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Exact controllability of a Rayleigh beam with a single boundary control

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Abstract

We prove exact boundary controllability for the Rayleigh beam equation \({\varphi_{tt} -\alpha\varphi_{ttxx} + A\varphi_{xxxx} = 0, 0 < x < l, t > 0}\) with a single boundary control active at one end of the beam. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment \({\varphi_{xx}(l, t)}\) or the rotation angle \({\varphi_{x}(l, t)}\) at an end of the beam. In each case, exact controllability is obtained on the space of optimal regularity for L 2(0, T) controls for \({T > 2l\sqrt{\frac{\alpha}{A}}}\). In certain cases, e.g., the clamped case, the optimal regularity space involves a quotient in the velocity component. In other cases, where the regularity for the observed problem is below the energy level, a quotient space may arise in solutions of the observed problem.

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Author notes

  1. A. Ozkan Ozer

    Present address: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada

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  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    A. Ozkan Ozer & Scott W. Hansen

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  1. A. Ozkan Ozer
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  2. Scott W. Hansen
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Correspondence to Scott W. Hansen.

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Ozkan Ozer, A., Hansen, S.W. Exact controllability of a Rayleigh beam with a single boundary control. Math. Control Signals Syst. 23, 199–222 (2011). https://doi.org/10.1007/s00498-011-0069-4

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  • DOI: https://doi.org/10.1007/s00498-011-0069-4

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