Optimal control problems with incomplete and different integral time domains in the objective and constraints

Optimal control problem with incomplete and different integral time domains is a class of very common practical engineering problems. In traditional way, the integral items are transformed to the transient items and treated as artificial states to reduce the complexity of programming. However, its main disadvantage is time wasting for the considered problems. In this paper, an efficient computational method is therefore proposed for this type of problem, where the integral time domains can be either fixed or variable. By employing the control vector parameterization and a timescaling transformation, the original problem is converted to an approximate optimal parameter selection problem. Moreover, new gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be easily applied to solve the generated approximate problem. For illustration, three classical numerical examples are tested. The research results, which save 10–22 % of time, show the effectivity of the proposed approach.