Stabilizability of Linear Systems Over a Commutative Normed Algebra with Applications to Spatially-Distributed and Parameter-Dependent Systems

The problem of achieving stabilization by using state feedback is considered for linear systems given by a pair of matrices whose entries belong to a real or complex commutative normed algebra. This framework is applicable to various types of linear systems, including spatially-distributed systems, systems depending on parameters, and infinite-dimensional systems. Necessary and sufficient conditions for stabilizability are derived in terms of solutions to an associated Riccati equation defined in the Gelfand-transform domain. Necessary and sufficient conditions for stabilizability are also given in terms of a local rank criterion involving the Gelfand transform of the system coefficients. The results are applied to the problem of positioning a long seismic cable.