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On the exponential stability of a certain Lurie system. (English) Zbl 1091.34534

The author gives some necessary and sufficient conditions which ensure the existence of a bounded solution, which is globally exponentially stable and periodic (or almost-periodic) for some systems. For example, it is considered the following system \[ \frac{dx}{dt}=Ax-bf(\sigma)+P(t),\qquad \sigma=c^Tx, \] where \(c,b\in \mathbb{R}^n,\) \(P(t)\) is bounded, \(A\) a special upper triangular matrix, \(f\) is continuous and \(f(0)=0,\) which is a special Lurie system.

MSC:

34D23 Global stability of solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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