**Posted Aug 9, 1996**

## Some Preliminary Results on Information State System Stability

### By J. W. Helton and M. R. James

Download postscript file

The purpose of this paper is to present some preliminary results
on the stability of the information state system.
The information state system underlies the (infinite
dimensional) dynamics of
an *H* controller
for a nonlinear system.
Thus it is important to understand its stability and the
structure of its
equilibrium points.
We analyse the important case
corresponding to the mixed sensitivity problem.
We prove the
existence of an
equilibrium information state,
convergence under very general conditions
to such an equilibrium state $p_e$ and
uniqueness of this state (up to an irrelevant constant).
In this case
the equilibrium $p_e$ is usually singular in the sense that
it takes on the value
$-\infty$ except on a low dimensional subset of its domain.
This meshes with the article
\cite{HJ2}
which analysed the effect of using $p_e$ to initialize the
information state controller and gave explicit formulas
which in many cases produce a dramatic
reduction in the amount of computation required to implement
the controller.
What this article suggests is that indeed $p_e$
is the only equilibrium initialization possible.