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Robustness of stability of time-varying index-1 DAEs

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Abstract

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations on the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples.

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Notes

  1. I thank the Associate Editor who handled this paper for pointing this out.

Abbreviations

\({\mathbb {N}},\ {\mathbb {N}}_0,\ {\mathbb {R}}_+\) :

The set of natural numbers \({\mathbb {N}}\), \({\mathbb {N}}_0={\mathbb {N}}\cup \{0\}\), \({\mathbb {R}}_+=[0,\infty )\)

\({{\mathrm{im}}}A, \ker A\) :

The image and kernel of the matrix \(A\in {\mathbb {R}}^{m\times n}\), resp.

\( {\mathbf {Gl}}_n({\mathbb {R}})\) :

The set of all invertible \(n\times n\) matrices over \({\mathbb {R}}\)

\(\Vert x\Vert \) :

\(:= \sqrt{x^{\top } x}\), the Euclidean norm of \(x \in {\mathbb {R}}^n\)

\(\Vert A\Vert \) :

\(:=\sup \left\{ \Vert Ax\Vert \left| \Vert x\Vert =1 \right. \right\} \), induced matrix norm of \(A \in {\mathbb {R}}^{n\times m}\)

\({\mathcal {C}}^k({\mathcal {I}}; {\mathcal { S}})\) :

The set of \(k\)-times continuously differentiable functions \(f:{\mathcal {I}}\rightarrow {\mathcal {S}}\) from a set \({\mathcal {I}}\subseteq {\mathbb {R}}\) to a vector space \({\mathcal { S}}\), \(k\in {\mathbb {N}}_0\)

\({\mathcal {B}}({\mathcal {I}}; {\mathcal {S}})\) :

The set of continuous and bounded functions \(f:{\mathcal {I}}\rightarrow {\mathcal {S}}\) from a set \({\mathcal {I}}\subseteq {\mathbb {R}}\) to a vector space \({\mathcal {S}}\)

\({1\!\!1}_{\mathcal {M}}(t)\) :

= \(\left\{ \begin{array}{ll} 1, &{}\quad \hbox {if}\ t\in {\mathcal {M}},\\ 0, &{} \quad \hbox {otherwise,}\end{array}\right. \) for \(t\in {\mathbb {R}}_+\) and \({\mathcal {M}}\subseteq {\mathbb {R}}_+\)

\({{\mathrm{dom}}}f\) :

The domain of the function \(f\)

\(\Vert f\Vert _\infty \) :

\(:=\sup \left\{ \Vert f(t)\Vert \left| t\in {{\mathrm{dom}}}f \right. \right\} \) the infinity norm of the function \(f\)

\(f\left. \right| _{{\mathcal {M}}}\) :

The restriction of the function \(f\) on a set \({\mathcal {M}}\subseteq {{\mathrm{dom}}}f\)

\(L^2({\mathcal {I}}; {\mathcal {S}})\) :

The set of measurable and square integrable functions \(f:{\mathcal {I}}\rightarrow {\mathcal {S}}\) from a set \({\mathcal {I}}\subseteq {\mathbb {R}}\) to a vector space \({\mathcal {S}}\)

\(\Vert f\Vert _{L^2[t_0,\infty )}\) :

\(:=\left( \int _{t_0}^\infty \Vert f(t)\Vert ^2\, \mathrm{\, d} t \,\right) ^{1/2}\) the \(L^2\)-norm of the function \(f\in L^2([t_0,\infty );{\mathcal {S}})\), \(t_0\in {\mathbb {R}}\)

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Acknowledgments

I am indebted to Achim Ilchmann (TU Ilmenau), Stephan Trenn (TU Kaiserslautern), and Volker Mehrmann (TU Berlin) for several constructive discussions. I also thank the three anonymous referees for their valuable comments which helped to improve the paper. This work was supported by the DFG grant Il25/9 and partially supported by the DAAD.

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  1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 , Hamburg, Germany

    Thomas Berger

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Berger, T. Robustness of stability of time-varying index-1 DAEs. Math. Control Signals Syst. 26, 403–433 (2014). https://doi.org/10.1007/s00498-013-0123-5

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