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Periodic differential equations with selfadjoint monodromy operator. (English. Russian original) Zbl 1024.34049
Sb. Math. 192, No. 3, 455-478 (2001); translation from Mat. Sb. 192, No. 3, 137-160 (2001).
The linear \(p\)-periodic differential equation \[ u'= A(t)u,\quad A(t+ p)= A(t),\qquad t\in\mathbb{R},\tag{1} \] 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.
MSC:
34G10 Linear differential equations in abstract spaces
34A30 Linear ordinary differential equations and systems, general
47N20 Applications of operator theory to differential and integral equations
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