Conditions of Existence of the Solution of the Observation Optimization for the Kalman-Bucy Filter

Abstract

In this paper, we are concerned with a problem of optimization of the linear observations which are used in the stationary Kalman-Bucy filter. Especially, we consider the optimization of the gain matrix in the observation. In the previous works the author, by introducing a change of variables using eigenvalues and eigenvectors of a symmetric matrix, the condition of optimality of these variables was obtained. In this paper, it is shown that the expression of the optimal gain obtained in the previous works is unique, and there does not exist any other gain matrix which produces the same value of the cost function. It will be also shown in this paper that under what condition the optimal gain fails to exist and that how one can avoid the situation by modifying the weight matrix in the cost function.