A shooting method for the numerical solution of optimal periodic control problems

A shooting type numerical optimization technique is developed by taking into account the particular characteristics of the periodic optimal control problem. The algorithm first closes a starting trajectory to one that satisfies all of the first order necessary conditions except the transversality condition associated with free period. Then a one-dimensional family is traced in the direction of improved cost criterion. A previously developed second order sufficiency condition is used to insure weak local optimality. The points where families of periodic paths cross (bifurcation points), which are related to the eigenvalues of the monodromy matrix, are used as starting points to trace new families. In this way a systematic investigation of the optimality of the numerous families that fill the state space is possible. A numerical example, which is composed of a two-degree of freedom Hamiltonian system, is constructed to yield periodic paths and to illustrate the method. Since this problem contains certain symmetries in the initial condition space that are used to simplify the numerical effort, an extensive investigation is performed demonstrating numerous theoretical results.