Stability regions

Stability regions

AAU

For a linear d-dimensional time-invariant dynamic equation

the stability region is given by

and determines the subset of the complex plane in which the spectrum of A is contained in order to guarantee exponential stability.

The classical cases are

Ordinary differential equations:

negative half plane

Difference equations:

unit circle with center -1

Hybrid equations:

Constant step-sizes:

continuous transition between the difference and differential equations case in the limit as h tends to 0.

Stability regions for dynamic equations on time scales

(cf. Pötzsche, Siegmund and Wirth: A spectral characterization of linear time-invariant systems on time scales, Discrete and Continuous Dynamical Systems (Series A) 9(5), 1223-1241, 2003)

Christian Pötzsche (Feb 2011)