CRITERIA FOR THE ABSOLUTE STABILITY OF NONLINEAR SYSTEMS WITH RESPECT TO A LINEAR OUTPUT, AND THEIR APPLICATIONS. II A. L. Likhtarnikov and V. A. Yakubovich UDC 517.9:62-50 Introduction. This paper is a continuation of [I] and is devoted to the application of the abstract absolute-stability criteria [I] to nonlinear delay systems. The stability problems for delay systems, which arose first in technical areas, are already under investigation for several decades (see [2, 3] and others); an impressive literature deals with sucb problems (the bibliography given in [4], which is far from being complete, comprises more than 600 titles). The notion of Lyapunov functional was introduced and successfully used in [5] for systems with delay, as a generalization of the Lyapunov functions. Starting with [5], Lyapunov functionals are widely applied whenever stability problems for such systems are solved. Frequency criteria for the absolute stability of nonlinear systems with delay were obtained in [3, 6-9]; instability and dichotomy criteria can be found in [9]. An extensive literature on this subject is given in [3]. The goal of our paper is to demonstrate that the abstract theory of absolute stability [10, 11] can be successfully used in the investigation of delay systems. Thus, we are concerned with the system consisting of the vector equation h d~: (t} _ ~ l .bx ( t ~ ) 4b j ~ ( t Tj)} ~/o (t) (0 1 ) d t " J=o and the inequality Tj , t ' G [ x ( t ) ' x ( t ~ l ) . . . . . ~ ' ( t--~k), ~ ( t ) , . . . , ~ ( t ~ ) ] d t ~ ? , ( 0 . 2 ) o where x ( t ) and ~ ( t ) a re v e c t o r f u n c t i o n s , Aj and bj a r e c o n s t a n t m a t r i c e s , G [ ' ] i s a H e r m i t i a n form in its arguments, 0 = t 0 0 and any function [(.) ~ L~[(0, T) ~ C '~] , the Cauchy problem (0.1), (1.1) admits a unique solution x(.)~L~[(0, T)-~C ~] [this is readily established by "step" integration of (0.1)]. It is natural to define the process [I] of system (0.1), (i.i) as the pair h=h(t) ={x(t--T), [(t-x)}, t~0. Then, the process h is an element of the space ~L~cl(O,~oo)-+C~+m]. We let ~T denote the Hilbert space L2[(0, T)~Qn+~]. (A vectorfunction h(-):[0, +~)~C ~+'~ belongs to ~Y~ if and only if the restriction h(.)l(0,T) = PT h belongs to ~r VT>0.) It is readily seen that the spaces 9~ form a family of compatible Hilbert spaces [10], with the role of accessible directed set played by the semiaxis T = (0, +~. Following the scheme of [I, 10], we isolate in 9~ the subspace of stable processes ~y~t =L~[(0, +~) ~ C~+~] and consider the Hilbert space ~: = L2[(O, + ~ ) ~ C"] X L2[(-z, O) C "+'~] X C ' , whose elements will be referred to as perturbations. Under our assumptions, (i .2) We shall distinguish below two cases, the real and the complex case, to apply the theory of [1] and formulate absolute-stability criteria. 753 The complex case was described above. In the real case all the given matrices as well as all the quantities appearing in (0.I), (0.2), and (1.1) are real. The definitions of the spaces ~,~z, 9 , and so on in the real case are similar: C n, C m should be replaced by R n, R TM, etc. Note that the real case is more interesting: as it turns out, the direct proofs of the necessity of a number of absolute-stability criteria are significantly more complicated. However, when the abstract theory of [I] is applied, the difference between the two cases is almost imperceptible. For the sake of definiteness we shall treat below only the complex case and restrict ourselves to brief comments concerning the real case. 2. Consider the set ~ of those processes which satisfy relation (0.1) with f~ ~ 0 and zero initial conditions: Clearly, ~ is a linear subspace of ~ (a linear nonperturbed block [I]). Remark 1.1. Subspace ~ can be also defined using the relation x(p) = X(P)'~(P), where x(p) and ~(p) are the Laplace transforms of the functions x(t) and ~(t), and X(P) is the transfer matrix-function of the linear block (1.1):