In this paper, we consider a class of optimal control problems with free terminal time and continuous inequality constraints. First, the problem is approximated by representing the control function as a piecewise-constant function. Then the continuous inequality constraints are transformed into terminal equality constraints for an auxiliary differential system. After these two steps, we transform the constrained optimization problem into a penalized problem with only box constraints on the decision variables using a novel exact penalty function. This penalized problem is then solved by a gradient-based optimization technique. Theoretical analysis proves that this penalty function has continuous derivatives, and for a sufficiently large and finite penalty parameter, its local minimizer is feasible in the sense that the continuous inequality constraints are satisfied. Furthermore, this local minimizer is also the local minimizer of the constrained problem. Numerical simulations on the range maximiCommunicated by George Leitmann. J Optim Theory Appl (2012) 154:30-53 31 zation for a hypersonic vehicle reentering the atmosphere subject to a heating constraint demonstrate the effectiveness of our method.
For the solution of optimal control problem involving an index-1 differential-algebraic equation, an efficient function evaluation algorithm is proposed in this paper. In the evaluation procedure, the state equation is propagated forwards, then, adjoint sensitivity is propagated backwards. Thus, it is computationally more efficient than forward sensitivity propagation when the number of constraints is less than that of optimization variables. In order to reduce Newton iterations, the adjoint sensitivity is derived utilizing the implicit function theorem, and the integration procedure is accelerated by incorporating a predictor-corrector strategy. This algorithm is integrated with a nonlinear programming solver Ipopt to solve sequentially the point-to-point optimal control for a Delta robot with constrained motor torque. Numerical experiments demonstrate the efficiency of this algorithm.
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