Stability for trajectories of periodic evolution families in Hilbert spaces

Abstract

Let $q$ be a positive real number and let $A(\cdot)$ be a $q$-periodic linear operator valued function on a complex Hilbert space $H$, and let $D$ be a dense linear subspace of $H$. Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be the evolution family generated by the family $\{A(t)\}$. We prove that if the solution of the well-posed inhomogeneous Cauchy Problem $$\displaylines{ \dot{u}(t) = A(t)u(t)+e^{i\mu t}y, \quad t>0, \cr u(0) = 0, }$$ is bounded on ${\mathbb{R}}_+$, for every $y\in D$, and every $\mu\in\mathbb{R}$, by the positive constant $K\|y\|$, $K$ being an absolute constant, and if, in addition, for some $x\in D$, the trajectory $U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval $(0, q)$, then $$ \sup_{z\in \mathbb{C}, |z|=1}\sup_{n\in\mathbb{Z}_+} \|\sum_{k=0}^nz^kU(q, 0)^kx\|:=N(x)<\infty. $$ The latter discrete boundedness condition has a lot of consequences concerning the stability of solutions of the abstract nonautonomous system $\dot u(t)=A(t)u(t)$. To our knowledge, these results are new. In the special case, when $D=H$ and for every $x\in H $, the map $U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval $(0, q)$, the evolution family $\mathcal{U}$ is uniformly exponentially stable. In the autonomous case, (i.e. when $U(t, s)=U(t-s, 0)$ for every pair $(t, s)$ with $t\ge s\ge 0$), the latter assumption is too restrictive. More exactly, in this case, the semigroup $\mathbf{T}:=\{U(t, 0)\}_{t\ge 0}$, is uniformly continuous.

Keywords

• Periodic evolution families
• uniform exponential stability
• boundedness
• strongly continuous semigroup
• periodic and almost periodic functions