# American Institute of Mathematical Sciences

2014, 7(6): 1321-1334. doi: 10.3934/dcdss.2014.7.1321

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

 1 Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24060

Received  April 2013 Revised  November 2013 Published  June 2014

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.
Citation: David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
##### References:
 [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. [2] S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems,, Cambridge University Press, (1995). [3] 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. doi: 10.1109/7.135434. [4] 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, 326 (1998), 453. doi: 10.1016/S0764-4442(97)89791-1. [5] W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass,, Angew. Math. & Phys., 12 (1961), 369. doi: 10.1007/BF01600687. [6] F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 36 (1998), 1962. doi: 10.1137/S0363012996302366. [7] M. Grobbelaar-Van Dalsen, Uniform stability for the Timoshenko beam with tip load,, J. Math. Anal. & and Appl., 361 (2010), 392. doi: 10.1016/j.jmaa.2009.06.059. [8] B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 39 (2001), 1736. doi: 10.1137/S0363012999354880. [9] J. Humar and M. Ruban, Dynamics of Structures,, CRC Press, (2002). [10] A. E. Ingham, Some trigonometric inequalities in the theory of series,, Mathem. Zeitschrift, 41 (1936), 367. doi: 10.1007/BF01180426. [11] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping,, Ann. Mat. Pura & Appl., 152 (1988), 281. doi: 10.1007/BF01766154. [12] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rat. Mech. & Anal., 103 (1988), 193. doi: 10.1007/BF00251758. [13] 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. doi: 10.1109/9.328811. [14] B. P. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control. & Opt., 33 (1995), 440. doi: 10.1137/S0363012992239879. [15] D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems,, J. Math. Anal. & Appl., 18 (1967), 542. doi: 10.1016/0022-247X(67)90045-5. [16] N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients,, Int. J. Tomogr. & Stat., 10 (2008), 125.

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##### References:
 [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. [2] S. Avdonin and S. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems,, Cambridge University Press, (1995). [3] 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. doi: 10.1109/7.135434. [4] 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, 326 (1998), 453. doi: 10.1016/S0764-4442(97)89791-1. [5] W. E. Boyce and G. H. Handelman, Vibrations of rotating beams with tip mass,, Angew. Math. & Phys., 12 (1961), 369. doi: 10.1007/BF01600687. [6] F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 36 (1998), 1962. doi: 10.1137/S0363012996302366. [7] M. Grobbelaar-Van Dalsen, Uniform stability for the Timoshenko beam with tip load,, J. Math. Anal. & and Appl., 361 (2010), 392. doi: 10.1016/j.jmaa.2009.06.059. [8] B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control. & Opt., 39 (2001), 1736. doi: 10.1137/S0363012999354880. [9] J. Humar and M. Ruban, Dynamics of Structures,, CRC Press, (2002). [10] A. E. Ingham, Some trigonometric inequalities in the theory of series,, Mathem. Zeitschrift, 41 (1936), 367. doi: 10.1007/BF01180426. [11] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping,, Ann. Mat. Pura & Appl., 152 (1988), 281. doi: 10.1007/BF01766154. [12] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rat. Mech. & Anal., 103 (1988), 193. doi: 10.1007/BF00251758. [13] 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. doi: 10.1109/9.328811. [14] B. P. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control. & Opt., 33 (1995), 440. doi: 10.1137/S0363012992239879. [15] D. L. Russell, Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems,, J. Math. Anal. & Appl., 18 (1967), 542. doi: 10.1016/0022-247X(67)90045-5. [16] N. Yebari and M. D. Aouragh, Uniform stabilization of a hybrid system of elasticity with variable coefficients,, Int. J. Tomogr. & Stat., 10 (2008), 125.
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