The representation of regular linear systems on Hilbert spaces

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: Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity
abstract: 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: 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: Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the Form ∞ 0 e -x x n f(x) dx

title: A kernel associated with certain multiply connected domains and its applications to factorization theorems

title: An example in linear quadratic optimal control
abstract: We construct a simple example of a quadratic optimal control problem for an infinite-dimensional linear system based on a shift semigroup. This system has an unbounded control operator. The cost is quadratic in the input and the state, and the weighting operators are bounded. Despite its extreme simplicity, this example has all the unexpected features discovered recently by O. Staffans (and also by M. Weiss and G. Weiss). More precisely, in the formula linking the optimal feedback operator to the optimal cost operator, as well as in the Riccati equation, the weighting operator of the input has to be replaced by another operator, which can be derived from the spectral factorization of the Popov function.