TY - JOUR
T1 - Discretization Methods for Semilinear Parabolic Optimal Control Problems
AU - Chryssoverghi , Ion
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 437
EP - 458
PY - 2006
DA - 2006/03
SN - 3
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/912.html
KW - optimal control, parabolic systems, discretization, piecewise bilinear controls, penalized gradient projection method, relaxed controls.
AB - <p style="text-align: justify;">We consider an optimal control problem described by
semilinear parabolic partial differential equations, with control and
state constraints. Since this problem may have no classical solutions,
it is also formulated in the relaxed form. The classical control problem
is then discretized by using a finite element method in space and the
implicit Crank-Nicolson midpoint scheme in time, while the controls are
approximated by classical controls that are bilinear on pairs of blocks.
We prove that strong accumulation points in $L^2$ of sequences of optimal
(resp. admissible and extremal) discrete controls are optimal (resp.
admissible and weakly extremal classical) for the continuous classical
problem, and that relaxed accumulation points of sequences of optimal
(resp. admissible and extremal relaxed) discrete controls are optimal
(resp. admissible and weakly extremal relaxed) for the continuous
relaxed problem. We then apply a penalized gradient projection method to
each discrete problem, and also a progressively refining version of the
discrete method to the continuous classical problem. Under appropriate
assumptions, we prove that accumulation points of sequences generated by
the first method are admissible and extremal for the discrete problem,
and that strong classical (resp. relaxed) accumulation points of
sequences of discrete controls generated by the second method are
admissible and weakly extremal classical (resp. relaxed) for the
continuous classical (resp. relaxed) problem. For nonconvex problems
whose solutions are non-classical, we show that we can apply the above
methods to the problem formulated in Gamkrelidze relaxed form. Finally,
numerical examples are given.</p>