Periodic differential equations with selfadjoint monodromy operator (Q2759319)

The linear \(p\)-periodic differential equation NEWLINE\[NEWLINEu'= A(t)u,\quad A(t+ p)= A(t),\qquad t\in\mathbb{R},\tag{1}NEWLINE\]NEWLINE with the continuous operator coefficient \(A(t):\mathbb{H}\to \mathbb{H}\) is considered. Here, \(\mathbb{H}\) is a Hilbert or a finite-dimensional Euclidean space. Let \(u(t)\) be the evolution operator of (1), that is, \(u(t)= U(t)a\) is a solution to (1) with the initial condition \(u(0)= a\). The author presents conditions on \(A(t)\) which guarantee that the monodromy operator \(U(p)\) is selfadjoint and positive definite and next gives conditions for the stability and asymptotic stability of (1). Results are applied to several problems of the dynamics of an incompressible viscous fluid.