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Asymptotic stability for intermittently controlled nonlinear oscillators. (English) Zbl 0809.34067
The following quasi-variational system \(\nabla{\mathcal L}(t,u,u')'- \nabla_ u{\mathcal L}(t,u,u')= Q(t,u,u')\), \(t\in J= [T,\infty)\), \(u: J\to \mathbb{R}^ N\) is considered, where \({\mathcal L}(t,u,p)= G(u,p)- F(t,u)\), \(F\) represents a restoring potential and \(Q\) is a general nonlinear damping. Intermittent damping means that restrictions or controls are placed on the damping term only on a sequence \(\{I_ n\}\) of nonoverlapping intervals. It is an old problem to guarantee that the rest state \(u= 0\) is a global attractor, i.e., \(u(t)\), \(u'(t)\to 0\) as \(t\to\infty\) for every bounded solution \(u\). The authors show that under appropriate conditions on the measures \(| I_ n|\) and on the damping term \(Q(t,u,p)\), (maybe unbounded or close to zero) the rest state \(u= 0\) becomes a global attractor. The paper is concluded by nice examples showing that the conditions are sharp.
Reviewer: L.Hatvani (Szeged)

MSC:
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
35A15 Variational methods applied to PDEs
70H03 Lagrange’s equations
70K20 Stability for nonlinear problems in mechanics
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