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

Consider the first order linear system

x ' =a(t)x+b(t)u,u ' =-c(t)x-a(t)u,t,(*)

where a,b and c are T-periodic real-valued piece-wise continuous functions defined on . The system (*) is said to be stable if all solutions are bounded on , unstable if all nontrivial solutions are unbounded on , and conditionally stable if there exists a nontrivial solution bounded on . The author obtain new stability criteria for (*).


MSC:
34D20Stability of ODE
37J25Stability problems (finite-dimensional Hamiltonian etc. systems)
References:
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[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 λ-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
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