The technical stability of parametrically excitable distributed processes (Q1097747)

The technical stability - in a finite interval of time - of parametrically excitable processes with distributed parameters, i.e., processes described by partial differential equations with time-dependent (particularly time-periodic) coefficients, is investigated. Using the comparison method [e.g: the author, Differ. Uravn. 20, No.11, 2009-2011 (1984; Zbl 0592.35029)] in conjunction with Lyapunov's second method, the sufficient conditions for technical stability with respect to a specified measure are obtained. The determination of the corresponding differential inequalities of the comparison [the author, Ukr. Math. J. 34, 509-513 (1983; Zbl 0555.34046)] rests on the extremal properties of Rayleigh's ratios for selfadjoint operators in Hilbert space. This approach is connected with the solution of the eigenvalue problem. The results obtained are used to establish the sufficient conditions using the specified measure in the problem of a clamped support [\textit{C. S. Hsu} and \textit{T. H. Lee}, Instability continuous systems, Symp. Herrenalb 1969, 112-118 (1971; Zbl 0232.73042)] loaded with some longitudinal force, particularly one which is time-periodic. At the same time the domain of technical stability is connected with the small parameter and the conditions of positive definiteness of Lyapunov's functional and the boundedness of the corresponding eigenvalues.