Application of extended Gershgorin theorems to certain distributed-parameter control problems

The paper presents a survey of theory and application of an extended version of Gershgorin´s theorems towards infinite matrix representations of distributed parameter systems (DPS). The paper gives an answer to the question: How are the eigenvalues of the actually infinite-dimensional control system related to those of its finite Galerkin approximation? Modal expansions are covered as a special case. Two theorems will be provided based on different assumptions on diagonal dominance. As a result, all eigenvalues of the DPS will be enclosed in Gershgorin disks. This provides a sufficient stability criterion, whenever the spectrum of the DPS consists only of eigenvalues. The basic results will be applied to a number of control problems, including numerical examples.