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) there exists 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.\]$.