A Strong Variation Algorithm for Numerical Solutions of Optimal Control Problems with Control-Variable-Constraints

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  • 操作量拘束を含む最適制御問題に関する強変分を用いた数値解法アルゴリズム
  • ソウサリョウ コウソク オ フクム サイテキ セイギョ モンダイ ニ カンスル

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Abstract

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.<br>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;<br>ui+1ε(t)={ui(t) for t∈I1k r(t) for t∈I2k<br>where 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.<br>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.

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