Radial Basis Function Networks Simulation of Age-Structure Population

  • Tibor Kmet
  • Maria Kmetova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11324)


Based on radial basis function networks (RBFN) we propose a new method to solve optimal control problems for systems governed by hyperbolic partial differential equations. RBFN are able to approximate continuous function with given precision [1]. This property of RBFN was used to estimate optimal control, optimal state trajectory and optimal co-state trajectory by adaptive critic design. A new algorithm was verified and compared with indirect methods on the harvesting model of age-dependent population.


Radial basis function networks Hyperbolic optimal control problem Adaptive critic synthesis Age-dependent population model Numerical simulations 


  1. 1.
    Park, J., Sandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991)CrossRefGoogle Scholar
  2. 2.
    Kmet, T., Kmetova, M.: Neural networks simulation of distributed control problems with state and control constraints. In: Villa, A.E.P., Masulli, P., Pons Rivero, A.J. (eds.) ICANN 2016. LNCS, vol. 9886, pp. 468–477. Springer, Cham (2016). Scholar
  3. 3.
    Kmet, T., Kmetova, M.: Echo state networks simulation of SIR distributed control. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2017. LNCS (LNAI), vol. 10245, pp. 86–96. Springer, Cham (2017). Scholar
  4. 4.
    Wang, D., Liu, D., Wei, Q., Zhao, D., Jin, N.: Optimal control of unknown nonaffine nonlinear discrete-time systems based on adaptive dynamic programming. Automatica 48, 1825–1832 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Borzi, A., Schulz, V.: Multigrid methods for pde optimization.
  6. 6.
    Herty, M., Kurganov, A., Kurichkin, D.: Numerical methods for optimal control problems governed by nonlinear hyperbolic systems of pdes. Commun. Math. Sci. 13(1), 15–48 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Knowles, G.: Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20, 414–427 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mittelmann, H.D.: Solving elliptic control problems with interior point and sqp methods: control and state constraints. J. Comput. Appl. Math. 120, 175–195 (2000)MathSciNetCrossRefGoogle Scholar
  9. 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 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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. Van Nostrand, New York (1992)Google Scholar
  12. 12.
    Nam, M.d., Thanh, T.C.: Approximation of function and its derivatives using radial basis function networks. Appl. Math. Model. 27, 197–220 (2003)Google Scholar
  13. 13.
    Parzlivand, F., Shahrezaee, A.M.: Numerical solution of an inverse reaction-diffusion problem via collocation methods based on radial basis function. Appl. Math. Model. 39, 3733–3744 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rad, J.A., Kazen, S., Parand, K.: Optimal control of a parabolic distributed parameter system via radial basis functions. Commun. Nonlinear Sci. Number Simulat. 19, 2559–2567 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Antolin, A.G., Garcia, J.P., Gomez, J.L.S.: Radial basis function networks and their application in communication systems. In: Andina, D., Pham, D.T. (eds.) Comput. Intell., pp. 109–130. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001)CrossRefGoogle Scholar
  17. 17.
    McKendrick, A.: Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1926)CrossRefGoogle Scholar
  18. 18.
    Kwon, H.D., Lee, J., Yang, S.D.: Optimal control of an age-structured model of hiv infection. Appl. Math. Comput. 219, 2766–2779 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R., Mischenko, E.F.: The Mathematical Theory of Optimal Process. Nauka, Moscow (1983). (in Russian)Google Scholar
  20. 20.
    Bryson Jr., A.E.: Dynamic Optimization. Addison-Wesley Longman Inc., New York (1999)Google Scholar
  21. 21.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, New York (1989)Google Scholar
  22. 22.
    Brokate, M.: Pontryagin’s principle for control problems in age-dependent population dynamics. J. Math. Biol. 23(1), 75–101 (1985)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hritonenko, N., Yatsenko, Y.: The structure of optimal time- and age-dependent harvesting in the lotka-mckendric population model. Math. Biosci. 208, 48–62 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsJ. Selye UniversityKomarnoSlovakia

Personalised recommendations