Abstract
We derive a stability criterion relative to a given measure on a finite time interval for distributed processes under parametric excitation. The corresponding theorem is proved by the comparison method combined with Lyapunov second method. Treating time as a parameter, we use the extremal properties of the Rayleigh quotient for self-adjoint operators in a Hilbert space, which in turn involves solving the eigenvalue problem generated by the linear operators corresponding to the original problem. The results are applied to establish sufficient conditions of technical stability relative to a given measure in the nonlinear problem of a hinged pole under the action of a continuous longitudinal force.
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Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 107–116, 1986.
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Matviichuk, K.S. Stability conditions of nonlinear distributed processes under parametric excitation. J Math Sci 58, 476–482 (1992). https://doi.org/10.1007/BF01100078
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DOI: https://doi.org/10.1007/BF01100078