Finite-time stabilization of an overhead crane with a flexible cable
The paper is concerned with the finite-time stabilization of a hybrid PDE-ODE system which may serve as a model for the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is assumed to be described by the wave equation with constant coefficients. Using a nonlinear feedback law inspired by those given by Haimo (SIAM J Control Optim 24(4):760–770, 1986) for a second-order ODE, we prove that a finite-time stabilization occurs for the full system platform \(+\) cable. The global well-posedness of the system is also established by using the theory of nonlinear semigroups.
KeywordsFinite-time stability PDE-ODE system Nonlinear feedback law Nonlinear semigroups Transparent boundary conditions Wave equation
Mathematics Subject Classification93C20 93D15
This work was done when Iván Moyano was visiting Centre de Robotique, MINES ParisTech, as a postdoctoral student. Iván Moyano thanks Centre de Robotique, MINES ParisTech, for its hospitality. The authors were supported by the ANR project Finite4SoS (ANR-15-CE23-0007). LR was also partially supported by the MathAmSud project Icops.
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