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Control via decoupling of a class of second order linear hybrid systems

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  • We study a terminal state control (reachability) problem for certain elastic systems of ``hybrid" type consisting of a single space dimension distributed parameter part coupled, at one endpoint of the relevant spatial, $x$, interval, to a lumped mass component. Two such systems are studied in detail. The first is a vibrating string system fixed at $x = 0$ and attached to a point mass at the right hand endpoint $x = L$. The second example concerns an Euler - Bernoulli beam system ``clamped" at $x = 0$ and attached, at $x = L$, to a mass with both translational and rotational inertia. In both cases the controls act on the mass, which is modeled by a finite dimensional system of differential equations. Analysis of the reachability problem is facilitated by a preliminary ``feedback type" transformation of the control variable which decouples the point mass from the distributed system. In both examples a concluding analysis is required to show that the component of the control generated by feedback lies in the same space as the originally applied control.
    Mathematics Subject Classification: Primary: 74C05, 74K10, 74K20, 90C25, 93D15.


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  • [1]

    M. D. Aouragh and N. Yebari, Riesz basis approach and exponential stabilization of a nonhomogeneous flexible beam with a tip mass, Int. J. Math. & Stat., 7 (2010), 46-53.


    S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, New York, Melbourne, 1995.


    M. S. Azam, N. Singh, A. Iyer and Y. P. Kakad, Detumbling and reorientation maneuvers and stabilization of NASA SCOLE system, IEEE Trans. Aerosp. & Electr. Syst., 28 (1992), 80-91.doi: 10.1109/7.135434.


    C. Baiocchi, V. Komornik and P. Loreti, Théorèmes du type Ingham et application à la théorie du contrôle, C. R. Acad. Sci. Paris Sér. I, Math., 326 (1998), 453-458.doi: 10.1016/S0764-4442(97)89791-1.


    W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass, Angew. Math. & Phys., 12 (1961), 369-392.doi: 10.1007/BF01600687.


    F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 36 (1998), 1962-1986.doi: 10.1137/S0363012996302366.


    M. Grobbelaar-Van Dalsen, Uniform stability for the Timoshenko beam with tip load, J. Math. Anal. & and Appl., 361 (2010), 392-400.doi: 10.1016/j.jmaa.2009.06.059.


    B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control. & Opt., 39 (2001), 1736-1747.doi: 10.1137/S0363012999354880.


    J. Humar and M. Ruban, Dynamics of Structures, CRC Press, Boca Raton, 2002.


    A. E. Ingham, Some trigonometric inequalities in the theory of series, Mathem. Zeitschrift, 41 (1936), 367-379.doi: 10.1007/BF01180426.


    W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura & Appl., 152 (1988), 281-330.doi: 10.1007/BF01766154.


    W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rat. Mech. & Anal., 103 (1988), 193-236.doi: 10.1007/BF00251758.


    Ö. Morgül, B. P. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.doi: 10.1109/9.328811.


    B. P. Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control. & Opt., 33 (1995), 440-454.doi: 10.1137/S0363012992239879.


    D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems, J. Math. Anal. & Appl., 18 (1967), 542-560.doi: 10.1016/0022-247X(67)90045-5.


    N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients, Int. J. Tomogr. & Stat., 10 (2008), 125-140.

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