Advertisement

Discrete Methods for Optimal Control Problems

  • Ion Chryssoverghi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

We consider a constrained optimal control problem, which we formulate in classical and in relaxed form. In order to approximate this problem numerically, we apply various discretization schemes on either of these two forms and study the behavior in the limit of discrete optimality and necessary conditions for optimality. We then propose discrete mixed gradient penalty methods that use classical or relaxed discrete controls and progressively refine the discretization, thus reducing computing time and memory. In addition, when the discrete adjoint state is not defined or difficult to calculate, we propose discrete methods that use approximate adjoints and derivatives. The result is that in relaxed methods accumulation points of generated sequences satisfy continuous strong relaxed optimality conditions, while in classical methods they satisfy weak optimality conditions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chryssoverghi, I., Bacopoulos, A.: Discrete approximation of relaxed optimal control problems. Journal of Optimization Theory and Applications, 65, 3 (1990) 395–407.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chryssoverghi, I., Bacopoulos, A.: Approximation of relaxed nonlinear parabolic optimal control problems. JOTA, 77, 1 (1993) 31–50.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chryssoverghi, I., Bacopoulos, A., Kokkinis, B., Coletsos, J.: Mixed Frank-Wolfe penalty method with applications to nonconvex optimal control problems. JOTA, 94, 2 (1997) 311–334.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chryssoverghi, I., Bacopoulos, A., Coletsos, J., Kokkinis, B.: Discrete approximation of nonconvex hyperbolic optimal control problems with state constraints. Control & Cybernetics, 27, 1 (1998) 29–50.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chryssoverghi, I., Coletsos, J., Kokkinis, B.: Discrete relaxed method for semilinear parabolic optimal control problems. Control & Cybernetics, 28, 2 (1999) 157–176.zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chryssoverghi, I., Coletsos, J., Kokkinis, B.: Approximate relaxed descent method for optimal control problems. Control & Cybernetics, 30, 4 (2001) 385–404.zbMATHGoogle Scholar
  7. 7.
    Dontchev, A.L., Hager, W. Veliov, V.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal., 38 (2000) 202–226.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (1997)zbMATHGoogle Scholar
  9. 9.
    Roubícek, T.: A convergent computational method for constrained optimal relaxed control problems. JOTA, 69 (1991) 589–603.zbMATHCrossRefGoogle Scholar
  10. 10.
    Roubícek, T.: Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin (1997)zbMATHGoogle Scholar
  11. 11.
    Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)zbMATHGoogle Scholar
  12. 12.
    Warga, J.: Steepest descent with relaxed controls. SIAM J. on Control, 15, 4 (1977) 674–689.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ion Chryssoverghi
    • 1
  1. 1.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations