B.S.T.J. Briefs: A Condition for the L8-Stability of Feedback SystemsContaining a Single Time-Varying Nonlinear Element

In the automatic control literature, a feedback system is frequently said to be stable if, regardless of the initial state of the system, each bounded input applied at / = 0 produces a bounded output. The purpose of this brief is to present a sufficient condition for the feedback system of Fig. 1 to be stable in this sense. Our discussion is restricted to cases in which gi,/, u, and v (in Fig. 1) denote real-valued measurable functions of t defined for t ^ 0. The block labeled p is assumed to represent a memoryless time-varying element that introduces the constraint u(t) = j/f(t),t], in which ip(.r,t) is a function of x and I with the properties that ^ ( 0 , 0 = 0 for t ^ 0 and there exist a positive constant /i and a real constant a such that « ^ ^ / * 0 for all real x 0. The block labeled K represents the linear time-invariant portion of the forward path, and is assumed to introduce the constraint v(t) = [ l;(t T)U(T)(IT - {h(t), t ^ 0 (1)