OBSERVER’S TRAJECTORY TRACKING OBJECT BYPASSING OBSTACLE ON THE SHORTEST CURVE

Motion of some object is considered. The object t moves from the initial point t∗ to the final one t ∗ . But since absent of the direct path, he should bypass an obstacle a connected bodily set G. It is supposed that t moves by the most short trajectory T = Tt , and the trajectory T is a convex curve. The observer’s f task is to find the trajectory Tf that provides tracking the object on the most part of the object’s motion and, if possible, the lesser object’s stealth of motion along the trajectory T . The latency is defined by the distance that the observer must pass to see the object in the field of vision. The object and observer start at the same initial instant, and their velocities are equal. In the paper, examples of the trajectories Tf in R 2 are presented, on which the observer can see the object’s trajectory T ; also, the value of the object’s latency is shown for the invisible parts of the trajectory T . The variant of Tf in R 3 is shown.

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature.

Estimation of Toxic Gases’ Density Increment in the Integral Mathematical Model of Initial Fire Stage in Buildings

In the framework of an integral mathematical model of initial fire stage in a building the differential equation for toxic gases’ volume-averaged density has been considered. Representations for toxic gases’ density increment in the vicinity of initial instant have been found, and estimates for this increment have been given.

Multiple Fixed Pole-Based Rational Approximation for Fractional Order Systems

Our topic is the rational approximation of fractional order systems under Riemann–Liouville definition. This is a venerable, vast, fundamental area which attracts ongoing attention in coming years. In this work, the multiple fixed-pole scheme is developed. First, new schemes with different relative degree are developed to approximate fractional operators. Then, the fractional order is extended to the case of α>1. A discussion is made on the uniformity between the differentiator-based method and the integrator-based method. Afterward, the multiplicity of pole/zero is further generalized. In this framework, the nonzero initial instant and nonzero initial state are considered. Four examples are finally provided to show the feasibility and effectiveness of the developed algorithms.

Principal Components Analysis for the Interpretation of Humidification Phenomena—Preliminary Results

Moisture is one of the major causes of building decay, compromising the indoor air quality and the durability of building components. Infrared thermography is a non-destructive technique that can be used to prevent damage caused by the presence of water. In this study, the potential of Principal Component Analysis (PCA) was assessed as a quantitative technique for processing the thermal images captured during 24 hours of water absorption by a lightweight concrete specimen. PCA showed that the first principal component is the one that better explains the phenomenon. The initial instant was the most interesting to identify the water level in the sample and the final instant allowed defining the first 8h as the most relevant.

Research of K∞-Robust Systems with Constrained Control

At the present, there are no satisfactory engineering solutions related to the synthesis of robust regulators taking into account the constraints on control. In this connection, it is important to study the influence of saturation effect of the controller on the robust properties of systems. In this paper, this problem is considered in connection with K∞-robust control systems with a high gain. It is shown that in control limit systems, in particular, K∞-robust systems at the initial instant of time, the control assumes an excessively large value. This ensures the robustness of the dynamic mode. The reason for this feature is related to the fact that at the initial instant of time the initial conditions has a great importance. The main reason for the deterioration of robust properties is due to the tight control in the initial time interval. Provision of robustness of the static mode does not require great control efforts. For the first time, using computer graphics in a three-dimensional coordinate system, taking into account the time, a visual representation of the sections of phase trajectories pertaining to different types of movements is given: rapid motion from an arbitrary initial state that ensures hitting the imaging point into the degenerate trajectory; slow motion along this trajectory; steady-state motion within the specified accuracy. For the limit control systems, an integral robustness estimate is proposed, which consists in calculating the integral of the absolute value of the slow motion trajectory. This indicator characterizes the discrepancy (dispersion) of the real trajectory with respect to the limiting trajectory. The reliability of theoretical reasoning is confirmed by solving a model problem in the block-vision environment of Matlab/Simulink.

Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration

A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the c.d.f. of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively.

Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 2. Steady and unsteady drift of individual vortices on a beta-plane

Drift of individual β-plane vortices confined to one layer of a two-layer fluid under the rigid-lid condition is considered. For this purpose, the theory of two-layer quasi-geostrophic singular vortices is employed. On a β-plane, any non-zonal displacement of a singular vortex results in the development of a regular flow. An individual singular β-plane vortex cannot be steady on its own: the vortex moves coexisting with a regular flow, be the drift steady or not. In this paper, both kinds of drift of a singular vortex are considered. A new steady exact solution is presented, a hybrid regular–singular modon. This hybrid modon consists of a dipole component and a circularly symmetric rider. The dipole is regular, and the rider is a superposition of the singular vortex and a regular circularly symmetric field. The unsteady drift of a singular vortex residing in one of the layers is considered under the condition that, at the initial instant, the regular field is absent. The development of barotropic and baroclinic regular β-gyres is examined. Whereas the barotropic and baroclinic modes of the singular vortex are comparable in magnitudes, the baroclinic β-gyres attenuate with time, making the trajectory of the vortex close to that of a barotropic monopole on a β-plane.

Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

The following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.