Using a non-linear transformation of the state space, the dynamic behaviour of an adaptive thirdorder relay system is analysed. The control strategy yields state paths close to the time-optimal trajectories.

A new method is developed for identifying real-valued coefficients r(x), p(x), and q(x) for which all solutions of the fourth-order differential equation

are L2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

Here eigenvalue problems A(λ)u = 0 and their approximations Ai(λ)νi = 0 are studied where the densely denned closed semi-Fredholm operators A and Ai depend holomorphically on the parameter λ. Two different kinds of approximations are established. One is based on a generalisation of the spectral projection and the other on a suitable linearisation of the problem. To this end generalised eigenvectors and suitable product spaces are introduced which provide a representation formula for the principal part of A -1 and A -1, respectively, in the neighbourhood of poles. The convergence of the methods is shown in the framework of discrete convergence theory. The results generalise the corresponding results for linearly dependent A, Ai in two directions: both methods are available for arbitrary holomorphic dependence on the parameter λ, and the first method provides convergent approximations to the whole generalised eigenspace which works also in cases of linear parameter dependence when the usual method fails.

In an unbounded domain Ω we study the asymptotic decay (for | x |→∞) of functions u ∊ L2(Ω) which are solutions of the following problem –Δu + cu = 0. c denotes a strictly positive function. Upper bounds are easily found via the maximum principle. When c is rotationally invariant lower bounds are obtained via asymptotic expansion. In the general case we use a method of ‘commutation’ of operators. In particular we consider the case where . Applications to the asymptotic decay of the bound states of a Hamiltonian are given.

The operator equation TS = 1 is studied for power bounded operators T, S on Hilbert space, and its relation to *—representations of the bicyclic semigroup is explored.

If Ω is a bounded domain in Rn satisfying certain conditions, Ωk denotes its intersection with a k-dimensional hyperplane, 1 ≦ k ≦ n, it is shown that the embedding of the Sobolev space Ws,p(Ω), s>0, into Lq(Ωk) is of type lm if for q<p<∞. The same result is obtained for the space of Bessel potentials Ls,p(Ω). Piecewise polynomial and Fourier approximations of functions and interpolation theorems areused.

This paper gives sufficient conditions ensuring that a non-linear control system of the form

is controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

It is shown that exactly six non-isomorphic distributively generated near rings can be defined on the infinite dihedral group. One has the null multiplication, two have trivial multiplications in which the group of left annihilators has index two, and three are distributive and have their product sets equal to a subgroup of order 2.

In a previous paper [2], a theory of fractional integration was developed for certain spaces Fp,μ of generalised functions. In this paper we extend this theory by relaxing some of the restrictions on the various parameters involved. In particular we show how a generalised Erdelyi-Kober operator can be defined on Fʹp,μ for 1 ≦ p ≦ ∞ and for all complex numbers μ except for those lying on a countable number of lines of the form Re μ = constant in the complex μ-plane. Mapping properties of these generalised operators are obtained and several applications mentioned.

Eisenstein series are entire modular forms Ek of even integral weight k≧ 4 with Fourier expansions given by (1.1). There are numerous identities, such as E8 = , relating these series. These are usually proved by arguments making use of the dimensions of vector spaces of modular forms, and not directly. The paper shows how such identities can be proved by elementary methods by studying chains of solutions of Diophantine equations of the form xξ+yη = n.

It is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

This report deals with the asymptotic behaviour of solutions of the wave equation in a domain Ω ⊆Rn. The boundary, Γof Ωft consists of two parts. One part reflects all energy while the other part absorbs energy to a degree. If the energy-absorbing part is non-empty we show that the energy tends to zero as t→∞. With stronger assumptions we are able to obtain decay rates for the energy. Certain relationships with controlability are discussed and used to advantage.