Problem: Consider the following interval differential equation
dx
--(t) = [A] x(t) (1)
dt
where [A] is an interval matrix that is independent of t. Assume that it is
known that the system is stable for all matrix A in [A].
If A(t) is a t-varying matrix that stays inside [A] for all t. Is the system
dx
--(t) = A(t) x(t) (2)
dt
necessarily stable? If not, what are the conditions on the elements of [A]
that guarantee the stability of equation (2)?
Where could we find some references about this topic?
Luc Jaulin, LSS, Supélec, Plateau de Moulon, 91 192, Gif-sur-Yvette.
www.istia.univ-angers.fr/~jaulin/
It is not sufficient that all matrices in [A] are stable; it is not difficult to construct counterexamples of the form A(t)=Q(t) D Q(t)T where D is a marginally stable diagonal matrix and Q(t) a suitable family of rotations.
What is needed is that there is a single linear transformation S such that S-1[A]S has logarithmic norm <0, for some underlying norm. At least this seems to be the weakest known tractable sufficient condition. See, e.g., Corollary 4.5 in
A. Neumaier, Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory, Computing 52 (1994), 315-336. http://solon.cma.univie.ac.at/~neum/papers.html#ell, where also a quantitative enclosure of the solution is given under these conditions.
A simple way to proceed is to take for S a matrix that diagonalizes the midpoint of A, or in case of complex eigenvalues produces real diagonal blocks of the form
c s
-s c
The the symmetric part B of S-1[A]S will have a diagonal midpoint,
and if B is an H-matrix with negative diagonal entries, the
logarithmic 2-norm is < 0. This will always work if the width of [A]
is not large.
Arnold Neumaier
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