Controlling Trajectories Globally via Spatiotemporal Finite-Time Optimal Control

The problems of (i) maximizing or minimizing Lagrangian mixing in fluids via the introduction of a spatiotemporally varying control velocity and (ii) globally controlling the finite-time location of trajectories beginning at all initial conditions in a chaotic system are considered. A particular form of solution to these is designed which uses a new methodology for computing a spatiotemporally dependent optimal control. An L²-error norm for trajectory locations over a finite-time horizon is combined with a penalty energy norm for the control velocity in defining the global cost function. A computational algorithm for cost minimization is developed, and theoretical results on global error and cost presented. Numerical simulations (using velocities which are specified, and obtained as data from computational fluid dynamics simulations) are used to demonstrate the efficacy and validity of the approach in determining the required spatiotemporally defined control velocity.