# CANONICAL FORMS AND SOLVABLE SINGULAR SYSTEMS OF DIFFERENTIAL EQUATIONS

@article{Campbell1983CANONICALFA,
title={CANONICAL FORMS AND SOLVABLE SINGULAR SYSTEMS OF DIFFERENTIAL EQUATIONS},
author={Stephen L. Campbell and Linda R. Petzold},
journal={Siam Journal on Algebraic and Discrete Methods},
year={1983},
volume={4},
pages={517-521}
}
• Published 1983
• Mathematics
• Siam Journal on Algebraic and Discrete Methods
In this paper we investigate the relationship between solvability and the existence of canonical forms for the linear system of differential equations $E ( t ) x' ( t ) + F ( t ) x ( t ) = f ( t )$. We show that if E, F are analytic on the interval $[ 0 \,\, T ]$, then the differential equation is solvable if and only if it can be put into a certain canonical form. We give examples to show that this is not true if E, F are only differentiable.
111 Citations
On nilpotent singular systems
• Mathematics
• 2001
Abstract We derive some results for linear differential-algebraic equations (DAEs) of the form N(t) z ′(t)+ z (t)= h (t) , where N(t) is a smooth nilpotent matrix for all t concerned and such thatExpand
Feedback Elimination of Impulse Terms from the Solutions of Differential-Algebraic Equations
Abstract We consider a controlled linear system of differential-algebraic equations with infinitely differentiable coefficients that is allowed to have an arbitrarily high unsolvability index. It isExpand
An algorithm for the reduction of linear DAE
• Mathematics, Computer Science
• ISSAC '95
• 1995
This work studies linear Differential Algebraic Equations, DAE, with time varying coefficients, and proposes a new generalization of the global index and a definition for the singularities of the initial system. Expand
Systems of singular differential equations with pulse action
• Mathematics
• 2013
The paper deals with the singular systems of ordinary differential equations with impulsive action under the assumption that the considered systems can be reduced into the central canonical form. AnExpand
Characterization of Classes of Singular Linear Differential-Algebraic Equations
• Mathematics
• 2005
We study linear, possibly over- or under-determined, differentialalgebraic equations that have the same solution behavior as linear differential-algebraic equations with well-dened strangenessExpand
CLASSICAL AND GENERALIZED SOLUTIONS OF TIME-DEPENDENT LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS
• Mathematics
• 1996
Abstract A coordinate-free reduction procedure is developed for linear time-dependent differential-algebraic equations that transforms their solutions into solutions of smaller systems of ordinaryExpand
Canonical forms for linear differential-algebraic equations with variable coefficients
• Mathematics
• 1994
Abstract We give a new set of local characterizing quantities for the treatment of linear differential-algebraic equations with variable coefficients. This leads to new global invariances under whichExpand
ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS
• Mathematics
• 1984
In this paper we study the numerical solution of the differential/algebraic systems $F(t,y,y') = 0$. Many of these systems can be solved conveniently and economically using a range of ODE methods.Expand
On stability of time-varying linear differential-algebraic equations
• Mathematics, Computer Science
• Int. J. Control
• 2013
A stability theory for time-varying linear differential algebraic equations (DAEs) is developed, and Lyapunov’s direct method is derived as well as the converse of the stability theorems. Expand
Invariants, Canonical Forms, and Minimal Realizations for Singular Linear Multivariable Systems
• Mathematics
• 1987
Abstract In this paper, we investigate the invariants, canonical forms, and the minimal realizations оГ singular linear multivariable systems. Two new sets of invariants and its canonical forms haveExpand

#### References

SHOWING 1-10 OF 14 REFERENCES
On Singular Implicit Linear Dynamical Systems
We investigate properties of existence, unicity, representation, of the (causal) solutions of implicit linear systems (or “generalized systems”) when the underlying matrix pencil is singular. WeExpand
Differential/Algebraic Equations are not ODE's
This paper outlines a number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form ${\bf F}(t,{\bf y},{\bf y}') = {\bf 0}$.Expand
One canonical form for higher-index linear time-varying singular systems
One form of the singular systemAx′+Bx=f is considered. The analytic solution, perturbation, and numerical solution of this form are examined. A class of systems which may be transformed into thisExpand
Inversion of multivariable linear systems
A new algorithm for constructing an inverse of a multivariable linear dynamical system is presented. This algorithm, which is considerably more efficient than previous methods, also incorporates aExpand
A Method of Global Blockdiagonalization for Matrix-Valued Functions
Let $A(x)$ be an $n \times n$ analytic matrix function of the vector variable x. Let the eigenvalues of $A(x)$ belong to two disjoint sets for every fixed x. Then there exists an invertible analyticExpand
Index two linear time-varying singular systems o] differential equations, this Journal
• Index two linear time-varying singular systems o] differential equations, this Journal
• 1983
Differential equations are not ODE's
• SIAM J. Sci. Stat. Comp
• 1982
Differential systems and matrix pencils
• Proc. Conference on Matrix Pencils
• 1982