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Adjoint Pairs of Differential-Algebraic Equations and Their Lyapunov Exponents

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Abstract

This paper is devoted to the analysis of adjoint pairs of regular differential-algebraic equations with arbitrarily high tractability index. We consider both standard form DAEs and DAEs with properly involved derivative. We introduce the notion of factorization-adjoint pairs and show their common structure including index and characteristic values. We precisely describe the relations between the so-called inherent explicit regular ODE (IERODE) and the essential underlying ODEs (EUODEs) of a regular DAE. We prove that among the EUODEs of an adjoint pair of regular DAEs there are always those which are adjoint to each other. Moreover, we extend the Lyapunov exponent theory to DAEs with arbitrarily high index and establish the general class of DAEs being regular in Lyapunov’s sense. The Perron identity which is well known in the ODE theory does not hold in general for adjoint pairs of Lyapunov regular DAEs. We establish criteria for the Perron identity to be valid. Examples are also given for illustrating the new results.

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Notes

  1. In contrast, when looking for adjoint operators of the operator representing the given ODE, one supposes a compact interval and, additionally, boundary conditions.

  2. In functional-analytic terms, the operator representing the DAE (13) with properly stated leading term is the closure of operator of the standard DAE (9).

  3. Correspondingly, the operator L representing a linear DAE on a compact interval \({\mathcal {I}}\) does not at all depend on the special proper factorization of the leading term, and the same is true for the adjoint operator \(L^{*}\), [26].

Abbreviations

\({\mathbb {K}}\) :

Set of real numbers \({\mathbb {R}}\) and set of complex numbers \({\mathbb {C}}\)

\({\mathcal {L}}({\mathbb {K}}^s,{\mathbb {K}}^n)\) :

Set of \({\mathbb {K}}\)-valued \(n\times s \)—matrices and linear operators from \({\mathbb {K}}^s\) to \({\mathbb {K}}^n\)

\({\mathcal {C}}({\mathcal {I}}, X)\) :

Space of continuous functions mapping \({\mathcal {I}}\) into X

\({\mathcal {C}}^1({\mathcal {I}}, X)\) :

Space of continuously differentiable functions mapping \({\mathcal {I}}\) into X

\({\mathcal {C}}^1_M({\mathcal {I}}, X)\) :

\(\{x\in {\mathcal {C}}({\mathcal {I}}, X):Mx\in {\mathcal {C}}^1({\mathcal {I}}, Y)\), with \(M\in {\mathcal {L}}(X,Y)\}\)

\(K^*\) :

Adjoint matrix

\(K^-\) :

Generalized inverse, \(KK^-K=K\), \(K^-KK^-=K^-\)

\(K^+\) :

Moore–Penrose inverse

\(K^{*\,-}\) :

\([K^{*}]^{-}\)

\(K^{-\,*}\) :

\([K^{-}]^{*}\)

\(K^{-\,*\,-}\) :

\([[K^{-}]^{*}]^{-}\)

\(\mathrm{ker}\,K\) :

Nullspace (kernel) of K

\(\mathrm{im}\,K\) :

Image (range) of K

\(\langle \cdot ,\cdot \rangle \) :

Scalar product in \({\mathbb {K}}^m\)

\((\cdot ,\cdot )\) :

Scalar product in function spaces

\(|\cdot |\) :

Vector and matrix norms

\(\Vert \cdot \Vert \) :

Norms on function spaces, operator norms

\(\oplus \) :

Direct sum

\(\chi ^u(f)\) :

The upper Lyapunov characteristic exponent of f

\(\chi ^l(f)\) :

The lower Lyapunov characteristic exponent of f

DAE:

Differential-algebraic equation

ODE:

Ordinary differential equation

IVP:

Initial value problem

IERODE:

Inherent explicit regular ODE

EUODE:

Essential underlying ODE

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Acknowledgments

The authors would like to thank an anonymous referee who read the manuscript very carefully and gave useful comments that led to the improvement of the paper. V.H. Linh’s research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) Project 101.02-2014.05.

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Authors and Affiliations

  1. Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    Vu Hoang Linh

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Roswitha März

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  1. Vu Hoang Linh
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  2. Roswitha März
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Correspondence to Vu Hoang Linh.

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Dedicated to the memory of Katalin Balla (1947–2005).

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Linh, V.H., März, R. Adjoint Pairs of Differential-Algebraic Equations and Their Lyapunov Exponents. J Dyn Diff Equat 29, 655–684 (2017). https://doi.org/10.1007/s10884-015-9474-6

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