Abstract
The paper studies dynamics modelling and control design for elastic systems with distributed parameters. The constitutive laws are specified by an integral equality according with the method of integro-differential relations. The original initial-boundary value problem is reduced to a constrained minimization problem for a nonnegative quadratic functional. A numerical algorithm is developed to solve direct and inverse dynamic problems in linear elasticity based on the Ritz method and finite element technique. The minimized functional is used to define an energy type criteria of solution quality. The efficiency of the approach is demonstrated on the example of a thin rectilinear elastic rod. The control problem is to find motion of a rod from an initial state to a terminal one at a fixed time with the minimal mean energy. The control input is presented by piecewise polynomial displacements on one end of the rod. It is possible to find the exact solution of the problem for a specific relation of the space-time mesh steps. The results of numerical analysis are presented and discussed.
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References
N.U. Ahmed, K.L. Teo, Optimal Control of Distributed Parameter Systems (North-Holland, New York, 1981)
M.J. Balas, Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite-dimensional controllers. J. Math. Anal. Appl. 114(1), 17–36 (1986)
S.P. Banks, State-space and Frequency-domain Methods in the Control of Distributed Parameter Systems (Peregrinus, London, 1983)
A.G. Butkovsky, Distributed Control Systems (Elsevier, New York, 1969)
F.L. Chernous’ko, I.M. Ananievski, S.A. Reshmin, Control of Nonlinear Dynamical Systems: Methods and Applications (Springer, Berlin, 1996)
P.D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-reaction Processes (Birkhäuser, Boston, 2001)
R. Curtain, H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory (Springer, New York, 1995)
G. Farin, Curves and Surfaces for Computer-aided Geometric Design, 4th edn. (Academic Press, San Diego, 1997)
M. Gerdts, G. Greif, H.J. Pesch, Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time. Math. Comput. Simul. 79(4), 1020–1032 (2008)
M. Gugat, Optimal nodal control of networked hyperbolic systems: evaluation of derivatives. Adv. Model. Optim. 7(1), 9–37 (2005)
G.V. Kostin, Construction of an optimal control for the motion of elastic bodies using integrodifferential relations. J. Comput. Syst. Sci. Int. 46(4), 532–542 (2007)
G.V. Kostin, V.V. Saurin, Integrodifferential Relations in Linear Elasticity (De Gruyter, Berlin, 2012)
W. Krabs, Optimal Control of Undamped Linear Vibrations (Heldermann, Lemgo, 1995)
J.E. Lagnese, G. Leugering, E.J.P.G. Schmidt, Modeling, Analysis, and Control of Dynamic Elastic Multi-link Structures (Birkhäuser, Boston, 1984)
D. Leineweber, E.I. Bauer, H. Bock, J. Schloeder, An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects. Comput. Chem. Eng. 27(2), 157–166 (2003)
G. Leugering, Domain decomposition of optimal control problems for dynamic networks of elastic strings. Comput. Optim. Appl. 16(1), 5–27 (2000)
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, New York, 1971)
J.L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30(1), 1–68 (1988)
A. Zuyev, O. Sawodny, Stabilization and observability of a rotating Timoshenko beam model. Math. Probl. Eng. (Art. ID 57238), 19 (2007)
Acknowledgments
This work was supported by the Russian Foundation for Basic Research, project nos. 12-01-00789, 13-01-00108, 14-01-00282, the Leading Scientific Schools Grants NSh-2710.2014.1, NSh-2954.2014.1.
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Kostin, G., Saurin, V. (2016). A Variational Approach to Modelling and Optimization in Elastic Structure Dynamics. In: Neittaanmäki, P., Repin, S., Tuovinen, T. (eds) Mathematical Modeling and Optimization of Complex Structures. Computational Methods in Applied Sciences, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-23564-6_15
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DOI: https://doi.org/10.1007/978-3-319-23564-6_15
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