The authors study the feedback stabilization of linear systems given by a pair of matrices (F,G) with entries in a commutative normed ${\bbfK}$- algebra $B\sb 0$ with identity. The problem is to find a suitable feedback matrix L over $B\sb 0$ such that the closed loop system is stable. The system is transformed into another system $(\hat F,\hat G)$ over a commutative $C\sp*$-algebra, where ${\hat{\ }}$ means the Gelfand transform. Necessary and sufficient conditions for the stabilizability of $(\hat F,\hat G)$ are obtained in terms of the corresponding Riccati equation. If the image $\hat B\sb 0$ or $\hat B $(B: the completion of $B\sb 0)$ under the Gelfand transform is *-closed, it is shown that the stabilizability of (F,G) is equivalent to that of $(\hat F,\hat G)$. Another condition for the stabilizability of (F,G) is stated in terms of local stabilizability of the system which is equivalent to the local rank condition for $(\hat F,\hat G)$. An example is shown as to the positioning of a seismic cable which is written by a discrete-time linear equation.

Reviewer: T.Nambu