# Accession Number:

## ADA043852

# Title:

## A Note on Weak Stabilizability of Contraction Semigroups,

# Descriptive Note:

# Corporate Author:

## CALIFORNIA UNIV LOS ANGELES DEPT OF SYSTEM SCIENCE

# Personal Author(s):

# Report Date:

## 1977-07-01

# Pagination or Media Count:

## 17.0

# Abstract:

A recent result on weak stabilizability is that the system x sub n Ax Bu, where A is the infinitesimal generator of a contraction semigroup over a Hilbert Space H, and B is linear bounded is weakly stabilizable if i A has a compact resolvent and ii A,B is approximately controllable. In this note, we show that condition i is superfluous and ii can be weakened to iii the weakly unstable states are approximately controllable, which actually turns out to be a necessary condition. Indeed, if i is verified, iii is necessary and sufficient for strong stabilizability. Moreover, a simple, direct proof is given using semigroup theoretic techniques.

# Descriptors:

# Subject Categories:

- Theoretical Mathematics