ANZIAM J.
43 (2002), 463478

Numerical solution of an optimal control problem with variable time points in the objective function

K. L. Teo
Department of Applied Mathematics
Hong Kong Polytechnic University
Hung Hom
Koowloon
Hong Kong


W. R. Lee
Department of Mathematics and Statistics
Curtin University of Technology
GPO Box U 1987
Perth
WA 6845
Australia


L. S. Jennings
Center for Applied Dynamics and Optimization
The University of Western Australia
WA 6907
Australia



S. Wang
Center for Applied Dynamics and Optimization
The University of Western Australia
WA 6907
Australia


and

Y. Liu
Department of Applied Mathematics
Hong Kong Polytechnic University
Hung Hom
Koowloon
Hong Kong



Abstract

In this paper, we consider the numerical
solution of a class of optimal control problems
involving variable time points in their cost
functions. The control enhancing transform is
first used to convert the optimal control problem
with variable time points into an equivalent
optimal control problem with fixed multiple
characteristic time (MCT). Using the control
parametrization technique, the time horizon is
partitioned into several subintervals. Let the
partition points also be taken as decision
variables. The control functions are
approximated by piecewise constant or piecewise
linear functions in accordance with these
variable partition points. We thus obtain a
finite dimensional optimization problem. The
control parametrization enhancing control
transform (CPET) is again used to convert
approximate optimal control problems with
variable partition points into equivalent
standard optimal control problems with MCT, where
the control functions are piecewise constant or
piecewise linear functions with prefixed
partition points. The transformed problems are
essentially optimal parameter selection problems
with MCT. The gradient formulae for the objective
function as well as the constraint functions with
respect to relevant decision variables are
obtained. Numerical examples are solved using the
proposed method.

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