The state controllability for generalised dynamical systems with constrained control is discussed in this paper. The main results of the paper are the following:

(1) a necessary and sufficient condition of the state controllability in the sense of control energy or amplitude constrained for generalised dynamical systems is obtained;

(2) a control function u(t) is constructed such that

a) u(t) satisfies constrained energy or amplitude condition,

b) the state driven by u(t) moves from an arbitrary x(0−) = x0 to x(T(x0)) = 0,

c) the trajectory driven by u(t) has no impulsive behaviour within (0, T(x0)].

An exact algebraic representation for the 2D elastodynamic Green's tensor is derived. A new displacement potential decomposition is employed which yields, in conjunction with the Pekeris–Cagniard–de Hoop method, the exact representation. The first motions of the major arrivals are evaluated in terms of their polarizations, radiation patterns, geometrical spreading and wave-front singularities. The tensorial components of the Rayleigh wave on the free surface are found and solutions for dipolar line source discussed. We also investigate diffracted phases first noticed by Lapwood in his 1949 paper [13].

Two generalised shallow-shell bending elements are developed for the analysis of doubly-curved shallow shells having arbitrarily shaped plan-forms. Although both elements are formulated using the concept of iso-deflection contour lines, one element uses the three displacement components U, V and W as the basic unknowns, while the displacement component W and the stress function ф serve as the unknowns in the other element. A number of illustrative examples are included to demonstrate the accuracy and relative convergence of the proposed shallow-shell elements when employed for static analysis purposes.

A simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.

It is usually stated that the Laplace transform cannot be applied to most superexponential functions; indeed this limitation is sometimes represented as a deficiency of the technique. It is shown here that a generalisation serves to overcome much of the force of this objection. The generalisation is based on one first proposed over fifty years ago, but which is not widely known, nor was it ever worked out in detail.

In this work the asymptotic behavior of the partial sums of the divergent asymptotic moment series , where μi are the moments of the weight functions w(x) = xαe−x, α > −1, and w(x) = xαEm (x), α > −1, m + α > 0, on the interval [0, ∞), is analyzed. Expressions for the converging factors are derived by the author for the infinite range integras with w(x) as given above.

In this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. By defining appropriate sub-and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solutions.

Recently in several papers the boundary element method has been applied to non-linear problems. In this paper we extend the analysis to strongly nonlinear boundary value problems. We shall prove the convergence and the stability of the Galerkin method in Lp spaces. Optimal order error estimates in Lp space then follow. We use the theory of A-proper mappings and monotone operators to prove convergence of the method. We note that the analysis includes the u4 -nonlinearity, which is encountered in heat radiation problems.

In the framework of Mond-Weir duality a new equivalence between nonlinear programming and a matrix game is given. Finally, certain conclusions about convex programming with nested maxima and matrix games are also included.

In this paper, the generalised complementarity problem studied by Parida and Sen [13] is further extended. The extended problem appears to be more general and unifying. Characterisations of solutions to this extended problem are given. Some existence results derived by these characterisations are presented. An application of the extended problem to the quasi-variational inequalities of obstacle type is considered.

Nonlinear convective roll cells that develop in thin layers of magnetized ferrofluids heated from above are examined in the limit as the wavenumber of the cells becomes large. Weakly nonlinear solutions of the governing equations are extended to solutions that are valid at larger distances above the curves of marginal stability. In this region, a vortex flow develops where the fundamental vortex terms and the correction to the mean are determined simultaneously rather than sequentially. The solution is further extended into the nonlinear region of parameter space where the flow has a core-boundary layer structure characterized by a simple solution in the core and a boundary layer containing all the harmonics of the vortex motion. Numerical solutions of the boundary layer equations are presented and it is shown that the heat transfer across the layer is significantly greater than in the conduction state.

Consider the neutral delay differential equation

where p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result.

Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation

has no real roots.

A detailed analytical and numerical study is made of the deformation of highly elastic circular cylinders and tubes produced by steady rotation about the axis of symmetry. Explicit results are obtained through the use of Ogden's strain–energy function for incompressible isotropic elastic materials which, as well as being analytically convenient, is capable of reproducing accurately the observed isothermal behaviour of vulcanized rubber over a wide range of deformations. The three problems of steady rotation considered here concern (i) a tube shrink-fitted to a rigid spindle, (ii)an unconstrained tube, and (iii) a solid cylinder. In each case a set of restictions on the material constans appearing in the strain–energy function is stated which ensures that a tubular of cylindrical shape-preserving deformation exists for all angular spees and that, for problems (i) and (iii), there is no other solution. In connection with problems (ii) and (iii) values of the material constans are also given which correspond to the bifuraction or non-existence of soultions. Enegry consideration are used to determine the local stability of the various solutions obtained.

Conservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Fréchet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

We show how a short and elementary proof can be provided for a recently-published inequality ([6], [4]) which has found a number of applications.

In this paper we study the effect of forced and free convection heat transfer on flow in an axisymmctric tube whose radius varies slowly in the axial direction. Asymptotic series expansions in terms of a small parameter ∈, which is a measure of the radius variation, are obtained for the velocity components, pressure and temperature on the assumption that the Reynolds number (R) is of order one. The effect of the free convection parameter or Grashof number (G) on the axial velocity, temperature distribution, shear stress and heat flux at the wall are discussed quantitatively for a locally constricted tube.

This paper presents a method for the inverse fractional matching problem. We show that the dual of this inverse problem can be transformed into the circulation flow problem on a directed bipartite graph which can be solved easily. We also give an algorithm to obtain the primal optimum solution of the inverse problem from its dual optimum solution by solving a shortest path problem. Furthermore, we generalize this method to solve the inverse symmetric transportation problem.

Recursive parametric series solutions are developed for polynomial ODE systems, based on expanding the system components in series of a form studied by Weiss. Individual terms involve first-order driven linear ODE systems with variable coefficients. We consider Lotka-Volterra systems as an example.