An input-to-state stability theory, which subsumes results of circle criterion type, is developed in the context of a class of infinite-dimensional systems. The generic system is of Lur'e type: a feedback interconnection of a well-posed infinite-dimensional linear system and a nonlinearity. The class of nonlinearities is subject to a (generalized) sector condition and contains, as particular subclasses, both static nonlinearities and hysteresis operators of Preisach type.

It has been observed that for many stable feedback control systems, the introduction of arbitrarily small time delays into the loop causes instability. In this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems. We consider the class of all matrix-valued transfer functions which are bounded on some right half-plane and which have a limit at + along the real axis. Such transfer functions are called regular. Under the assumption that a regular transfer function is stabilized by unity output feedback, we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop. These conditions are given in terms of the high-frequency behavior of the openloop system. Moreover, we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators. In particular, we show that if a plant with infinitely many poles in the closed right half-plane is stabilized by a controller, then the stability is not robust with respect to delays. We show that the instability created by small delays is itself robust to small delays. Three examples are given to illustrate these results.

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It is well-known that closing the loop around an exponentially stable, nite-dimensional, linear, time-invariant plant with square transfer-function matrix G(s) compensated by a controller of the form (k=s)? 0 , where k 2 R and ? 0 2 R m m , will result in an exponentially stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of G(0)? 0 have positive real parts and (ii) the gain parameter k is positive and su ciently small. In this paper we consider a rather general class of in nite-dimensional linear systems, called regular systems, for which convenient representations are known to exist, both in time and in frequency domain. The purpose of the paper is twofold: (i) we extend the above result to the class of exponentially stable regular systems and (ii) we show how the parameters k and ? 0 can be tuned adaptively. The resulting adaptive tracking controllers are not based on system identi cation or plant parameter algorithms, nor is the injection of probing signals required.

It is shown that a set of six natural conditions is necessary and sufficient for the existence of a finite-dimensional stabilizing sampled-data controller for a well-posed infinite-dimensional system. The underlying stability concept for the sampled-data system is reminiscent of the notion of input-to-state stability from nonlinear control theory.

An input-to-state stability theory, which subsumes results of circle criterion type, is developed in the context of continuous-time Lur'e systems. The approach developed is inspired by the complexified Aizerman conjecture.

Abstract. Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-andhold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space X and the control space U are Hilbert spaces, the system is of the formẋ(t) = Ax(t) + Bu(t), where A is the generator of a strongly continuous semigroup on X, and the continuous time feedback is u(t) = F x(t). The answer to the above question is known to be "yes" if X and U are finitedimensional spaces. In the infinite-dimensional case, if F is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is "yes", if B is a bounded operator from U into X. Moreover, if B is unbounded, we show that the answer "yes" remains correct, provided that the semigroup generated by A is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.

Using a frequency-domain analysis, it is shown that the application of a feedback controller of the form k=(z ?1) or kz=(z ?1), where k 2 R, to a power-stable in nite-dimensional discrete-time system with square transfer-function matrix G(z) will result in a power-stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of G(1) have positive real parts and (ii) the gain parameter k is positive and su ciently small. Moreover, for single-input single-output systems we show how the gain parameter gain k can be tuned adaptively. The resulting adaptive tracking controllers are universal in the sense that they apply to any power-stable system with Re G(1) > 0, and in particular they are not based on system identi cation or plant parameter estimation algorithms, nor is the injection of probing signals required. Finally, we apply these discrete-time results to obtain adaptive sample-data low-gain controllers for the class of regular systems, a rather general class of in nite-dimensional continuous-time systems, for which convenient representations are known to exist, both in state-space and in frequency domain. We emphasize that our results guarantee not only asymptotic tracking at the sampling instants, but also in the sampling interval.