance-based sensitivity analysis of the random QoI. Based on expansions via generalised polynomial chaos, the stochastic Galerkin method yields a new deterministic system of DAEs of high dimension. We define sensitivity measures by system norms, i.e., the H∞-norm of the transfer function associated with the Galerkin system for different combinations of outputs. To ameliorate the enormous computational effort required to compute norms of high-dimensional systems, we apply balanced truncation, a particular method of model order reduction (MOR), to obtain a low-dimensional linear dynamical system that produces approximations of system norms. MOR of DAEs is more sophisticated in comparison to systems of ordinary differential equations. We show an a priori error bound for the sensitivity measures satisfied by the MOR method. Numerical results are presented for two stochastic models given by DAEs.