Abstract
Optimal control problem with partial derivative equation (PDE) constraint is a numerical-wise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a new approach to solve this type of problem by space discretization and transform it into a standard optimal control for a multi-agent system. This resulting problem is formulated from a microscopic perspective while the solution only needs limited the macroscopic measurement due to the approach of Hamilton-Jacobi-Bellman (HJB) equation approximation. For solving the problem, only an HJB equation (a PDE with only terminal boundary condition) needs to be solved, although the dimension of that PDE is increased as a drawback. A pollutant elimination problem is considered as an example and solved by this approach. A numerical method for solving the HJB equation is proposed and a simulation is carried out.
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References
Luenberger D G, Optimization by Vector Space Methods, John Wiley & Sons, New York, 1969.
Athans M and Falb P L, Optimal Control: An Introduction to the Theory and Its Applications, Courier Corporation, New York, 2013.
Bellman R, Dynamic programming and Lagrange multipliers. Proceedings of the National Academy of Sciences, 1956, 42(10): 767–769.
Tröltzsch F, Optimal control of partial differential equations. Graduate Studies in Mathematics, 2010, 112.
Aziz A K, Wingate J W, and Balas M J, Control Theory of Systems Governed by Partial Differential Equations, Academic Press, New York, 2014.
Bergounioux M and Kunisch K, Primal-dual strategy for state-constrained optimal control problems, Computational Optimization and Applications, 2002, 22(2): 193–224.
Neitzel I, Prüfert U, and Slawig T, Solving time-dependent optimal control problems in COMSOL multiphysics. European COMSOL Conference, 2008.
Olfati-Saber R and Murray R M, Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 2004, 49(9): 1520–1533.
Ren W and Beard R W, Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 2005, 50(5): 655–661.
Ji M, Muhammad A, and Egerstedt M, Leader-based multi-agent coordination: Controllability and optimal control, American Control Conference, 2006, 1358–1363.
Yang Y, Dimarogonas V D, and Hu X, Optimal leader-follower control for crowd evacuation, 52nd IEEE Conference on Decision and Control. IEEE, 2013.
Yang Y, Dimarogonas V D, and Hu X, Shaping up crowd of agents through controlling their statistical moments, European Control Conference (ECC). IEEE, 2015.
Øksendal B K and Sulem A, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2005.
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This research was supported by the Swedish Research Council (VR).
This paper was recommended for publication by Editor HONG Yiguang.
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Yang, Y., Hu, X. Optimal control using microscopic models for a pollutant elimination problem. J Syst Sci Complex 30, 86–100 (2017). https://doi.org/10.1007/s11424-017-6185-6
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DOI: https://doi.org/10.1007/s11424-017-6185-6