On partial stability and detectability of functional differential systems with aftereffect

We consider a general class of the nonlinear non-stationary system of functional differential equations with aftereffect that admits a “partial” (with respect to a part of the variables) zero equilibrium position. We obtain conditions under which stability (asymptotic stability) with respect to a part of the variables of the “partial” equilibrium position implies its stability (asymptotic stability) in all variables. We analyze these conditions from the standpoint of the problem of partial detectability of the system in question and introduce the concept of its partial zero dynamics. We also study an application to the problem of partial stabilization of controllable systems.