Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity
We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) , like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Foures and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.

title: Admissibility of unbounded control operators
abstract: For linear systems described by $\dot x(t) = Ax(t) + Bu(t)$, where A generates a semigroup on the state space X and B is an unbounded operator, some necessary as well as some sufficient conditions are given for B to be admissible, i.e., for any t, the state $x(t)$ should be in X and should depend continuously on the input $u \in L^p $. This approach begins with an axiomatic description of such a system in terms of a functional equation. The results are applied to the wave equation on a bounded interval.

title: Two conjectures on the admissibility of control operators
abstract: We are searching for necessary and/or sufficient conditions for the admissibility of unbounded control operators for semigroups on Hilbert spaces, with respect to input functions of class L 2. Our first conjecture is that admissibility of an unbounded input element 6 for a semigroup with generator A is equivalent to a certain decay rate of ∥(sI - A)-1 b∥ as Re s → ∞. The second conjecture states that a control operator B defined on a Hilbert space U is admissible if and only if, for any v ∈ U, Bv is an admissible input element. It is proved that both conjectures hold in many important particular cases (e.g., the first conjecture is true if the semigroup is normal).

title: The resolvent growth assumption for semigroups on Hilbert spaces
abstract: Let X be a Hilbert space and let T = (T1),,, be a C, semigroup of linear operators on X, with generator A. The aim of this paper is to show how information about the behavior of T, for large t can be obtained from information about the behavior of the resolvent (slA)-’ for s with large imaginary part, and vice versa. In particular, the type of U can easily be determined if we know on which vertical lines in the complex plane the resolvent is bounded. We start by defining a sequence of numbers (oU,), which give information about the behavior of the semigroup for large t.

title: Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability
abstract: GEORGE WEISS Department of Theoretical Mathematics, The Weizmann Institute, Rehovot 76100, Israel Received September 24, 1987 We prove that a strongly continuous semigroup of linear operators on a Hilbert space is weakly P-stable for some PE Cl, co) if and only if the semigroup is exponentially stable. As an application, we prove that the Cauchy problems associated with the semigroup are well posed on the infinite time interval (- co, 0) if and only if the semigroup is exponentially stable.

title: Repetitive Control Systems: Old and New Ideas
abstract: Many signals in engineering are periodic, or at least they can be well approximated by a periodic signal over a large time interval. This is true, for example, for most signals associated with engines, electrical motors and generators, converters, or machines performing a task over and over again. Thus, it is a natural control problem to try to track a periodic signal with the output of a plant, or (what is almost the same), to try to reject a periodic disturbance acting on a control system. We examine this problem in Sections 1 and 3 of this paper (Section 2 is for background). In Section 4 we shall indicate a way of generalizing these ideas to cope with superpositions of periodic signals of arbitrary periods.

title: Calculation of certain multiple generating functions

title: On Certain Redundant Systems which Operate at Discrete Times
abstract: This paper discusses the effect of using redundant components on the operation of systems which perform a given task repetitively. It is assumed that each component of the system has a constant probability of failure at every cycle. A question which immediately arises is: “What is the probability distribution for the number of the cycle of first system failure?” The solution to this problem generalizes considerations leading to the geometric distribution. The effects of replacements on the failure cycle distribution are analyzed.

title: Complex Methods in Harmonic Analysis

title: Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the Form ∞ 0 e -x x n f(x) dx

title: A derivation of the main results of the theory of Hp spaces