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Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

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Abstract

Let u μ, x, s (., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X

$$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$

Here, x is a vector in X, μ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family \({\mathcal{U} = \{U(t, s): t \geq s\}}\) generated by the family {A(t)}, we prove that if for each μ, each s ≥ 0 and every x the solution u μ, x, s (·, 0) is bounded on 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.

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Authors and Affiliations

  1. Department of Mathematics, West University of Timisoara, Bd. V. Parvan No. 4, 300223, Timisoara, România

    Constantin Buşe & Olivia Saierli

  2. Laboratoire SAMM EA543, Universite Paris 1 Pantheon-Sorbonne, centre P. M. F. 90 rue de Tolbiac, 75634, Paris cedex 13, France

    Dhaou Lassoued

  3. Western Kentucky University, Bowling Green, KY, 42101-1078, US

    Thanh Lan Nguyen

  4. Department of Computer Sciences, Tibiscus University of Timisoara, Str. Lascăr Catargiu, No. 4-6, 300559, Timisoara, România

    Olivia Saierli

Authors
  1. Constantin Buşe
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  2. Dhaou Lassoued
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  3. Thanh Lan Nguyen
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  4. Olivia Saierli
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Correspondence to Constantin Buşe.

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Buşe, C., Lassoued, D., Nguyen, T.L. et al. Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach. Integr. Equ. Oper. Theory 74, 345–362 (2012). https://doi.org/10.1007/s00020-012-1993-5

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  • DOI: https://doi.org/10.1007/s00020-012-1993-5

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