Strong variation techniques have been found to be efficient in the theoretical study of optimal control problems and it can be expected that such techniques may play an important role in a field of computational methods. Strong variations, however, had not attracted attention before “step size adjustment method” using such variations was introduced in differential dynamic programming.
This paper proposes a strong variation algorithm for numerical solutions of controlvariable-constrained optimal control problems. In this algorithm a new “step size adjustment method” is presented. The control interval is divided into subintervals
Ik(
k=1, …,
m) and
Ik into
I1k,
I2k (|
I1k|=(1-ε)|
Ik|, |
I2k|=ε|
Ik|, and ε∈[0, 1]). While adjusting the value of ε and the number of the subintervals
m, a new control
ui+1 is determined by calculating
ui+1ε according to the following formula;
ui+1ε(t)={
ui(t) for
t∈
I1k r(t) for
t∈
I2kwhere
ui and
r are the present control and a control maximizing the related Hamiltonian function, respectively. The idea of the switching scheme is based on the concept of chattering controls first proposed by Gamkrelidze.
It is proved that accumulation points generated by the algorithm, if exist, satisfy maximum principle. Some examples are given to show the usefulness of the algorithm. Incidental conditions concerning the size of
Ik are considered and the related problems in programming are discussed.
View full abstract