On numerical solution of singularly perturbed optimal control problems (Q1673918)

Singularly perturbed (SP) optimal control problems in which optimal controls may take the form of rapidly oscillating functions, i. e., problems with fast and slow dynamics depending on a parameter \(\varepsilon\), are considered. It is demonstrated that a straightforward application of standard optimal control methods is computationally infeasible for small values of \(\varepsilon\). To overcome this difficulty, an averaged optimal control problem and a ``discrete-time'' version of it are considered. A convergence result establishing that a near-optimal solution of the SP problem can be constructed on the basis of an optimal solution of the time-discrete version of the averaged problem is provided. An iterative algorithm for finding numerical solutions of the time-discrete version of the averaged optimal control problem, in which the equations describing the fast dynamics are independent of the slow state variables, is proposed. The algorithm is based on a fixed-point iteration scheme using a forward and a backward sweep for the adjoints, where periodic optimal control problems have to be solved during the forward sweep. Finally, the numerical results are presented.