Finding the Best Coefficients in the Objective Function of a Linear Quadratic Control Problem

Finding the best weights of the state variables and the control variables in the objective function of a linear-quadratic control problem is considered. The weights of these variables are considered as two diagonal matrices with appropriate size and so the objective function of the control problem becomes a function of the diagonal elements of these matrices. The optimization problem which is discussed in this paper is to minimize the objective function of the control problem as a function of these diagonal elements, when these elements are positive and the their sum in each matrix is a constant. This problem is named "the substitution between objectives in economic planning literature. In this paper, it is proved that the optimal solution of this problem is in one of the extreme points of the feasible set subject to the positive definiteness of these diagonal matrices. A method for selecting this extreme point is offered and then the unconstrained optimization problem is solved by the steepest descent method. The resulting optimal solution is the selected extreme point. Finally, the procedure is used to solve a numerical example.