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Control of an axially moving viscoelastic Kirchhoff string. (English) Zbl 1460.74068

Summary: The control problem of axially moving strings occurs in a large class of mechanical systems. In addition to the longitudinal displacement, the strings are subject to undesirable transversal vibrations. In this work, in order to suppress these vibrations, we consider a control by a hydraulic touch-roll actuator at the right boundary. We prove uniform stability of the system using a viscoelastic material and an appropriate boundary control force applied to the touch rolls of the actuator. The features of the present work are: taking into account the mass flow entering in and out at the boundaries due to the axial movement of the string and overcoming the difficulty raised by the Kirchhoff coefficient which does not allow us to profit from the dissipativity of the system (as in the existing works so far). We shall make use of an inequality which is new in this theory.

MSC:

74M05 Control, switches and devices (“smart materials”) in solid mechanics
74K05 Strings
74H45 Vibrations in dynamical problems in solid mechanics
74D05 Linear constitutive equations for materials with memory
35Q74 PDEs in connection with mechanics of deformable solids
93C20 Control/observation systems governed by partial differential equations
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