Optimum control of certain linear systems with quadratic loss. I

Abstract

The asymptotic properties of the particular class of N-stage, discrete-time, multidimensional linear control systems, subject to quadratic loss in both state and control, are investigated from the deterministic and stochastic points of view. A third point of view, the adaptive, will be reported on in a subsequent paper. It is shown that this class of control systems has an optimal policy, when optimality is defined as minimum of expected total quadratic loss. Expressions for the optimal policy, the minimum expected total loss, and the system-state are obtained. The asymptotic properties (as the number of stages increases indefinitely) are discussed. It is seen that the loss functional of a homogeneous deterministic system (as defined in this paper) always converges, and the loss functional of a stochastic process is asymptotic to KN (K is a known constant).