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Article Dans Une Revue Integral Equations and Operator Theory Année : 2012

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.

Dates et versions

hal-00943318 , version 1 (07-02-2014)

Identifiants

Citer

Dhaou Lassoued, Constantin Buse, Lan Nguyen Thanh, Olivia Saierli. Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach. Integral Equations and Operator Theory, 2012, 74 (3), pp.345-362. ⟨10.1007/s00020-012-1993-5⟩. ⟨hal-00943318⟩
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