The periodic Riccati differential equation and the periodic regulator

The problem of controlling small state deviations from a periodic path generated by the solution to an autonomous optimal control problem with periodic boundary conditions is considered by solving the associated infinite-time accessary minimum problem. Some of the properties of the resulting solution to the periodic Riccati differential equation with periodic boundary condition are derived using Floquet theorem. Given a controllability condition, the periodic Riccati solution required for the periodic regulator is characterized in terms of the eigenvalues and periodic eigenvectors associated with the monodromy matrix, the transition matrix associated with the perturbed Hamiltonian system evaluated over the period.