Optimal control of a rotating body beam
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 157-178.

In this paper we consider the problem of optimal control of the model for a rotating body beam, which describes the dynamics of motion of a beam attached perpendicularly to the center of a rigid cylinder and rotating with the cylinder. The control is applied on the cylinder via a torque to suppress the vibrations of the beam. We prove that there exists at least one optimal control and derive a necessary condition for the control. Furthermore, on the basis of iteration method, we propose numerical approximation scheme to calculate the optimal control and give numeric examples.

DOI : 10.1051/cocv:2002007
Classification : 49K20, 35L75, 74K10
Mots clés : rotating body beam, optimal control, numerical approximation scheme 
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     author = {Liu, Weijiu},
     title = {Optimal control of a rotating body beam},
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     pages = {157--178},
     publisher = {EDP-Sciences},
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     mrnumber = {1925025},
     zbl = {1053.49023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002007/}
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Liu, Weijiu. Optimal control of a rotating body beam. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 157-178. doi : 10.1051/cocv:2002007. http://www.numdam.org/articles/10.1051/cocv:2002007/

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] J. Baillieul and M. Levi, Rotational elastic dynamics. Physica D 27 (1987) 43-62. | MR | Zbl

[3] J. Baillieul and M. Levi, Constrained relative motions in rotational mechanics. Arch. Rational Mech. Anal. 115 (1991) 101-135. | MR | Zbl

[4] S.K. Biswas and N.U. Ahmed, Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations. Math. Control Signals Systems 2 (1989) 1-18. | MR

[5] B. Chentouf and J.F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping. ESAIM: COCV 4 (1999) 515-535. | Numdam | MR | Zbl

[6] J.-M. Coron and B. D'Andréa-Novel, Stabilization of a rotating body beam without damping. IEEE Trans. Automat. Control 43 (1998) 608-618. | Zbl

[7] C.J. Damaren and G.M.T. D'Eleuterio, Optimal control of large space structures using distributed gyricity. J. Guidance Control Dynam. 12 (1989) 723-731. | Zbl

[8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976). | MR | Zbl

[9] H. Laousy, C.Z. Xu and G. Sallet, Boundary feedback stabilization of a rotating body-beam system. IEEE Trans. Automat. Control 41 (1996) 241-245. | MR | Zbl

[10] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). | MR | Zbl

[11] J.L. Lions and E. Magenes, Non-homogeneous Boundary value Problems and Applications, Vol. I. Springer-Verlag, Berlin, Heidelberg, New York (1972). | MR | Zbl

[12] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl

[13] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. (4) CXLVI (1987) 65-96. | MR | Zbl

[14] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd Ed. Springer-Verlag, New York (1997). | MR | Zbl

[15] C.Z. Xu and J. Baillieul, Stabilizability and stabilization of a rotating body-beam system with torque control. IEEE Trans. Automat. Control 38 (1993) 1754-1765. | MR | Zbl

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