Abstract
We investigate a new approach for solving boundary control problems for dynamical systems that are governed by transport equations, when the control function is restricted to binary values. We consider these problems as hybrid dynamical systems embedded with partial differential equations and present an optimality condition based on sensitivity analysis for the objective when the dynamics are governed by semilinear convection-reaction equations. These results make the hybrid problem accessible for continuous non-linear optimization techniques. For the computation of optimal solution approximations, we propose using meshfree solvers to overcome essential difficulties with numerical dissipation for these distributed hybrid systems. We compare results obtained by the proposed method with solutions taken from a mixed inter programming formulation of the control problem.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
- Switching Cost
- Mixed Integer Programming
- Numerical Dissipation
- Order Optimality Condition
- Optimal Boundary Control
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Amin, S., Hante, F.M., Bayen, A.M.: On stability of switched linear hyperbolic conservation laws with reflecting boundaries. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 602–605. Springer, Heidelberg (2008)
Xu, X., Antsaklis, P.: Optimal control of switched autonomous systems. In: Proc. IEEE Conf. Decision and Control, Las Vegas, NV (December 2002)
Barton, P.I.: Modeling, Simulation and Sensitivity Analysis of Hybrid Systems. In: Proc. of the IEEE Int. Symposium on Computer-Aided Control System Design, Anchorage, Alaska, September 25–27 (2000)
Bayen, A.M., Raffard, R.L., Tomlin, C.J.: Network congestion alleviation using adjoint hybrid control: Application to highways. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 95–110. Springer, Heidelberg (2004)
Bressan, A.: Hyperbolic Systems of Conservation Laws. Oxford University Press, New York (2000)
ILOG, Inc.: ILOG CPLEX Version 9.1, Sunnyvale, CA, USA (2007)
Dyer, P., McReynolds, S.R.: The Computation and Theorie of Optimal Control. Series Mathematics in Science and Engineering, vol. 65. Academic Press, New York (1970)
Egerstedt, M., Wardi, Y., Axelsson, H.: Transition-Time Optimization for Switched-Mode Dynamical Systems. IEEE Transactions on Automatic Control 51(1), 110–115 (2006)
Farjoun, Y., Seibold, B.: Solving One Dimensional Scalar Conservation Laws by Particle Management (January 2008) arXiv:0801.1495 [math.NA]
Fügenschuh, A., Herty, M., Klar, A., Martin, A.: Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks. SIAM Journal on Optimization 16, 1155–1176 (2006)
Geißler, B., Kolb, O., Lang, J., Leugering, G., Martin, A., Morsi, A.: Mixed Integer Linear Models for the Optimization of Dynamical Transport Networks (submitted, 2008)
Hante, F.M., Leugering, G., Seidman, T.I.: Modeling and Analysis of Modal Switching in Networked Transport Systems. Applied Mathematics and Optimization (in print) (2008) DOI:10.1007/s00245-008-9057-6
Kleinert, T., Lunze, J.: Modelling and state observation of Simulated Moving Bed processes based on an explicit functional wave form description. Mathematics and Computers in Simulation 68(3), 235–270 (2005)
Leugering, G.: Optimization and control of transport processes on networked systems. In: Conf. on Control of Physical Systems and Partial Differential Equations, Paris, June 16–20 (2008)
Martin, A., Möller, M., Moritz, S.: Mixed Integer Models for the Stationary Case of Gas Network Optimization. Math. Program., Ser. B 105, 563–582 (2006)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. 2. corr. printing. Springer, Berlin (1997)
Sager, S., Bock, H.G., Diehl, M., Reinelt, G., Schloder, J.P.: Numerical Methods for Optimal Control with Binary Control Functions Applied to a Lotka-Volterra Type Fishing Problem. In: Seeger, A. (ed.) Recent Advances in Optimization. LNEMS, vol. 563, pp. 269–289. Springer, Heidelberg (2006)
Sun, D., Strub, I., Bayen, M.: Comparison of the performance of four Eulerian network flow models for strategic air traffic flow management. Networks and Heterogeneous Media 2(4), 569–594 (2007)
The Mathworks, Inc.: Matlab Release 7.5.0 (R2007b), Natick, MA, USA (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hante, F.M., Leugering, G. (2009). Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions. In: Majumdar, R., Tabuada, P. (eds) Hybrid Systems: Computation and Control. HSCC 2009. Lecture Notes in Computer Science, vol 5469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00602-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-00602-9_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00601-2
Online ISBN: 978-3-642-00602-9
eBook Packages: Computer ScienceComputer Science (R0)