38620513
Regular linear systems with feedback
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.
122167029
Admissible observation operators for linear semigroups
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.
8267029
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.
117037404
Two conjectures on the admissibility of control operators
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).
115711723
The representation of regular linear systems on Hilbert spaces
119810829
The resolvent growth assumption for semigroups on Hilbert spaces
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.
124018984
Complex Methods in Harmonic Analysis
116350194
Invariant systems of conjugate harmonic functions associated with compact Lie groups
54083834
Interpolation of operators with change of measures
Introduction. 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 [4]. The idea of interpolating between measure seems to have been implicit for some time. The general situation has, however, only recently been investigated. (See e.g. [2] and [8].) 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 [5], and Pitt [6]. §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
119816439
An example in linear quadratic optimal control
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.
125361717
Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the Form ∞ 0 e -x x n f(x) dx