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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 (F ^,G ^) over a commutative C * -algebra, where ^ means the Gelfand transform. Necessary and sufficient conditions for the stabilizability of (F ^,G ^) are obtained in terms of the corresponding Riccati equation. If the image B ^ 0 or B ^(B: the completion of B 0 ) under the Gelfand transform is *-closed, it is shown that the stabilizability of (F,G) is equivalent to that of (F ^,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 (F ^,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
MSC:
93D15Stabilization of systems by feedback
93B25Algebraic theory of control systems
93C05Linear control systems
44A15Special transforms (Legendre, Hilbert, etc.)
46H25Normed modules and Banach modules, topological modules
93B17System transformation
93C25Control systems in abstract spaces