Vibrational stabilizability of distributed parameter systems.

*(English)*Zbl 0554.93059Stabilizability of linear distributed parameter systems by introduction of zero mean oscillations in the system coefficients is analysed. By means of a finite dimensional approximation a condition for vibrational stabilizability of a second order (with respect to time) partial differential equation is given. As an example of vibrational stabilization of nonlinear systems the stabilization condition for the hydrodynamic equations of the Bénard problem is derived.

Reviewer: A.Tylikowski

##### MSC:

93D15 | Stabilization of systems by feedback |

93C05 | Linear systems in control theory |

93C20 | Control/observation systems governed by partial differential equations |

93C10 | Nonlinear systems in control theory |

35B35 | Stability in context of PDEs |

76E30 | Nonlinear effects in hydrodynamic stability |

70J25 | Stability for problems in linear vibration theory |

70K20 | Stability for nonlinear problems in mechanics |

##### Keywords:

linear distributed parameter systems; zero mean oscillations; vibrational stabilizability; nonlinear systems; Bénard problem
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\textit{S. M. Meerkov}, J. Math. Anal. Appl. 98, 408--418 (1984; Zbl 0554.93059)

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##### References:

[1] | Meerkov, S.M, Principle of vibrational control: theory and applications, IEEE trans. automat. control, AC-25, 755-762, (1980) · Zbl 0454.93021 |

[2] | Meerkov, S.M, Condition of vibrational stabilizability for a class of nonlinear systems, IEEE trans. automat. control, AC-27, 485-487, (1982) · Zbl 0491.93034 |

[3] | Bellman, R; Bentsman, J; Meerkov, S.M, Vibrational control of systems with arrhenius dynamics, J. math. anal. appl., 90, (1983) · Zbl 0525.93034 |

[4] | Chelomey, V.N, On increased stability of systems with vibrations, Dokl. acad. nauk SSSR, 110, 3, (1956) |

[5] | Bogoliubov, N.N; Mitropolsky, Yu.A, Asymptotic methods in the theory of nonlinear oscillations, (1974), Nauka Moscow, (in Russian) |

[6] | Wolf, G.H, The dynamic stabilization of the Rayleigh-Taylor instability and the corresponding dynamic equilibrium, Z. phys., 227, 291-300, (1969) |

[7] | Boris, J.P, Dynamic stabilization of the imploding-shell Rayleigh-Taylor instability, Comments plasma phys. cont. fusion, 3, 1-13, (1977) |

[8] | Berge, G, Equilibrium and stability of MHD-fluids by dynamic techniques, Nuclear fusion, 12, 99-117, (1972) |

[9] | Osovets, S.M, Dynamic methods of retention and stabilization of hot plasma, Soviet phys. uspehi, 112, 637-683, (1974) |

[10] | Gresho, P.M; Sani, R.L, The effects of gravity modulation on the stability of a heated fluid layer, J. fluid mech., 40, 783-806, (1970) · Zbl 0191.56502 |

[11] | Meerkov, S.M, Averaging of trajectories of slow dynamic systems, J. differential equations, 9, 1239-1245, (1973) · Zbl 0306.34058 |

[12] | Chandrasekhar, S, Hydrodynamic and hydromagnetic stability, (1968), Oxford Univ. Press (Clarendon) London/New York · Zbl 0142.44103 |

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