[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
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