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Stability criteria for linear periodic Hamiltonian systems. (English) Zbl 1195.34079
Consider the first order linear system $$x^{\prime }=a(t)x+b(t)u,\quad u^{\prime }=-c(t)x-a(t)u,\quad t\in \Bbb R,\tag*$$ where $a,b$ and $c$ are $T$-periodic real-valued piece-wise continuous functions defined on $\Bbb R$. The system ($*$) is said to be stable if all solutions are bounded on $\Bbb R$, unstable if all nontrivial solutions are unbounded on $\Bbb R$, and conditionally stable if there exists a nontrivial solution bounded on $\Bbb R$. The author obtain new stability criteria for ($*$).

##### MSC:
 34D20 Stability of ODE 37J25 Stability problems (finite-dimensional Hamiltonian etc. systems)
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##### References:
 [1] Cheng, S. S.: Lyapunov inequalities for differential and difference equations, Fasc. math. 23, 25-41 (1991) · Zbl 0753.34017 [2] Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations, (1955) · Zbl 0064.33002 [3] Guseinov, G. Sh.; Kaymakcalan, B.: Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. math. Appl. 45, 1399-1416 (2003) · Zbl 1055.39029 · doi:10.1016/S0898-1221(03)00095-6 [4] Guseinov, G. Sh.; Zafer, A.: Stability criteria for linear periodic impulsive Hamiltonian systems, J. math. Anal. appl. 335, 1195-1206 (2007) · Zbl 1128.34005 · doi:10.1016/j.jmaa.2007.01.095 [5] Krein, M. G.: Foundations of the theory of $\lambda$-zones of stability of canonical system of linear differential equations with periodic coefficients, Amer. math. Soc. transl. Ser. 2 120, 1-70 (1955) · Zbl 0516.34049 [6] Miller, R. K.; Michel, A. N.: Ordinary differential equations, (1982) · Zbl 0552.34001