Stability and the matrix Lyapunov equation for differential systems with point, distributed and/or mixed point-distributed delays

This paper establishes sufficient conditions for stability of linear and time-invariant delay differential systems including their various usual subclasses (i.e., point, distributed and mixed point-distributed delay systems). Sufficient conditions for stability are obtained in terms of the Schur's complement of operators and the frequency domain Lyapunov equation. The basic idea in the analysis consists in the use of modified Laplace operators which split the characteristic equation into two separate multiplicative factors whose roots characterize the system stability. The method allows a simple derivation of stabilizing control laws.