The main purpose of this note is to present a method for solving the linear-quadratic optimal regulator for discrete, linear general two-dimensional system with constant coefficients. The quadratic optimal regulator problem can be formulated: find of sequence of control vectors in fixed rectangle, which transfer the system to given final state vector and minimizes the quadratic performance index, with constraints of control and state vector. This problem, by transformation for systemand performance index is reduced to equivalent mathematical programming problem. Necessary and sufficient conditions are established for the existence of a solution to this problem. With slight modifications the considerations can be extended for 2-D systems with variable coefficient and n-D linear systems.
