an:06283982
Zbl 1308.47049
Barbu, Dorel; Blot, Joel; Buse, Constantin; Saierli, Olivia
Stability for trajectories of periodic evolution families in Hilbert spaces
EN
Electron. J. Differ. Equ. 2014, Paper No. 01, 13 p. (2014).
00331612
2014
j
47D06 35B15 35B10
periodic evolution families; uniform exponential stability; boundedness; strongly continuous semigroup; periodic and almost periodic functions
The authors consider a \(q\)-periodic evolution family \(\mathcal{U}\) with propagator \(A\) on a Hilbert space \(H\). Under the assumptions that
(A) the trajectories \(\mathcal{U}(\cdot,0)x\) for \(x\) belonging to a dense subspace \(D\) of \(H\) satisfy a Lipschitz condition on \(]0,q[\) and
(B) the solutions of \(\dot{u}(t)=A(t)u(t)+\exp(i\mu t)x\) are uniformly bounded in \(\mu\in \mathbb{R}\) and \(x\in D\) with \(\|x\|\leq 1\),
they derive a discrete boundedness condition for \(\mathcal{U}\). The proof is based on Fourier-expansions of the functions involved and suitable estimates for their Fourier coefficients. The method of proof restricts the results to the Hilbert space case. It is remarkable that the boundedness condition for \(\mathcal{U}\) implies the strong stability of the trajectories \(\mathcal{U}(t,\cdot)x\) for \(x\in D\) and, if (A) is assumed for all \(x\) in \(H\), the exponential stability of \(\mathcal{U}\). Similar results are obtained in the autonomous case, i.e., when \(\mathcal{U}(t,s)=T(t-s)\) for a \(C_0\)-semigroup \(T\). In fact, in this case the condition (B) can be weakened by assuming the uniform boundedness just for \(x\) in the unit ball of the domain of \(A\), equipped with its graph norm.
Sascha Trostorff (Dresden)