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Stabilizability of linear systems over a commutative normed algebra with applications to spatially distributed and parameter-dependent systems. (English) Zbl 0564.93054
The authors study the feedback stabilization of linear systems given by a pair of matrices (F,G) with entries in a commutative normed $𝕂$- algebra ${B}_{0}$ with identity. The problem is to find a suitable feedback matrix L over ${B}_{0}$ such that the closed loop system is stable. The system is transformed into another system $\left(\stackrel{^}{F},\stackrel{^}{G}\right)$ over a commutative ${C}^{*}$-algebra, where $\stackrel{^}{\phantom{\rule{4pt}{0ex}}}$ means the Gelfand transform. Necessary and sufficient conditions for the stabilizability of $\left(\stackrel{^}{F},\stackrel{^}{G}\right)$ are obtained in terms of the corresponding Riccati equation. If the image ${\stackrel{^}{B}}_{0}$ or $\stackrel{^}{B}$(B: the completion of ${B}_{0}\right)$ under the Gelfand transform is *-closed, it is shown that the stabilizability of (F,G) is equivalent to that of $\left(\stackrel{^}{F},\stackrel{^}{G}\right)$. 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 $\left(\stackrel{^}{F},\stackrel{^}{G}\right)$. An example is shown as to the positioning of a seismic cable which is written by a discrete-time linear equation.
Reviewer: T.Nambu
##### MSC:
 93D15 Stabilization of systems by feedback 93B25 Algebraic theory of control systems 93C05 Linear control systems 44A15 Special transforms (Legendre, Hilbert, etc.) 46H25 Normed modules and Banach modules, topological modules 93B17 System transformation 93C25 Control systems in abstract spaces