DC-dominant property of cone-preserving transfer functions

We consider a class of square MIMO transfer functions that map a proper cone in the space of L2L2 input signals to the same cone in the space of output signals. Transfer functions in this class have the “DC-dominant” property: the maximum radius of the operator spectrum is attained by a DC input signal and, hence, the dynamic stability of the feedback interconnection of such transfer functions is guaranteed solely by static gain analysis. Using this property, we prove that cone-preserving linear delay differential equations are robustly stable against arbitrary constant delay values. This provides an alternative proof of the delay-independent mean-square stability of a multi-dimensional geometric Brownian motion.