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

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.