Optimal control
semilinear parabolic systems
state constraints
relaxed controls
discretization
$\theta$scheme
discrete penalized conditional descent method
DiscretizationOptimization Methods for Nonlinear Parabolic Relaxed Optimal Control Problems with State Constraints
Ion Chryssoverghi
Chryssoverghi
Ion
J. Coletsos
Coletsos
J.
Juergen Geiser
Geiser
Juergen
B. Kokkinis
Kokkinis
B.
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 22 pp.
DiscretizationOptimization Methods for Nonlinear Parabolic Relaxed Optimal Control Problems with State Constraints
Ion Chryssoverghi
,
J. Coletsos,
Juergen Geiser
,
B. Kokkinis
Preprint series:
Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976), 22 pp.
MSC 2000
 49M25 Discrete approximations

49M05 Methods based on necessary conditions
Abstract
We consider an optimal control problem described by a semilinear
parabolic partial differential equation, with control and
state constraints, where the state constraints and cost involve also the state
gradient. Since this problem may have no classical solutions, it is
reformulated in the relaxed form. The relaxed control problem is
discretized by using a finite element method in space involving numerical
integration and an implicit thetascheme in time for space approximation,
while the controls are approximated by blockwise constant relaxed
controls. Under appropriate assumptions, we prove that relaxed
accumulation points of sequences of optimal (resp. admissible and extremal)
discrete relaxed controls are optimal (resp. admissible and extremal)
for the continuous relaxed problem. We then apply a penalized
conditional descent method to each discrete problem, and also a
progressively refining version of this method to the continuous
relaxed problem. We prove that accumulation points of sequences
generated by the first method are extremal for the discrete problem,
and that relaxed accumulation points of sequences of discrete
controls generated by the second method are admissible
and extremal for the continuous relaxed problem. Finally, numerical
examples are given.
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