Suppose T is a continuous linear operator between two Hilbert spaces X and Y and let K be a closed convex nonempty cone in X. We investigate the possible existence of $\delta > 0$ such that $\delta B_y \bigcap T(K) \subseteq T(B_x \bigcap K)$, where $Bx, By$ denote the closed unit balls in $X$ and $Y$ respectively. This property, which we call openness relative to $K$, is a generalization of the classical openness of linear operators. We relate relative openness to Jameson's property (G), to the strong conical hull intersection property, to bounded linear regularity, and to metric regularity. Our results allow a simple construction of two closed convex cones that have the strong conical hull intersection property but fail to be boundedly linearly regular.

A controller which minimises the $H_2$ norm of the closed loop transfer function for a linear time invariant plant, subject to an output disturbance decoupling constraint is found. The problem formulation results in an optimization over a parameter $Q(s)$ which is constrained to be a diagonal transfer matrix. The solution method relies on an $H_2$ isometric isomorphism between transfer matrices and transfer vectors.

It is well known that various properties of constrained optimization, such as converse Karush-Kuhn-Tucker and duality, remain valid when convex hypotheses are much relaxed, e.g. to invex. But convex does not need derivatives, whereas invex does. However, there is a property intermediate between convexifiable (by transformation of the domain) and invex, which gives a nondifferentiable extension of invex. Its properties will be surveyed.

A new property, namely strict property $(M)$, that implies the Opial property is introduced. We discuss relations between this property and some other well known properties. We also prove that Cesaro sequence spaces have strict property $(M)$.

In this paper we present an integral operator that arises in the context of linear dynamical systems, which describes the time evolution of the state probability density function. We propose a finite rank approximation to this integral operator and show that this finite rank operator converges in norm to the integral operator. We discuss Markov chains arising from this finite rank approximation, and show that the eigenvalues of the transition matrices of these Markov chains converge to the eigenvalues of the integral operator as the number of divisions in the statediscretization is increased.

If every functional which exposes a subset of the unit ball of a Banach space does so uniformly strongly (uniformly weakly) then the space is uniformly rotund (weakly uniformly rotund).

In the identification of linear systems the aim is to estimate the impulse response to within a given tolerance based on a finite number of noisy observations of the output. Whether this is possible depends upon the model set, that is, the set of impulse responses to which that of the system is assumed to belong. We give conditions on the model set which ensure that such identification is possible and also briefly review recent results concerning the complexity of identification, that is, the minimum number of required output samples.

This paper examines the use of the Remez Exchange Algorithm for Multi-channel FIR filter design.

This paper provides a generalisation of certain classical Fourier convergence and asymptotic Toeplitz matrix properties to the case where the underlying orthonormal basis is not the conventional trigonometric one, but a rational generalisation which encompasses the trigonometric one as a special case. These generalised Fourier and Toeplitz results have particular application in dynamic system estimation theory. Specifically, the results allow a unified treatment of the accuracy of least squares system estimation using a range of model structures, including those that allow the injection of prior knowledge of system dynamics via the specification of fixed pole or zero locations.

An important theorem of Gross and Tucker characterises a directed graph $E$ which admits a free action of a group as a skew product. Here we extend this to directed graphs admitting free actions of semigroups, $S$, under certain hypotheses. Indeed, the main criterion we employ can be completely characterised by properties of the quotient map $E \rightarrow E/S$.

We study the algebraic consequences of the traditional descriptions of dynamics based on differential equations. We thus show how $C*$-dynamical systems consisting of an action of a locally compact group on a $C*$-algebra provide a common framework for studying the time evolution and symmetry groups of systems in classical and quantum physics.

This paper is essentially a survey paper and falls into two parts. The first part is a brief overview of the representation theory of groups. The second part looks at some recent results in the area; these were chosen so as to maximise their relevance to the audience at the time. Both parts are necessarily short and superficial. We refer the interested reader to the references for more details.

This survey article (also containing a few new results) is devoted to a consideration of various properties of some well known classes of operators, such as decomposable operators, generalized scalar operators, well-bounded operators and AC-operators, for example, when attention is restricted to the particular (but important) setting of Fourier $p$-multiplier operators on the real line group. The aim is to highlight some relevant phenomena and to pose several natural questions which occur in this setting.

In this paper we extend the theory of real-valued increasing convex along rays functions for functions mapping into the semi-extended real line. We give a full description of the Fenchel-Moreau conjugate function to an increasing positively homogeneous of the first degree function.

A calculus of discontinuous functions employed in recent models of interfacial flow phenomena is examined. It is found that this calculus is not the calculus of distribution theory as the models suppose. Misapplication of the calculus leads to an incorrect formulation of the energy conservation equation, the entropy production rate and the phenomenological laws at the interface. The appropriate calculus is derived from first principles and applied to the interfacial flow modeL The resulting phenomenological laws at the interface are more physically reasonable than those given in the existing models.

We examine a class of maps which have fixed points for all Banach spaces. Included in the class are affine mappings and Banach contractions. The emphasis is on examples, with an eye to applications.

A fundamental limitation exists in the achievable tracking performance of systems in which the plant has a single input and two outputs (SITO). For a SITO plant in a unity feedback configuration, we consider the optimal reference tracking controller in the $L_2$ sense. The results are compared to the limiting cost of cheap control problems for SITO plants.