On contractive semigroups and uniform asymptotic stability

Abstract

Th is paper call s attention to the equ ivalence be twee n two we ll-known ma th e ma t.i ca l ideas : cont .. ac ti on mappings (in the sense of Banach) and asy mptoti c s tab ili t. y. The equiva le nce is form a li zed by de fin ing a Aow (represe nting the poss ible movements ove r tim e of some sys te m through it s s tat e s pace) as a co nt inuou s o ne-pa ra me te r semigroup of operators on a me tri c s pace, and the n showing th at these opera to rs a re all contrac tions (in suitab ly re vi se d me tri cs) if and o n ly if the re is a un iform ly asympt.otic a ll y s tab le equ ili brium point. Gene ra liza t. ions to o.the r ope rator se mi grollps a re a lso given.