Abstract
We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d3m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the case of simple exemplaric networks.
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J.-D. Benamou, “A domain decomposition method for the optimal control of systems governed the Helmholtz equation,” in Mathematical and Numerical Aspects of Wave Propagation, G. Cohen (Ed.), SIAM, pp. 653-662, 1995.
J.-D. Benamou, “A domain decomposition method for control problems,” in DD9 Proceedings, Bergen, P. Bjørstad (Ed.), John Wiley & Sons, 1996.
J.-D. Benamou, “A domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations,” SIAM J. Numerical Analysis, vol. 33, no. 6, pp. 2401–2416, 1996.
J.-D. Benamou, “Analyse numerique-Décomposition de domaine pour le contrôle de systèmes gouvernés par des équations d’evolution,” Indiana University Mathematics Journal, preprint, 1996.
B. Després, “Méthodes de décomposition all domaine pour les problèmes de propagation d’ondes en régimes harmoniques,” Ph.D. thesis, Paris 9, 1991.
P. Glowinski and P. Le Tallec, “Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method,” in The Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan and R. Glowinski (Eds.), SIAM, 1990.
R. Glowinski and J.L. Lions, “Exact and approximate controllability for distributed parameter systems I,” Acta Numerica, pp. 269-378, 1994.
R. Glowinski and J.L. Lions, “Exact and approximate controllability for distributed parameter systems II,” Acta Numerica, pp. 159-333, 1996.
J.E. Lagnese, G. Leugering, and E.J.P.G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser: Boston, 1994.
G. Leugering, “On domain decomposition of controlled networks of elastic strings and joint masses,” in Control of Distributed Parameter Systems, F. Kappel (Ed.), ISNM vol. 126, pp. 191–205, Birkhäuser-Verlag, Basel 1998.
G. Leugering, “Domain decomposition of optimal control problems of networks of strings and Timoshenko beams,” SIAM J. Control and Optimization, vol. 37, no. 6, pp. 1649–1675, 1999.
P.L. Lions, “On the Schwarz alternating method 3,” in The Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan and R. Glowinski (Eds.), SIAM: New York, 1990.
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag: Berlin, 1972.
A. Quarteroni and A. Valli, “Theory and application of Steklov-Poincar´e operators for boundary-value problems. The heterogenous operator case,” in The Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan and R. Glowinski (Eds.), SIAM: New York, 1994.
E.J.P.G. Schmidt, “On the modelling and exact controllability of networks of strings,” SIAM J. Control and Optimization, vol. 30, pp. 229–245, 1992.
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Leugering, G. Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings. Computational Optimization and Applications 16, 5–27 (2000). https://doi.org/10.1023/A:1008721402512
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DOI: https://doi.org/10.1023/A:1008721402512