On Certain Redundant Systems which Operate at Discrete Times
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: 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.

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

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: 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: Admissible observation operators for linear semigroups
abstract: Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=CTtx,t≧0, should be inLp and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=CTtx, even if the trajectory ofx is not in the domain ofC.

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: Regular linear systems with feedback
abstract: We consider a rather general class of infinite-dimensional linear systems, called regular linear systems, for which convenient representations are known to exist both in time and in the frequency domain. We introduce and study the concept of admissible feedback operator for such a system and of well-posedness radius. We show that the closed-loop system obtained from a regular linear system with an admissible feedback operator is again regular and we describe the relationship between the generating operators of the open-loop and closed-loop systems.