Stabilization and Riesz basis property for an overhead crane model with feedback in velocity and rotating velocity

Abstract

This paper studies a variant of an overhead crane model's problem, with a control force in velocity and rotating velocity on the platform. We obtain under certain conditions the well-posedness and the strong stabilization of the closed-loop system. We then analyze the spectrum of the system. Using a method due to Shkalikov, we prove the existence of a sequence of generalized eigenvectors of the system, which forms a Riesz basis for the state energy Hilbert space.

Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

“…Since several decades, there has been much interest to the boundary controllability and stabilization of hybrid systems with attached masses. To mention some examples, see previous studies for strings with interior or end masses and other works for beams with an attached point mass. Some recent progress are concerned to the boundary controllability of hybrid systems with internal point masses and variable coefficients …”

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

“…See also Guo [24,25] for applications to beam equations with boundary feedback controls. Sometimes, the use of Shkalikov's theory [42] is helpful when a spectral parameter appears in the boundary conditions of the associated spectral problem (see e.g., [12,23]). In the majority of cases, the application of the abstract result due to Xu and Yung [45] be the perfect solution to get the Riesz basis generation for string equations [16], for thermo-elastic systems [30], or for Timoshenko beam systems [31].…”

“…A few years later, Littman and Taylor [36] improved the first result for K 2 = 0 by giving a polynomial stability result under some assumptions in the initial datum. Augustin et al, [12] have studied the question on the stability and the Riesz basis generation for an overhead crane model with two end masses and subject to a feedback in velocity and rotating velocity. Under some conditions on the feedback parameters, the authors have proved the strong stability of the system due to LaSalle's invariance principle, and they obtained the Riesz basis property thanks to Shkalikov's theory.…”