Reduction of dimensionality in choosing optimal control

Abstract

Given that xðtÞ ¼ ðx 1 ðtÞ; . . . ; x n ðtÞÞ represents a trajectory of the considered dynamical system, we demonstrate that reduction of dimensionality of xðtÞ by using its principal components in choosing the optimal control uðtÞ 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.