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}
}
  • S. Campbell, L. Petzold
  • Published 1 December 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. 
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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. We…
Differential/Algebraic Equations are not ODE's
TLDR
A number of difficulties which can arise when numerical methods are used to solve systems of differential/algebraic equations of the form ${\bf F(t, t, y, y') = {\bf 0}$.
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 this…
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 a…
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 analytic…
Singular systems of differential equations II
Singular Systems of Differential Equations
Relative inverses of meromorphic operator functions and associated holomorphic projection functions
Differential equations are not ODE's
  • SIAM J. Sci. Stat. Comp
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Differential systems and matrix pencils
  • Proc. Conference on Matrix Pencils
  • 1982
...
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