Repetitive Control Systems: Old and New Ideas
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: 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.

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: 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 representation of regular linear systems on Hilbert spaces

title: On the Inversion of the Specific-Heat Function
abstract: In this paper we present a method for the inversion of specific heat data to find the vibrational frequency spectrum of a solid. Our results are given in terms of a power series in UJ, the frequency. In this series the coefficients depend on C v (T) and its d~rivatives at T=O. The series thus derived is valid only in an asymptotic sense, but represents an improvement over inversions by Montroll and Kroll which require a knowledge of functions formed from the data, and the evaluation of double integrals over these functions, A method for forming the successive values of Cv(n) (0) from experimental data is also presented. One of the most frequently measured physical characteristics of a solid is the specific heat at constant volume Cv (T), where T is the temperature. The behavior of this function has quite accurately been predicted by various characterizations of a lattice of atoms which have harmonic interactions. If it is assumed that the specific heat can be attributed solely to harmonic effects, then it can be written as an integral over the vibrational frequency spectrum 1 ;

title: Torsion of a Shaft with a Toroidal Cavity
abstract: This paper gives the solution to the torsion problem for a near‐cylindrical shaft of circular section having a toroidal cavity. Since the interpretation of the axially symmetric problem as a flow problem in 5 dimensions is well known, the first part of the paper is devoted to developing a method for handling the flow problem about an n dimensional ring‐shaped body of general cross section. In the second part of the paper this method is applied to the case of an n dimensional torus ring. The case n=3 then yields the solution to the classical flow problem, and n=5 gives the solution to the torsion problem under consideration.

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

title: Complex Methods in Harmonic Analysis