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Progressively Refining Discrete Gradient Projection Method for Semilinear Parabolic Optimal Control Problems

  • Ion Chryssoverghi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3401)

Abstract

We consider an optimal control problem defined by semilinear parabolic partial differential equations, with convex control constraints. Since this problem may have no classical solutions, we also formulate it in relaxed form. The classical problem is then discretized by using a finite element method in space and a theta-scheme in time, where the controls are approximated by blockwise constant classical ones. We then propose a discrete, progressively refining, gradient projection method for solving the classical, or the relaxed, problem. We prove that strong accumulation points (if they exist) of sequences generated by this method satisfy the weak optimality conditions for the continuous classical problem, and that relaxed accumulation points (which always exist) satisfy the weak optimality conditions for the continuous relaxed problem. Finally, numerical examples are given.

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References

  1. 1.
    Chryssoverghi, I.: Nonconvex optimal control problems of nonlinear monotone parabolic systems. Systems Control Lett. 8, 55–62 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chryssoverghi, I., Bacopoulos, A.: Approximation of relaxed nonlinear parabolic optimal control problems. J. Optim. Theory Appl. 77(1), 31–50 (1993)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), 311–334 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chryssoverghi, I., Coletsos, J., Kokkinis, B.: Discrete relaxed method for semilinear parabolic optimal control problems. Control Cybernet 28(2), 157–176 (1999)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (1997)zbMATHGoogle Scholar
  6. 6.
    Roubic̆ek, T.: A convergent computational method for constrained optimal relaxed control problems. Control Cybernet 69, 589–603 (1991)Google Scholar
  7. 7.
    Roubic̆ek, T.: Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin (1997)Google Scholar
  8. 8.
    Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)zbMATHGoogle Scholar
  9. 9.
    Warga, J.: Steepest descent with relaxed controls. SIAM J. Control 15(4), 674–682 (1977)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

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

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