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A control process is globally finite time null controllable if it is globally asymptotically stable and locally controllable to the origin. Sufficient conditions are stated for the system \[\dot x = f(t,x,u)\quad {\text{in }}C^1 (R \times R^n \times R^m )\] to be globally finite time null controllable. The conditions are stated in terms of the Jacobian of f and the controllability of a related linear equation.

The problem of Lurie is posed for systems described by a functional differential equation of neutral type. Sufficient conditions are obtained for absolute stability for the controlled system if it is assumed that the uncontrolled plant equation is uniformly asymptotically stable. Both the direct and indirect control cases are treated.

The paper studies the equation (1.1) in three cases: (i) $p \equiv 0$, (ii) $p( \ne 0)$ satisfies $| {p(t,x,y,z,w,u)} | \leqq (A_0 + | y | + | z | + | w | + | u |)\psi (t)$, where $\psi $ is a nonnegative function of t; (iii) $p( \ne 0)$ satisfies $| {p(t,x,y,z,w,u)} | \leqq A < \infty $, where A is a positive constant. In case (i) the asymptotic stability (in the large) of the solution $x = 0$ is studied; in case (ii) a general estimate and a boundedness result are deduced for solutions of (1.1); in case (iii) the ultimate boundedness of all solutions of (1.1) is proved in such a way that the ultimate bounding constant is independent of any solutions chosen. The major contribution of the paper is the successful generalization of the Routh–Hurwitz criteria for linear constant coefficient fifth order equations to the nonlinear case, a generalization that involves the use of Lyapunov functions. The results obtained seem to be as general as recent analogous treatments of fourth order equations.