Abstract
Malkin’s theorem on stability under permanently acting perturbations guarantees non-asymptotic stability in the presence of permanently acting perturbations in systems in which asymptotic stability took place without such perturbations. In this case, traditionally, the conclusion about asymptotic stability in the absence of permanently acting perturbations is obtained on the basis of an analysis of the location of the roots of the characteristic equation of the system of the first approximation of the equations of perturbed motion: the real parts of all roots of the characteristic equation must be negative. In this paper, we consider the application of Malkin’s theorem on stability under permanently acting perturbations to systems whose characteristic equation without perturbations has zero roots, and the equations of perturbed motion have a special structure that allows the application of the principle of reducing the theory of critical cases with regard to additional conditions on the initial perturbations of critical variables. It is shown that this approach is applicable to the problems of stability and stabilization of steady motions of the systems with geometric connections using the Shulgin equations in the Routh variables.
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Krasinskiy, A.Y. (2020). On Stability and Stabilization with Permanently Acting Perturbations in Some Critical Cases. In: Tarasyev, A., Maksimov, V., Filippova, T. (eds) Stability, Control and Differential Games. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-42831-0_29
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DOI: https://doi.org/10.1007/978-3-030-42831-0_29
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