Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach
Résumé
Let $u_{\mu, x, s}(\cdot, 0)$ be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space $X$ $$\left\{\begin{array}{lc} \dot{u}(t) = A(t)u(t)+e^{i\mu t}x, \quad t>s \\ u(s) = 0. \end{array} \right.$$ Here, $x$ is a vector in $X,$ $\mu$ is a real number, $q$ is a positive real number and $A(\cdot)$ is a $q$-periodic linear operator valued function. Under some natu\-ral assumptions on the evolution family $\mathcal{U}=\{U(t, s): t\ge s\}$ gene\-rated by the family $\{A(t)\},$ we prove that if for each $\mu$, each $s\ge 0$ and every $x$ the solution $u_{\mu, x, s}(\cdot, 0)$ is bounded on ${\bf R}_+$ by a positive constant, depending only on $x,$ then the family $\mathcal{U}$ is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.