A Note on Weak Stabilizability of Contraction Semigroups,
CALIFORNIA UNIV LOS ANGELES DEPT OF SYSTEM SCIENCE
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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.
- Theoretical Mathematics