Stability of matrices with sufficiently strong negative-dominant-diagonal submatrices
✍ Scribed by Herman J. Nieuwenhuis; Lambert Schoonbeek
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 908 KB
- Volume
- 258
- Category
- Article
- ISSN
- 0024-3795
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✦ Synopsis
A well-known sufficient condition for stability of a system of linear first-order differential equations is that the matrix of the homogeneous dynamics has a negative dominant diagonal. However, this condition cannot be applied to systems of secondorder differential equations. In this paper we introduce the concept of a (negative) dominant diagonal with a given strength factor. Using this, we present stability theorems which show that second-order systems are stable if the matrix of the homogeneous dynamics has submatrices with a sufficiently strong negative dominant diagonal. © Elsevier Science Inc., 1997 l. INTRODUCTION Consider the homogeneous system of linear first-order differential equations y = By, where y = (Yi) represents an n × 1 vector of variables and its time derivative, and where B =-(bij) is a real matrix of dimensions n × n.
As is well known, this system is asymptotically stable if and only if all eigenvalues of the matrix B have negative real parts. Asymptotic stability of the system means that any solution converges towards the equilibrium, i.e. the n × 1 vector of which all elements are equal to zero. We call a square matrix stable if all its eigenvalues have negative real parts. For a general reference, see e.g. [8], [9], [10], or [13].