The technical stability of parametrically excitable distributed processes

Abstract

The technical stability /1, 2/ - 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 /3–6/ in conjunction with Lyapunov's second method /7/, the sufficient conditions for technical stability /1–6/ with respect to a specified measure are obtained. The determination of the corresponding differential inequalities of the comparison /4/ rests on the extremal properties of Rayleigh's ratios for self adjoint operators in Hilbert space /8–12/. 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 /9/ 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 /11, 13, 14/. The technical stability of distributed-parameter systems for constantly acting perturbations have been investigated previously /1/, and the technical stability of processes with after-effect was examined using an axiomatic approach /2/. The problem of the technical stability of some systems which simultaneously contain distributed and lumped parameters was considered in /15/.