On the optimal control problems with characteristic time control constraints

In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples.