# [SciPy-user] numerical minimization of the spectral abscissa

Nils Wagner nwagner at mecha.uni-stuttgart.de
Wed Jul 21 02:24:20 CDT 2004

Dear experts,

Let us consider a dynamical system described by

\dot{x} = A(D) x

where A is a linear operator that depends on the
distribution of dissipative material encoded by
D. We imagine that when D=0 that the system
is conservative, i.e. the eigenvalues of A(0)
are purely imaginary. The obejctive is to choose
D in order that the spectrum of A(D) is moved
as far as possible into the left half-plane.
If \sigma(D) denotes the spectrum of A(D)
and \Re returns the real part of a complex number,
then our objective is to minimize the spectral
abscissa

\omega(D) \equiv max{\Re \lambda : \lambda \in \sigma(D)}

Can I solve such problems with the current version of scipy ?

A(D) is given by

[ zeros(n,n) , identity(n);
-M^{-1} K , -M^{-1} D]

the damping matrix is given by

D = G^T diag(d) G

where the geometry matrix is given by

G = -identity(n)+diag(ones(n-1),-1)

The stiffness matrix is given by

K = G^T diag(k) G

The damping parameters

d = array(([d1,d2,\dots,dn]))

should be optimized

Any pointer to this problem would be appreciated.

Nils