Stability criteria for linear differential difference systems

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Abstract
Stability properties of systems described by the linear differential difference equation [xdot](t) = Ax(t) + Bx(t − t) are studied. If the system x(t) = Bx(t − t) is stable enough and the term Ax(t) is ‘ small ’ enough, then the above differential difference systems may be proved to be stable. Based on this intuition, several stability criteria are derived. Studies are first carried out for systems without the term Ax(t) and then the results are extended to cover more general cases. The criteria are expressed in terms of the locations of the eigenvalues of the matrix B in the complex plane.