Method of orienting curves for solving optimal control problems with state constraints1

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Abstract
This paper is concerned with a class of optimal control problems with state and control constraints which have one state variable but several control variables. The variables can appear linearly or not linearly, and the performance index and the function in the state equation are able to be non-convex in their variables. In order to overcome difficulities in solving such problems that are caused by state constraints, the Method of Orienting Curves is developed. By this method, the optimal trajectory is constructed as a path which consists of parts of orienting curves, boundary arcs, and finally, of the final curve. This method can be applied tho solve a lot of practical problems, for example, optimal control of hydroelectric power plants, Zemerlo's navigation problem with state constraints …. Here, an inventory problems is considered as application example.