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Stability criteria for linear periodic Hamiltonian systems. (English) Zbl 1195.34079

Consider the first order linear system
\[ x^{\prime }=a(t)x+b(t)u,\quad u^{\prime }=-c(t)x-a(t)u,\quad t\in \mathbb R,\tag{\(*\)} \]
where \(a,b\) and \(c\) are \(T\)-periodic real-valued piece-wise continuous functions defined on \(\mathbb R\). The system (\(*\)) is said to be stable if all solutions are bounded on \(\mathbb R\), unstable if all nontrivial solutions are unbounded on \(\mathbb R\), and conditionally stable if there exists a nontrivial solution bounded on \(\mathbb R\). The author obtain new stability criteria for (\(*\)).

MSC:

34D20 Stability of solutions to ordinary differential equations
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
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References:

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