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)