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Neural Networks Solution of Optimal Control Problems with Discrete Time Delays and Time-Dependent Learning of Infinitesimal Dynamic System

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Artificial Neural Networks

Part of the book series: Springer Series in Bio-/Neuroinformatics ((SSBN,volume 4))

Abstract

The paper presented describes two possible applications of artificial neural networks. The first application is related to solve optimal control problems with discrete time delays in state and control variables subject to control and state constraints. The optimal control problem is transcribed into nonlinear programming problem which is implemented with feed forward adaptive critic neural network to find optimal control and optimal trajectory. The proposed simulation methods are illustrated by the optimal control problem of nitrogen transformation cycle model with discrete time delay of nutrient uptake. The second application deals with backpropagation learning of infinite-dimensional dynamical systems. The proposed simulation methods are illustrated by the back-propagation learning of continuous multilayer Hopfield neural network with discrete time delay using optimal trajectory as teacher signal. Results show that adaptive critic based systematic approach are promising in obtaining the optimal controlwith discrete time delays in state and control variables subject to control and state constraints and that continuous Hopfield neural networks are able to approximate signals generated from optimal trajectory.

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References

  1. Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39, 930–945 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bryson Jr., A.E.: Dynamic Optimization. Addison-Wesley Longman Inc., New York (1999)

    Google Scholar 

  3. Buskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variable, sensitivity analysis and real-time control. Journal of Computational and Applied Mathematics 120, 85–108 (2000)

    Article  MathSciNet  Google Scholar 

  4. Hopfield, J.J.: Neural Networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79, 2554–2554 (1982)

    Article  MathSciNet  Google Scholar 

  5. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. 81, 3088–3092 (1984)

    Article  Google Scholar 

  6. Hornik, M., Stichcombe, M., White, H.: Multilayer feed forward networks are universal approximators. Neural Networks 3, 256–366 (1989)

    Google Scholar 

  7. Hrinca, I.: An Optimal Control Problem for the Lotka-Volterra System with Delay. Nonlinear Analysis, Theory, Methods, Applications 28, 247–262 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gnecco, A.: A Comparison Between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization. Journal of Applied Mathematics 2012, article ID 806945 (2012)

    Google Scholar 

  9. Gollman, L., Kern, D., Mauer, H.: Optimal control problem with delays in state and control variables subject to mixed control-state constraints. Optim. Control Appl. Meth. 30, 341–365 (2009)

    Article  Google Scholar 

  10. Chai, Q., Loxton, R., Teo, K.L., Yang, C.: A class of optimal state-delay control problems. Nonlinear Analysis: Real World Applications 14, 1536–1550 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, Inc., Mineola (1989)

    Google Scholar 

  12. Kmet, T.: Material recycling in a closed aquatic ecosystem. I. Nitrogen transformation cycle and preferential utilization of ammonium to nitrate by phytoplankton as an optimal control problem. Bull. Math. Biol. 58, 957–982 (1996)

    Article  MATH  Google Scholar 

  13. Kmet, T.: Neural network solution of optimal control problem with control and state constraints. In: Honkela, T. (ed.) ICANN 2011, Part II. LNCS, vol. 6792, pp. 261–268. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Kmet, T., Kmetova, M.: Adaptive Critic Neural Network Solution of Optimal Control Problems with Discrete Time Delays. In: Mladenov, V., Koprinkova-Hristova, P., Palm, G., Villa, A.E.P., Appollini, B., Kasabov, N. (eds.) ICANN 2013. LNCS, vol. 8131, pp. 483–494. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  15. Makozov, Y.: Uniform approximation by neural networks. Journal of Approximation Theory 95, 215–228 (1998)

    Article  MathSciNet  Google Scholar 

  16. Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)

    Article  MATH  Google Scholar 

  17. Padhi, R., Balakrishnan, S.N., Randoltph, T.: A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems. Neural Networks 19, 1648–1660 (2006)

    Article  MATH  Google Scholar 

  18. Polak, E.: Optimization Algorithms and Consistent Approximation. Springer, Heidelberg (1997)

    Google Scholar 

  19. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Process. Nauka, Moscow (1983) (in Russian)

    Google Scholar 

  20. Rumelhart, D.F., Hinton, G.E., Wiliams, R.J.: Learning internal representation by error propagation. In: Rumelhart, D.E., McClelland, D.E. (eds.) PDP Research Group: Parallel Distributed Processing: Foundation, vol. 1, pp. 318–362. The MIT Press, Cambridge (1987)

    Google Scholar 

  21. Sandberg, E.W.: Notes on uniform approximation of time-varying systems on finite time intervals. IEEE Transactions on Circuits and Systems-1: Fundamental Theory and Applications 45, 305–325 (1998)

    Google Scholar 

  22. Sharif, H.R., Vali, M.A., Samat, M., Gharavisi, A.A.: A New Algorithm for Optimal Control of Time-Delay Systems. Applied Mathematical Science 5, 595–606 (2011)

    MATH  Google Scholar 

  23. Sun, D.Y., Huang, T.C.: A solutions of time-delayed optimal control problems by the use of modified line-up competition algorithm. Journal of the Taiwan Institute of Chemical Engineers 41, 54–64 (2010)

    Article  Google Scholar 

  24. Tokuda, I., Tokunaga, R., Aihara, K.: Back-propagation learning of infinite-dimensional dynamical systems. Neural Networks 16, 1179–1193 (2003)

    Article  Google Scholar 

  25. Werbos, P.J.: Approximate dynamic programming for real-time control and neural modelling. In: White, D.A., Sofge, D.A. (eds.) Handbook of Intelligent Control: Neural Fuzzy, and Adaptive Approaches, pp. 493–525 (1992)

    Google Scholar 

  26. Xia, Y., Feng, G.: A New Neural Network for Solving Nonlinear Projection Equations. Neural Network 20, 577–589 (2007)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, Constantine the Philosopher University, Tr. A. Hlinku 1, 949 74, Nitra, Slovakia

    Tibor Kmet

  2. Department of Mathematics, Constantine the Philosopher University, Tr. A. Hlinku 1, 949 74, Nitra, Slovakia

    Maria Kmetova

Authors
  1. Tibor Kmet
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  2. Maria Kmetova
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Correspondence to Tibor Kmet .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

    Petia Koprinkova-Hristova

  2. Technical University Sofia, Sofia, Bulgaria

    Valeri Mladenov

  3. Auckland University of Technology, Auckland, New Zealand

    Nikola K. Kasabov

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Kmet, T., Kmetova, M. (2015). Neural Networks Solution of Optimal Control Problems with Discrete Time Delays and Time-Dependent Learning of Infinitesimal Dynamic System. In: Koprinkova-Hristova, P., Mladenov, V., Kasabov, N. (eds) Artificial Neural Networks. Springer Series in Bio-/Neuroinformatics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-09903-3_15

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  • DOI: https://doi.org/10.1007/978-3-319-09903-3_15

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09902-6

  • Online ISBN: 978-3-319-09903-3

  • eBook Packages: EngineeringEngineering (R0)

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