# Complete Controllability and Spectrum Assignment in Infinite
Dimensional Spaces

##
Aleksander Markus and
Vadim Olshevsky

The concept of complete controllability of a system plays an important role
in the theory of systems with control. A large number of conditions is known,
which are equivalent to complete controllability. Some of these are generalized to
infinite dimensional case.

In the present paper two further criteria for complete controllability in infinite
dimensional spaces are derived.
We also obtained a new proof for the recent Takahashi theorem on the equivalence
in Hilbert spaces of complete
controllability and the assignability of the spectrum.

Examples are presented which show that in Banach spaces the assignability of
the spectrum does not always follow from complete controllability. Properties which
are dual to complete controllability are
investigated in this paper as well. They can
be applied to the study of observability in infinite dimensional spaces. Finally,
the connections with the theory of analytic operator functions are indicated.

Vadim Olshevsky's Home page

Vadim Olshevsky
olshevsk@isl.stanford.edu
Last modified: Fri Sep 15 1995