Consistent initialization and perturbation analysis for abstract differential-algebraic equations. (English) Zbl 1167.34022
The author studies differential algebraic equations of the form
\[ Ex'=Ax+f, \]
where \(E\) and \(A\) are infinite dimensional linear operators on certain Hilbert spaces. These equations are often called abstract differential algebraic equations (ADAEs) or, if they arise from coupling partial differential equation with classical differential equations and algebraic constraints, partial differential algebraic equations (PDAEs). One main result is a sufficient conditions for the consistency of an initial value. To check this condition it is assumed that the PDAE can be transformed into a certain decoupling form. Furthermore, the author shows the continuous dependence of solutions with respect to certain norms of the (consistent) initial values and the inhomogeneities. The results are then applied to an illustrative example of an electrical circuit with a transmission line. The paper is well structured and nicely written.
\[ Ex'=Ax+f, \]
where \(E\) and \(A\) are infinite dimensional linear operators on certain Hilbert spaces. These equations are often called abstract differential algebraic equations (ADAEs) or, if they arise from coupling partial differential equation with classical differential equations and algebraic constraints, partial differential algebraic equations (PDAEs). One main result is a sufficient conditions for the consistency of an initial value. To check this condition it is assumed that the PDAE can be transformed into a certain decoupling form. Furthermore, the author shows the continuous dependence of solutions with respect to certain norms of the (consistent) initial values and the inhomogeneities. The results are then applied to an illustrative example of an electrical circuit with a transmission line. The paper is well structured and nicely written.
Reviewer: Stephan Trenn (Ilmenau)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 34D10 | Perturbations of ordinary differential equations |
| 94C05 | Analytic circuit theory |
Keywords:
Abstract differential-algebraic equations; Infinite-dimensional linear systems theory; Consistent initialization; Perturbation analysis; Partial differential-algebraic equationsReferences:
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