Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability
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: 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: 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: The representation of regular linear systems on Hilbert spaces

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: H CLASSES OF HOLOMORPHIC FUNCTIONS IN TUBE DOMAINS.

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: Interpolation of operators with change of measures
abstract: This paper has two purposes. First, it is to give general interpolation theorems which allow one to change measures simultaneously with changing exponents of the Lp classes concerned. These theorems are generalizations of theorems of M. Riesz [7] and Marcinkiewicz . The idea of interpolating between measure seems to have been implicit for some time. The general situation has, however, only recently been investigated.  and .) Our second purpose is to apply these techniques. In §§3 and 4, we shall prove theorems for unbounded orthonormal systems which generalize known results for bounded orthonormal systems. These theorems are extensions of theorems of Paley, Marcinkiewicz and Zygmund , and Pitt . §1 will be devoted to necessary definitions and statements of needed known results. §2 will contain the general interpolation theorems. 1. Definitions and statement of general results. Let (M, 93J, p) and (N, yi, v) be two measure spaces. We will consider only measurable, real or complex-valued functions on these two spaces. In this paper we will establish a generalization of the well known convexity theorem of M. Riesz (Theorem (1.1) below) and give some applications of this new theorem to certain orthonormal series. In order to do this we will need to establish some notation and give a few definitions. Suppose T is a mapping of a class of functions on M into a class of functions on N. T is called a sublinear operator if it satisfies the following properties: (i) If/=/i+/2 and Tft (i=l, 2) are defined then Tf is defined; (ii) | T(fx+f2) | ^ | Tf\\ + | Tf2\ almost everywhere; (iii) For any scalar k, we have | T(kf)\ =\k\\ Tf\ almost everywhere. Let p, g2t 1 be two real numbers. We say that T is of type (p, q) in case T is defined for all functions/ in L"(M, W, ju) and there exists a positive number, K, independent of/such that

title: Invariant systems of conjugate harmonic functions associated with compact Lie groups

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.