Comptes Rendus
no. 5
Optimal control for a Timoshenko beam
[Contrôle optimal d'une barre de Timoshenko]

Présenté par : Évariste Sanchez-Palencia

Michail I. Zelikin1  ; Larissa A. Manita2
1 Moscow State (Lomonosov) University, Vorobjevy Gory, 119899 Moscow, Russia
2 Moscow State Institute of Electronics and Mathematics, Bolshoy Trehsviatitelskiy Per. 3/12, 109028 Moscow, Russia
Comptes Rendus. Mécanique, Volume 334 (2006) no. 5, pp. 292-297.

On considére une barre de Timoshenko en rotation, dont une extrêmité est reliée à un disque et l'autre est libre. Son mouvement est controlé par l'accéleration angulaire du disque. Nous étudions le problème de la minimisation de la moyenne quadratique de la deviation. Pour le problème de la minimisation du premier mode, nous démontrons qu'un contrôle optimal a une infinité de points de discontinuité en temps fini. Nous proposons une procédure pour construire d'un contrôle sous-optimale qui a un nombre fini de points de discontinuité.

We consider a linear model of a rotating Timoshenko beam, which is clamped at one end to a disk the other being free. The motion of the beam is controlled by the angular acceleration of the disk. We study the minimization problem of mean square deviation of the Timoshenko beam from a given position. For the minimization problem of the first mode we prove that optimal control is the chattering control, i.e., it has an infinite number of switches in a finite time interval. We construct a suboptimal control with a finite number of switches.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2006.03.011
Keywords: Computational solid mechanics, Timoshenko beam, Singular regimes, Chattering solutions
Mot clés : Mécanique des solides numérique, Barre de Timoshenko, Règimes singuliers, Solutions avec un nombre infini de points de discontinuité
Michail I. Zelikin 1 ; Larissa A. Manita 2

1 Moscow State (Lomonosov) University, Vorobjevy Gory, 119899 Moscow, Russia
2 Moscow State Institute of Electronics and Mathematics, Bolshoy Trehsviatitelskiy Per. 3/12, 109028 Moscow, Russia 
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Michail I. Zelikin; Larissa A. Manita. Optimal control for a Timoshenko beam. Comptes Rendus. Mécanique, Volume 334 (2006) no. 5, pp. 292-297. doi : 10.1016/j.crme.2006.03.011. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.03.011/

[1] S. Taylor; S. Yau Boundary control of a rotating Timoshenko beam, ANZIAM J., Volume 44 (2003), p. E143-E184

[2] M. Gugot Controllability of a slowly rotating Timoshenko beam, ESAIM Control Optim. Calc. Var., Volume 6 (2001), pp. 333-360

[3] W. Krabs; G.M. Sklyar On the controllability of a slowly rotating Timoshenko beam, J. Anal. Appl., Volume 18 (1999) no. 2, pp. 437-448

[4] W. Krabs; G.M. Sklyar On the stabilizability of a slowly rotating Timoshenko beam, J. Anal. Appl., Volume 19 (2000) no. 1, pp. 131-145

[5] M.I. Zelikin, L.A. Manita, Optimal chattering regimes in the control problem for Timoshenko beam, J. Appl. Math. Mech., submitted for publication

[6] M.I. Zelikin; V.F. Borisov Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering, Birkhäuser, Boston, 1994

[7] L.A. Manita Optimal operating modes with chattering switchings in manipulator control problems, J. Appl. Math. Mech., Volume 64 (2000) no. 1, pp. 17-24

[8] V.G. Boltyanskii Mathematical Methods of Optimal Control, Holt, Reinhart and Winston, 1971

[9] M.I. Zelikin; L.F. Zelikina The deviation of a functional from the optimal value under chattering exponentially decays as the number of switchings tends to infinity, Differential Equations, Volume 35 (1999) no. 11, pp. 1489-1493

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