On a Criterion of Yakubovich Type for the Absolute Stability of Nonautonomous Control Processes

We generalize a criterion of Yakubovich for the absolute stability of control processes with periodic coefficients to the case when the coefficients are bounded and uniformly continuous functions. 1. Introduction. In the past two decades, the study of differential equations with time-varying coefficients (nonautonomous differential equations) has benefited from the use of methods of the ergodic theory and topological dynamics. The well-known paper of Oseledets [15] showed how basic methods of the er-godic theory can be applied to the theory of the classical Lyapunov exponents of linear and nonautonomous differential systems. Then, Sacker and Sell [17] introduced the important concept of dichotomy spectrum for linear systems with time-varying coefficients. Using the dichotomy spectrum, we can study the basic notion of exponential dichotomy for such systems using the methods of topological dynamics. In more recent years, the concepts of rotation number [8] and pullback attractor [19] have been used with profit by workers in the field of nonautonomous differential systems. In the study of rotation numbers and exponential dichotomies, we use techniques of the ergodic theory and topo-logical dynamics. It is, therefore, not surprising that such techniques have had an increasing impact in the study of control-theoretic problems. To illustrate this point, the work of Colonius and Kliemann [3] on the reachability theory for an exposition of many results concerning their concept of control set. We also mention the monograph [9], where it is shown how exponential dichotomies and the rotation number for linear and nonautonomous Hamiltonian systems can be used to a good effect to study the nonautonomous versions of the classical linear regulator problem and the classical feedback control problem. In this paper, we study the absolute stability problem for linear control processes with an integral quadratic constraint, when the coefficients are aperi-odic and bounded functions of time. To orient the discussion, consider the control system