Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature

We consider a time-optimal control problem on the plane with a circular vectogram of velocities and a non-convex target set with a boundary having a finite number of points of discontinuity of curvature. We study the problem of identifying and constructing scattering curves that form a singular set of the optimal result function in the case when the points of discontinuity of curvature have one-sided curvatures of different signs. It is shown that these points belong to pseudo-vertices that are characteristic points of the boundary of the target set, which are responsible for the generation of branches of a singular set. The structure of scattering curves and the optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed, consisting in the fact that one pseudo-vertex can generate two different branches of a singular set. The equation of the tangent to the smoothness points of the scattering curve is derived. A scheme is proposed for constructing a singular set, based on the construction of integral curves for first-order differential equations in normal form, the right-hand sides of which are determined by the geometry of the boundary of the target in neighborhoods of the pseudo-vertices. The results obtained are illustrated by the example of solving the control problem when the target set is a one-dimensional manifold.

Singularity functions revisited: Clarifications and extensions for construction of shear–moment diagrams in beams

This paper provides clarification and extension of singularity functions for the construction of shear–moment diagrams in beams and the subsequent determination of beam deflections. The mathematical formulation for impulse, polynomial, and general-form singularity functions and their integral properties is reviewed, clarified, and provided graphically in tabular form. Several examples of various complexity are presented to assist the student at evaluating the constants of integration and properly interpreting the values of shear force and bending moment in the limit of approach to points of discontinuity. The main emphasis of the paper is to demonstrate the applicability of the singularity function method for any specified distributed load function.

Results on Existence and Uniqueness of Solution Of Impulsive Neutral Integro-Differential System

In this work, the solution of the impulsive neutral integro-differential system is analysed. The Du Bois–Reymond’s assumptions on solution variation of piecewise smooth functions are used to establish the existence of only the impulsive term as a solution of the system at the points of discontinuity. The theories of infinitesimal generator of a strongly continuous compact semigroup is used to formulate theorems on existence and uniqueness of system solution, and proves are provided using an approximate piecewise continuous, compact operator and a continuous positive non decreasing function . Results obtained are improvement on the qualitative analysis of impulsive neutral integro-differential system

Hybrid genetic method with gradient learning and predictionfor solving global optimization of multiextremal functions.

In the article in order to determine the most effective global optimum multiextremal multivariate functions in the general case containing the points of discontinuity of the first and second kind, proposed modification of the genetic method. Numerical evaluation of the effectiveness of finding the global optimum of the proposed modification of the genetic method in comparison with a standard genetic algorithm and its known modifications made to a select group multiextremal test functions.