Stephen Montgomery-Smith

Stability Radius and Internal Versus External Stability in Banach Spaces: An Evolution Semigroup Approach

In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include autonomous and nonautonomous systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of an (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are proven for Banach-space systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systems. In particular, for the nonautonomous setting, internal stability is shown to be equivalent to input-output stability for stabilizable and detectable systems. For the autonomous setting, an explicit formula for the norm of input-output operator is given.

On singular integral and martingale transforms

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on L p X (R 2 ) with p ∈ (1, ∞). Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals p * — 1 with p * := max{p, (p/(p - 1))}, where the novelty is the lower bound.

Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations

Let

On the distribution of Sidon series

u(x,t)

Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions

be the solution of the Schrodinger or wave equation with

Power-Bounded Operators and Related Norm Estimates

L_2

Set-functions and factorization

initial data. We provide counterexamples to plausible conjectures involving the decay in

Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables

t

Lyapunov theorems for Banach spaces

of the

Numerical Evaluation of Single Fiber Motion for Short-Fiber-Reinforced Composite Materials Processing

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