On stability of time-varying linear differential-algebraic equations

Abstract

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Well-known stability concepts of ordinary differential equations are generalised to DAEs and characterised. Lyapunov’s direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs, which are transferable into standard canonical form; in this case the existence of the generalised transition matrix is exploited.

Keywords:

• time-varying linear differential algebraic equations
• exponential stability
• Lyapunov’s direct method
• Lyapunov equation
• Lyapunov function
• standard canonical form

Acknowledgement

We are indebted to our colleagues Roswitha März (Humboldt University Berlin) and Eugene P. Ryan (University of Bath) for some constructive discussions. This work was supported by DFG grant IL 25/9.