Advertisement

Optimal Control of a Class of Size-Structured Systems

  • Oana Carmen Tarniceriu
  • Vladimir M. Veliov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

Optimality conditions of Pontryagin’s type are obtained for an optimal control problem for a size-structured system described by a first order PDE where the differential operator depends on a control and on an aggregated state variable. Typically in this sort of problems the state function is not differentiable, even discontinuous, which creates difficulties for the variational analysis. Using the method of characteristics (which are control and state dependent for the considered system) the problem is reformulated as an optimization problem for a heterogeneous control system, investigated earlier by the second author. Based on this transformation, the optimality conditions are obtained and a stylized meaningful example is given where the optimality conditions allow to obtain an explicit differential equation for the optimal control.

Keywords

Optimal Control Problem Adjoint System Gradient Projection Method Population Context Continuous Dependence Result 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ackleh, A., Deng, K., Wang, X.: Competitive exclusion and coexistence for a quasilinear size-structured population model. Mathematical Biosciences 192, 177–192 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abia, L.M., Angulo, O., Lopez-Marcos, J.C.: Size-structured population dynamics models and their numerical solutions. Discrete Contin. Dyn. Syst. Ser. B4(4), 1203–1222 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chryssoverghi, I.: Approximate Gradient Projection Method with General Runge-Kutta Schemes and Piecewise Polynomials Controls for Optimal Control Problems. Control and Cybernetics 34(2), 425–451 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Farkas, J.Z.: Stability conditions for a nonlinear size-structured model. Nonlinear Analysis 6(5), 962–969 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kato, N., Sato, K.: Continuous dependence results for a general model of size dependent population dynamics. J. Math. Anal. Appl. 272, 200–222 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kato, N.: A general model of size-dependent population dynamics with nonlinear growth rate. J. Math. Anal. Appl. 297, 234–256 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Nikol’skii, M.S.: Convergence of the gradient projection method in optimal control problems. Computational Mathematics and Modeling 18(2), 148–156 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Schwartz, L.: Analyse Mathématique. Hermann (1967)Google Scholar
  9. 9.
    Veliov, V.M.: Newton’s method for problems of optimal control of heterogeneous systems. Optimization Methods and Software 18(6), 689–703 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Oana Carmen Tarniceriu
    • 1
  • Vladimir M. Veliov
    • 2
    • 3
  1. 1.Faculty of Mathematics“Al. I. Cuza” UniversityRomania
  2. 2.Institute Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesBulgaria

Personalised recommendations