EXPONENTIAL STABILITY OF LINEAR SYSTEMS IN BANACH SPACES

In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class$[{H^2}(sigma )]$ and by this lemma obtains some fundamental results of exponential stability of $[{C_0}]$-semigroup of bounded linear operators in Banach spaces. Specially, if $[{omega _s} = sup { {mathop{rm Re}nolimits} lambda ;lambda in sigma (A) < 0} ]$ and $[sup { left| {{{(lambda - A)}^{ - 1}}} right|;{mathop{rm Re}nolimits} lambda ge sigma } < infty ]$ , where [sigma in ({omega _s},0)]) and A is the infinitesimal generator of a $[{C_0}]$-semigroup in a Banach space $X$, then $[(a)int_0^infty {{e^{ - sigma t}}left| {f({e^{tA}}x)} right|} dt < infty ]$, $[forall f in {X^*},x in X]$; (b) there exists $[M > 0]$ such that $[left| {{e^{tA}}x} right| le N{e^{sigma t}}left| {Ax} right|]$, $[forall x in D(A)]$; (c) thereexists a Banach space $[hat X supset X]$ such that $[left| {{e^{tA}}x} right|hat x le {e^{sigma t}}left| x right|hat x,forall x in X.]$.