Bohl exponent for time-varying linear differential-algebraic equations

Abstract

We study stability of linear time-varying differential-algebraic equations (DAEs). The Bohl exponent is introduced and finiteness of the Bohl exponent is characterised, the equivalence of exponential stability and a negative Bohl exponent is shown and shift properties are derived. We also show that the Bohl exponent is invariant under the set of Bohl transformations. For the class of DAEs which possess a transition matrix introduced in this article, the Bohl exponent is exploited to characterise boundedness of solutions of a Cauchy problem and robustness of exponential stability.

Keywords:

• time-varying linear differential-algebraic equations
• transition matrix
• Bohl exponent
• Bohl transformation
• exponential stability
• robustness

Acknowledgements

I am indebted to Achim Ilchmann (Ilmenau University of Technology) and Stephan Trenn (University of Kaiserslautern) for various constructive discussions. This work was supported by DFG grant Il25/9.