Linear dynamically varying LQ control of nonlinear systems over compact sets

Linear-quadratic controllers for tracking natural and composite trajectorie s of nonlinear dynamical systems evoluting over compact sets are developed, Typically, such systems exhibit "complicated dynamics," i.e., have nontriv ial recurrence, The controllers, which use small perturbations of the nomin al dynamics as input actuators, are based on modeling the tracking error as a linear dynamically varying (LDV) system. Necessary and:sufficient condit ions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic Riccati equation (FARE), It is shown that, despite the lack of continuity of the asymptotic trajectory re lative to initial conditions,, the cost to stabilize about the trajectory, as given by the solution to the FARE, is continuous. An ergodic theory meth od for solving the FARE is presented. Furthermore, it is shown that wrappin g the LDV controller around the nonlinear system secures a stable tracking dynamics. Finally, an example of controlling the Henon map to follow an ape riodic orbit is presented.