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Well-posedness, regularity and exact controllability of the SCOLE model

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Abstract

The SCOLE model is a coupled system consisting of a flexible beam (modelled as an Euler–Bernoulli equation) with one end clamped and the other end linked to a rigid body. Its inputs are the force and the torque acting on the rigid body. It is well-known that the SCOLE model is not exactly controllable with L 2 input signals in the natural energy state space H c, because the control operator is bounded from the input space \({\mathbb{C}^2}\) to H c, and hence compact. We regard the velocity and the angular velocity of the rigid body as the output signals of this system. Using the theory of coupled linear systems (one infinite-dimensional and one finite-dimensional) developed by us recently in another paper, we show that the SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space \({\mathcal{X}\subset H^c}\).

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References

  1. Curtain RF, Zwart HJ (1995) An introduction to infinite-dimensional linear systems theory. Springer, New York

    MATH  Google Scholar 

  2. Fattorini HO (1968) Boundary control systems. SIAM J Control Optim 6: 349–385

    Article  MATH  MathSciNet  Google Scholar 

  3. Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J Control Optim 44: 1864–1892

    Article  MathSciNet  Google Scholar 

  4. Guo BZ (2002) On boundary control of a hybrid system with variable coefficients. J Optim Theory Appl 114: 373–395

    Article  MATH  MathSciNet  Google Scholar 

  5. Guo BZ, Ivanov SA (2005) On boundary controllability and observability of a one-dimensional non-uniform SCOLE system. J Optim Theory Appl 127: 89–108

    Article  MATH  MathSciNet  Google Scholar 

  6. Lions J-L, Magenes E (1972) Non-homogeneous boundary value problems and applications, vol I. Springer, New York. Transl. from French by P. Kenneth, Die Grundlehren der math. Wissenschaften, Band 181

  7. Littman W, Markus L (1988) Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann Mat Pura Appl 152: 281–330

    Article  MATH  MathSciNet  Google Scholar 

  8. Littman W, Markus L (1988) Exact boundary controllability of a hybrid system of elasticity. Arch Ration Mech Anal 103: 193–235

    Article  MATH  MathSciNet  Google Scholar 

  9. Malinen J, Staffans OJ (2006) Conservative boundary control systems. J Differ Equ 231: 290–312

    Article  MATH  MathSciNet  Google Scholar 

  10. Malinen J, Staffans OJ (2007) Impedance passive and conservative boundary control systems. Complex Anal Oper Theory 1: 279–300

    Article  MATH  MathSciNet  Google Scholar 

  11. Rao B (2001) Exact boundary controllability of a hybrid system of elasticity by the HUM method. ESAIM COCV 6: 183–199

    Article  MATH  Google Scholar 

  12. Salamon D (1987) Infinite-dimensional linear systems with unbounded control and observation: a functional analytical approach. Trans Am Math Soc 300: 383–431

    MATH  MathSciNet  Google Scholar 

  13. Staffans OJ (2005) Well-posed linear systems. Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge

    Book  Google Scholar 

  14. Triebel H (1978) Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam

    Google Scholar 

  15. Tucsnak M, Weiss G (2000) Simultaneous exact controllability and some applications. SIAM J Control Optim 38: 1408–1427

    Article  MATH  MathSciNet  Google Scholar 

  16. Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  17. Weiss G (1994) Transfer functions of regular linear systems. Part I. Characterizations of regularity. Trans Am Math Soc 342: 827–854

    Article  MATH  Google Scholar 

  18. Weiss G, Zhao X (2009) Well-posedness and controllability of a class of coupled linear systems. SIAM J Control Optim 48: 2719–2750

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhao X, Weiss G (2008) Strong stabilization of a non-uniform SCOLE model. In: Proceedings of the 17th IFAC World Congress, Seoul, Korea, pp 8761–8766

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Correspondence to Xiaowei Zhao.

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Zhao, X., Weiss, G. Well-posedness, regularity and exact controllability of the SCOLE model. Math. Control Signals Syst. 22, 91–127 (2010). https://doi.org/10.1007/s00498-010-0053-4

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  • DOI: https://doi.org/10.1007/s00498-010-0053-4

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