Reduction of dimensionality in choosing optimal control

摘要

Given that x(t) = (x_1(t),... ,x_n(t)) represents a trajectory of the considered dynamical system, we demonstrate that reduction of dimensionality of x(t) by using its principal components in choosing the optimal control u(t) is necessary to overcome the computational difficulties that might arise in obtaining a solution of the optimal control problem. We obtain explicit solutions for linear time-varying systems with quadratic cost functional and show how to overcome difficulties appearing in non-linear matrix differential equations of Riccati type. A gradient method of searching for the close-to-optimal trajectory and control in non-linear optimal control problems is also considered.