Discrete-time Setting Research Articles

Turnpike Properties for the Optimal Use of a Natural Resource

This paper is an attempt to deal rigorously with some fundamental mathematical problems arising from the consideration of natural resources in the neoclassical framework of optimal economic growth. The theory of optimal economic growth initiated by Ramsey [20] and developed by many others has benefited considerably from an optimal control formulation (see for example [1] and [22]). Incidentally it posed two interesting problems which seem to be typical of these economic models. The first one concerns the precise formulation of the Turnpike Property (see Cass [5]), the second one is the definition and the characterization of the optimality when the time interval considered is infinite and the formulation of existence conditions for such an optimal solution (see Koopmans [14]). It appeared that the two problems are linked together through the important role played in both problems by an optimal steady state suitably defined (see for example Gale [9] and McKenzie [17] for discrete time growth models and Haurie [11] for continuous time control systems). In the case of optimal exploitation of a natural resource, the Turnpike Property is well illustrated by the very simple model of optimal fish-harvesting studied by Cliff and Vincent [6] (this model is very close to the one proposed by Plourde [19]). Those authors showed that, for their particular model, any optimal trajectory on a sufficiently large interval would contain a singular arc where the fish population is maintained at its level of maximal yield, this singular arc being independent of the initial and terminal conditions. This result is similar to the turnpike property first presented in DOSSO [8] for efficient programmes of capital accumulation and then extended to the neoclassical optimal growth framework. However no formal proof has been given yet for an extension of the turnpike property to the case of an optimal growth model where both capital and natural resources enter into the production process. A first difference between capital and resource assets is that the former enter as stock inputs in the production process while the latter are often flow inputs. Another difference is that capital depreciates over time and is reproducible through investment while a natural resource is depleted by its use in the economic process and, when replenishable, reproduces itself according to a natural law. In Section 3 this extended turnpike property is proven for a continuous time model and expressed as a bound for the measure of the time spent away from the Von Neumann Set of the economy, a set which always contains the optimal steady state. This is essentially an adaptation of the classical result obtained by McKenzie in a discrete time setting without natural resources. When the Von Neumann set reduces to a

Reliability Prediction Studies of Complex Systems Having Many Failed States

In this paper, the theory of discrete Markov processes is used to develop methods for predicting the reliability and moments of the first time to failure of complex systems having many failed states. It is assumed that these complex systems operate in a repair environment and are composed of subsystems that have known constant failure and repair rates. Specifically, complex systems composed of any finite number of subsystems are considered. The complex system at any time is in an acceptable state or in a failed state. The methods presented for the reliability modeling of such complex systems assume a state behavior that is characterizable by a stationary Markov process (also called Markov chain) with finite-dimensional state space and a discrete time set. It is shown that once the matrix of the constant failure and repair rates of the subsystems is known, and the state assignment is made, then it is a straightforward matter to obtain the probabilistic description of the complex system.

Relaxation Processes and Inertial Effects I: Free Rotation about a Fixed Axis

The limitations of the usual concepts of relaxation processes for very short periods are discussed, and it is shown that in the limit of very high frequencies the product of the loss factor and frequency must tend to zero. For a consistent kinetic treatment the probability must be considered in phase space, not merely in configuration space; for Cartesian coordinates the stochastic (generalized Liouville) equations can be simplified by a Fourier transformation with regard to the velocities. By means of the simplified equations the dielectric properties are determined for a system of rigid dipoles rotating about fixed axes in a viscous medium. The results are equivalent to, but more compact than, those derived for the same problem by E. P. Gross; for strong viscous damping they agree with Debye's relaxation formulae, except at very high frequencies. The polarization can be described by an infinite discrete set of relaxation times with amplitudes which alternate in sign; the longest relaxation time is identical with that calculated by Debye. The logarithm of the function expressing the dielectric after-effect follows a law similar to the Ornstein-Furth generalization of Einstein's law of diffusion. The analytical solutions obtained by Gross for abrupt collisions between dipoles and heat-bath are also re-derived by a somewhat easier method.

Theory of Dielectric Relaxation for a Single-Axis Rotator in a Crystalline Field. II

Previous work on the relaxation times associated with a polar single-axis rotator where the dipole may occupy three or four orientational sites by turning in a lattice point has been extended to include the more general case where the energies of the sites (as well as the intervening barriers) are all different. The results for these more general models tend to confirm the view that the presence of barriers of different magnitudes between the equilibrium orientations of a dipole can be the source of a set of discrete dielectric relaxation times. The dielectric properties of some specialized three- and four-position models are then considered in detail. A very narrow or an extremely wide set of relaxation times can be obtained depending on the nature of the barrier system, and accordingly, a wide range of dielectric behavior can be accommodated. Despite the wide range of behavior, certain qualitative features (noted in Sec. VII in italics) appear to persist in many of the models. These features may be useful in applying the theory in a general way. The equations given will facilitate the calculation of the properties of three- and four-position models not already considered.