Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits

We study a semilinear differential-algebraic equation (DAE) with the focus on the Lagrange stability (instability). The conditions for the existence and uniqueness of global solutions (a solution exists on an infinite interval) of the Cauchy problem, as well as conditions of the boundedness of the global solutions, are obtained. Furthermore, the obtained conditions for the Lagrange stability of the semilinear DAE guarantee that every its solution is global and bounded, and, in contrast to theorems on the Lyapunov stability, allow to prove the existence and uniqueness of global solutions regardless of the presence and the number of equilibrium points. We also obtain the conditions of the existence and uniqueness of solutions with a finite escape time (a solution exists on a finite interval and is unbounded, i.e., is Lagrange unstable) for the Cauchy problem. We do not use constraints of a global Lipschitz condition type, that allows to use the work results efficiently in practical applications. The mathematical model of a radio engineering filter with nonlinear elements is studied as an application. The numerical analysis of the model verifies the results of theoretical investigations

Solvability conditions, consistency and weak consistency for linear differential-algebraic equations and time-invariant singular systems: The general case

Control Systems;operations research

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Index Reduction for Differential-Algebraic Equations with Mixed Matrices

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. The difficulty in solving numerically a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and S\"{o}derlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This paper presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants, and transforms a DAE into an equivalent DAE to which Mattsson--S\"{o}derlind's index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen, Norway, June 201

Dynamic state reconciliation and model-based fault detection for chemical processes

In this paper, we present a method for the fault detection based on the residual generation. The main idea is to reconstruct the outputs of the system from the measurements using the extended Kalman filter. The estimations are compared to the values of the reference model and so, deviations are interpreted as possible faults. The reference model is simulated by the dynamic hybrid simulator, PrODHyS. The use of this method is illustrated through an application in the field of chemical processe