An Exact Penalty Method for Free Terminal Time Optimal Control Problem with Continuous Inequality Constraints

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 maximization for a hypersonic vehicle reentering the atmosphere subject to a heating constraint demonstrate the effectiveness of our method. Research supported by a scholarship under the State Scholarship Fund of China and a grant from the Australian Research Council. Communicated by George Leitmann. C. Jiang · G.-R. Duan Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, P.R.China E-mail: [email protected] G.-R. Duan E-mail: [email protected] Q. Lin · C. Yu · K. L. Teo (B) Department of Mathematics and Statistics, Curtin University, Perth 6845, Australia E-mail: [email protected] C. Yu E-mail: [email protected] K. L. Teo E-mail: [email protected]