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
Chryssoverghi, I.: Nonconvex optimal control problems of nonlinear monotone parabolic systems. Systems Control Lett. 8, 55–62 (1986)
Chryssoverghi, I., Bacopoulos, A.: Approximation of relaxed nonlinear parabolic optimal control problems. J. Optim. Theory Appl. 77(1), 31–50 (1993)
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)
Chryssoverghi, I., Coletsos, J., Kokkinis, B.: Discrete relaxed method for semilinear parabolic optimal control problems. Control Cybernet 28(2), 157–176 (1999)
Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (1997)
Roubic̆ek, T.: A convergent computational method for constrained optimal relaxed control problems. Control Cybernet 69, 589–603 (1991)
Roubic̆ek, T.: Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin (1997)
Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)
Warga, J.: Steepest descent with relaxed controls. SIAM J. Control 15(4), 674–682 (1977)
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© 2005 Springer-Verlag Berlin Heidelberg
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Chryssoverghi, I. (2005). Progressively Refining Discrete Gradient Projection Method for Semilinear Parabolic Optimal Control Problems. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_28
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DOI: https://doi.org/10.1007/978-3-540-31852-1_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24937-5
Online ISBN: 978-3-540-31852-1
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