In this paper, we present sufficient conditions for global optimality of a general nonconvex smooth minimisation model problem involving linear matrix inequality constraints with bounds on the variables. The linear matrix inequality constraints are also known as “semidefinite” constraints which arise in many applications, especially in control system analysis and design. Due to the presence of nonconvex objective functions such minimisation problems generally have many local minimisers which are not global minimisers. We develop conditions for identifying global minimisers of the model problem by first constructing a (weighted sum of squares) quadratic underestimator for the twice continuously differentiable objective function of the minimisation problem and then by characterising global minimisers of the easily tractable underestimator over the same feasible region of the original problem. We apply the results to obtain global optimality conditions for optinusation problems with discrete constraints.

We construct a set of functions, say, composed of a cosine function and a sigmoidal transformation γr of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [−1, 1]. It is proven that if a function f is continuous and piecewise smooth on [−1, 1] then its series expansion based on converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r ≤ 1. Owing to the variational feature of according to the value of r, one can expect improvement of the traditional Fourier series approximation for a function on a finite interval. Several numerical examples show the efficiency of the present series expansion in comparison with the Fourier series expansion.

In this paper, a closed-form analytical solution for pricing convertible bonds on a single underlying asset with constant dividend yield is presented. A closed-form analytical formula has apparently never been found for American-style convertible bonds (CBs) of finite maturity time although there have been quite a few approximate solutions and numerical approaches proposed. The solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms, and thus is completely analytical and in a closed form. Although it is only for the simplest CBs without call or put features, it is nevertheless the first closed-form solution that can be utilised to discuss convertibility analytically. The solution is based on the homotopy analysis method, with which the optimal converting price has been elegantly and temporarily removed in the solution process of each order, and consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical solution for the optimal converting price and the CBs' price.

We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player and yet the new combined game is favourable—this is a counter-intuitive phenomenon known as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.

Given a pair of biorthogonal multiscaling functions, we present an algorithm for raising their approximation orders to any desired level. Precisely, let Φ(x) and (x) be a pair of biorthogonal multiscaling functions of multiplicity r, with approximation orders m and , respectively. Then for some integer s, we can construct a pair of new biorthogonal multiscaling functions Φnew(x) = [ΦT (x), φr+1 (x), φr+2(x),… φr+s(x)]T and new(x) = [ (x) T, r+1(x), r+2(x),… r+s(x)]T with approximation orders n (n > m) and ñ (ñ > ), respectively. In addition, corresponding to Φnew(x) and new(x) a biorthogonal multiwavelet pair ψnew(x) and new(x) is constructed explicitly. Finally, an example is given.

This paper analyses a model for combustion of a self-heating chemical (such as pool chlorine), stored in drums within a shipping container. The system is described by three coupled nonlinear differential equations for the concentration of the chemical, its temperature and the temperature within the shipping container. Self-sustained oscillations are found to occur, as a result of Hopf bifurcation. Temperature and concentration profiles are presented and compared with the predictions of a simpler two-variable approximation for the system. We study the period of oscillation and its variation with respect to the ambient temperature and the reaction parameter. Nonlinear resonances are found to exist, as the solution jumps between branches having different periods.

Planar hinged segmented bodies have been used to represent models of biomechanical systems. One characteristic of a segmented body moving under gravitational acceleration and torques between segments is the possibility that the body's segments spin through more than a revolution or past a natural limit, and a computational mechanism to stop such behaviour should be provided. This could be done by introducing angle constraints between segments, and computational models utilising optimal control are studied here. Three models to maintain angle constraints between segments are proposed and compared. These models are: all-time angle constraints, a restoring torque in the state equations and an exponential penalty model. The models are applied to a 2-D three-segment body to test the behaviour of each model when optimising torques to minimise an objective. The optimisation is run to find torques so that the end effector of the body follows the trajectory of a half-circle. The result shows the behaviour of each model in maintaining the angle constraints. The all-time constraints case exhibits a behaviour of not allowing torques (at a solution) which makes segments move past the constraints, while the other two show a flexibility in handling the angle constraints which is more similar to what occurs in a real biomechanical system.

This paper is concerned with robustness with respect to small delays for the exponential stability of abstract differential equations in Banach spaces. Some necessary and sufficient conditions are given in terms of the uniformly square integrability of the fundamental operator family and the uniform boundedness of its resolvent on the imaginary axis.

This paper deals with robust guaranteed cost control for a class of linear uncertain descriptor systems with state delays and jumping parameters. The transition of the jumping parameters in the systems is governed by a finite-state Markov process. Based on stability theory for stochastic differential equations, a sufficient condition on the existence of robust guaranteed cost controllers is derived. In terms of the LMI (linear matrix inequality) approach, a linear state feedback controller is designed to stochastically stabilise the given system with a cost function constraint. A convex optimisation problem with LMI constraints is formulated to design the suboptimal guaranteed cost controller. A numerical example demonstrates the effect of the proposed design approach.

In this paper, we consider a coupled, nonlinear, singular (in the sense that the reaction terms in the equations are not Lipschitz continuous) reaction-diffusion system, which arises from a model of fractional order chemical autocatalysis and decay, with positive initial data. In particular, we consider the cases when the initial data for the the dimensionless concentration of the autocatalyst, β, is of (a) O(x−λ) or (b) O(e−σ x) at large x (dimensionless distance), where σ > 0 and λ are constants. While initially the dimensionless concentration of the reactant, α, is identically unity, we establish, by developing the small-t (dimensionless time) asymptotic structure of the solution, that the support of β(x, t) becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support of β as t → 0 is given in both cases.