Rudolf Rupp

Simultaneous Stabilization of Three or More Plants: Conditions on the Positive Real Axis Do Not Suffice

The problem of the simultaneous stabilizability of a finite family of single-input, single-output time-invariant systems by a time- invariant controller is studied. The link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants. These necessary conditions are proved to be, in general, not sufficient. This result also disproves a long- standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller. The main result is to show that, unlike the case of two plants, the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles.

On Zero and One Points of Analytic Functions

Let f be an analytic function in the open unit disk that assumes the values 0 and 1 at least once and an unequal number of rimes in a disk of radius 1 10-5 entered at the origin. We prove that fmust assume the value 0 or 1 elsewhere in the unit disk. This result is motivated by a control engineering design problem which is also presented.

Composition of inner functions

According to K. Stephenson [12] an inner function I is called prime if, for any given composition I = u o υ with inner functions u and υ, at least one of them is a Möbius transform. Only a few examples of prime inner function.

A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set 𝐾 and holomorphic in the interior 𝐾∘ of 𝐾. In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on 𝐾.

Zerofree extension of continuous functions on a compact Hausdorff space

Abstract Let C R (X) denote, as usual, the Banach algebra of all real valued continuous functions on a compact Hausdorff space X endowed with the supremum norm. We present an elementary proof of the following extension result for C R (X) : For a given g∈C R (X) with zero set Z g and for the n -tuple (f 1 ,…,f n )∈C R n (X) without common zeros in Z g the following assertions are equivalent: (i) The restriction tuple (f 1 ,…,f n )| Z g has an extension to (F 1 ,…,F n )∈C R n (X) without common zeros in X . (ii) There exists an n -tuple (h 1 ,…,h n )∈C R n (X) such that the n -tuple (f 1 +h 1 g,…,f n +h n g)∈C R n (X) has no common zeros in X .

A covering theorem for a composite class of analytic functions

Let S denote, as usual, the class of normalized schlicht functions and let F denote the class of analytic functions in the open unit disk D, having no fixed point there. We prove the following covering theorem: For every function f.g, f ∊ S, g ∊ F , the image covers at least the open disk . As shown in the proof below, the class S can be replaced by a larger class S 0.

Stable rank boundary principle

Abstract (1) Let K denote a compact subset of the complex plane C . We present correct proof that the stable rank of A ( K ) is one. Hereby, A ( K ) is the algebra of all continuous functions on K which are analytic in the interior of K . (2) Let G denote a plane domain whose boundary consists of finitely many closed, nonintersecting Jordan curves. We show that for a fixed function of gϵ C ( Ḡ ), g≠0, the following assertions are equivalent: (i) Every unimodular element ( f, g ) is reducible to the principal component exp( C ( Ḡ )). (ii) The zero set Z g is polynomially convex, i.e., its complement C ⧹ Z g is connected.

Totally reducible elements in rings of analytic functions

Let A be an arbitrary ring of analytic functions properly containing the constants. We present a necessary condition on a function f ∈ A to be totally reducible in A. As a corollary it is shown that A does not have the unit-1-stable range. In the second part several sufficient conditions on a function f are given in order to be totally reducible in the disk algebra .

Stabilizable by a stable and by an inverse stable but not by a stable and inverse stable

The authors disprove conjectures on simultaneous stabilizability conditions by showing that, unlike the case of two plants, the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles. An example of a plant which has the even interlacing property but which is not stabilizable by a bistable controller is presented.<<ETX>>

Logarithms and exponentials in Banach algebras

Let