Switching Time Optimization for Bang-Bang and Singular Controls

Optimal control problems with the control variable appearing linearly are studied. A method for optimization with respect to the switching times of controls containing both bang-bang and singular arcs is presented. This method is based on the transformation of the control problem into a finite-dimensional optimization problem. Therein, first and second-order optimality conditions are thoroughly discussed. Explicit representations of first and second-order variational derivatives of the state trajectory with respect to the switching times are given. These formulas are used to prove that the second-order sufficient conditions can be verified on the basis of only first-order variational derivatives of the state trajectory. The effectiveness of the proposed method is tested with two numerical examples.