zbMATH — the first resource for mathematics

Vibrational stabilizability of distributed parameter systems. (English) Zbl 0554.93059
Stabilizability 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

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.