Robust Controllability of Linear Differential-Algebraic Equations with Unstructured Uncertainty

We consider the linear stationary systems of ordinary differential equations (ODEs) that are unsolvedwith respect to the derivative of the unknown vector-function and degenerate identically in the domain of definition. These systems are usually called differential-algebraic equations (DAEs). The measure of how a system of DAEs is unsolved with respect to the derivative is an integer which is called the index of the system of DAEs. The analysis is carried out under the assumption of existence of a structural form with separated differential and algebraic subsystems. We investigate the robust controllability of these systems (controllability in the conditions of uncertainty). The sufficient conditions for the robust complete and R-controllability of a system of DAEs with the indices 1 and 2 are obtained.     

Distributional solution theory for linear DAEs

A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but also having optimal order of convergence. The new class of (s,s)$(s,s)$-Gauss–Lobatto specialized partitioned additive Runge–Kutta (SPARK) methods uses greatly the structure of the DAEs and possesses all desired properties. In the second part we propose a unified approach for the solution of ordinary differential equations (ODEs) mixing analytical solutions and numerical approximations. The basic idea is to consider local models which can be solved efficiently, for example analytically, and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction. In order to preserve also symmetry we define the new class of symmetrized Runge–Kutta methods with local model (SRKLM).     

A Test Set for stiff Initial Value Problem Solvers in the open source software R: Package deTestSet

In this paper we present the R package deTestSet that includes challenging test problems written as ordinary differential equations (ODEs), differential algebraic equations (DAEs) of index up to 3 and implicit differential equations (IDEs). In addition it includes 6 new codes to solve initial value problems (IVPs). The R package is derived from the Test Set for Initial Value Problem Solvers available at http://www.dm.uniba.it/~testset which includes documentation of the test problems, experimental results from a number of proven solvers, and Fortran subroutines providing a common interface to the defining problem functions. Many of these facilities are now available in the R package deTestSet, which comprises an R interface to the test problems and to most of the Fortran solvers. The package deTestSet is free software which is distributed under the GNU General Public License, as part of the R open source software project.     

Numerical solution and structural analysis of two-dimensional integral-algebraic equations

The υ-smoothing property of a one-dimensional Volterra integral operator and some projectors (Liang and Brumer, SIAM J. Numer. Anal. 51, 2238–2259 (2013)) are extended for two-dimensional integral-algebraic equations (TIAEs). Using these concepts, we decompose the given general TIAEs into mixed systems of two-dimensional Volterra integral equations (TVIEs) consisting of second- and first-kind TVIEs. Numerical technique based on the Chebyshev polynomial collocation methods is presented for the solution of the mixed TVIE system. Global convergence results are established and the performance of the numerical scheme is illustrated by means of some test problems.     

On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

Controllability characterization of switched DAEs

We study controllability of switched differential algebraic equations (switched DAEs) with fixed switching signal. Based on a behavioral definition of controllability we are able to establish a controllability characterization that takes into account possible jumps and impulses induced by the switches. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multistep methods for differential algebraic equations

Multistep methods for the differential/algebraic equations (DAEs) in the form of

Controllability of switched DAEs: The single switch case

We study controllability of switched DAEs and formulate a definition of controllability in the behavioral sense. In order to characterize controllability for switched DAEs we first present new characterizations of controllability of non-switched DAEs based on the Wong-sequences. Afterwards a first result concerning the single-switch case is presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear Index-1 DAEs: Regular and Singular Problems

Several features and interrelations of projector methods and reduction techniques for the analysis of linear time-varying differential-algebraic equations (DAEs) are addressed in this work. The application of both procedures to regular index-1 problems is reviewed, leading to some new results which extend the scope of reduction techniques through a projector approach. Certain singular points are well accommodated by reduction methods; the projector framework is adapted in this paper to handle (not necessarily isolated) singularities in an index-1 context. The inherent problem can be described in terms of a scalarly implicit ODE with continuous operators, in which the leading coefficient function does not depend on the choice of projectors. The nice properties of projectors concerning smoothness assumptions are carried over to the singular setting. In analytic problems, the kind of singularity arising in the scalarly implicit inherent ODE is also proved independent of the choice of projectors. The discussion is driven by a simple example coming from electrical circuit theory. Higher index cases and index transitions are the scope of future research.     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

On One class of Hermite projectors

We conjecture that every ideal projector on \({\mathbb {C}}\left[ x_1,\ldots ,x_d\right] \) whose kernel is generated by precisely d polynomials is Hermite (i.e., the limit of Lagrange interpolation projectors). We validate this conjecture in case the d generators of the kernel have no roots at infinity.     

Stabilization of switched DAEs via fast switching

Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

The Laplace-Beltrami problem Δ_Γψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderón projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.     

On the Limits of Lagrange Projectors

This article addresses a question of Carl de Boor (In: Constructive theory of functions, Varna 2005, pp. 51–63, Marin Drinov Academic, Sofia, [2006]): What ideal projectors are the limits of Lagrange projectors? The results of this paper answer the question in the sense that for every ideal projector P, we prescribe finitely many computations that determine whether the projector P is a limit of Lagrange projectors.     

Specialized Runge-Kutta methods for index differential-algebraic equations

We consider the numerical solution of systems of semi-explicit index differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods specialized Runge-Kutta methods for index DAEs (SRK-DAE).

     

On the strength of a weak variant of the axiom of counting

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of , and that proves the consistency of .     

Hodograph‐Type Transformations for Linearization of Systems of Nonlinear Diffusion Equations

In this paper, we study the problem of linearization of nonlinear systems of equations which is a potential form of systems of nonlinear diffusion equations We construct a class of point transformations of the form which connects the nonlinear systems with linear systems of equations . These point transformations are hodograph‐type transformations which have the property that the new independent and dependent variables depend, respectively, on the old dependent and independent variables. All systems of equations admitting such transformations are completely classified.     

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.