Stability tests for one- two- and multidimensional linear systems

The paper reviews analytical stability tests for 1-dimensional linear systems since the early tests of E.J. Routh in his famous Adams Prize essay of 1877. The historical background of Routh´s stability test and criterion, as well as Fuller´s conjecture on its simplification, will be mentioned. In the historical review, the works of Hermite, Sylvester, Maxwell and others as related to the stability problem are also discussed. This review provides the context for a discussion of recent stability tests obtained for 2-dimensional and multidimensional linear systems. These tests are described and their computational complexity discussed in detail. In addition, the applications of stability testing to the study of 2- and multidimensional digital filters, numerical analysis of stiff-differential equations, realisation of mixed lumped- and distributed parameter systems, and the design of output feedback systems will be briefly mentioned. Comments on future research in this area conclude the review.