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# Interval FAQ Luc Jaulin -- Stability of x'(t) = A(t) x(t)?

From Luc Jaulin (jaulin@lss.supelec.fr), LSS, Supe'lec, France:

## What are the conditions on the elements of [A] that guarantee the stability of x'(t) = A(t) x(t) ?

Veit Hagenmeyer and myself would like to ask you kindly whether you know the answer or references to the following problem.

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/

### Response by Arnold Neumaier:

(neum@cma.univie.ac.at), University of Vienna.

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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