Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd denominator or vice versa.

Keywords: rotating Timoshenko beam, exact controllability, eigenvalues, moment problem

@article{COCV_2001__6__333_0, author = {Gugat, Martin}, title = {Controllability of a slowly rotating {Timoshenko} beam}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {333--360}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, zbl = {1031.93103}, mrnumber = {1824106}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__333_0/} }

TY - JOUR AU - Gugat, Martin TI - Controllability of a slowly rotating Timoshenko beam JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 SP - 333 EP - 360 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__333_0/ UR - https://zbmath.org/?q=an%3A1031.93103 UR - https://www.ams.org/mathscinet-getitem?mr=1824106 LA - en ID - COCV_2001__6__333_0 ER -

Gugat, Martin. Controllability of a slowly rotating Timoshenko beam. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 333-360. http://www.numdam.org/item/COCV_2001__6__333_0/

[1] Families of Exponentials. Cambridge University Press (1995). | MR | Zbl

and ,[2] Modelling of a rotating flexible beam, in Control of Distributed Parameter Systems, edited by H.E. Rauch. Pergamon Press, Los Angeles (1986) 383-387.

, , and ,[3] Wave Motion in Elastic Solids. Dover Publications, New York (1991).

,[4] A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems. J. Comput. Appl. Math. 114 (2000) 103-119. | MR | Zbl

,[5] Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987) 1417-1429. | MR | Zbl

and ,[6] On moment theory and contollability of one-dimensional vibrating systems and heating processes. Springer-Verlag, Heidelberg, Lecture Notes in Control and Informat. Sci. 173 (1992). | MR | Zbl

,[7] Controllability of a rotating beam. Springer-Verlag, Lecture Notes in Control and Inform. Sci. 185 (1993) 447-458. | MR | Zbl

,[8] On the controllability of a slowly rotating Timoshenko beam. J. Anal. Appl. 18 (1999) 437-448. | MR | Zbl

and ,[9] A classical approach to uniform null controllability of elastic beams. SIAM J. Control Optim. 36 (1998) 1073-1085. | MR | Zbl

,[10] Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl. 18 (1967) 542-560. | MR | Zbl

,[11] Spectral operators generated by Timoshenko beam model. Systems Control Lett. 38 (1999). | MR | Zbl

,[12] On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. (1921) xli.

,