The Solution of the Sensors Placement Problem for Perimeter Defense Using the Gradient Method

Abstract

The research is devoted to the mathematical aspects of creating a modern security system. The paper develops a method that numerically solves the problem of optimizing the deployment of sensors to counteract the breach of a protected perimeter. The protected perimeter is a segment on the plane, and a mobile object evading detection tries to cross it in a given time. This object is considered as a material point, controlled to minimize the risk functional of detection by the primary hydroacoustic field. The problem is formalized as a maximin problem, the sensors should be placed in the admissible area so that the minimum possible value of the functional of the mobile object was as large as possible. For the solution the authors have developed a program in C++ language. The paper presents the results of numerical modeling obtained using the gradient method and the solution of the boundary value problem of the maximum principle of L.S. Pontryagin to find locally optimal trajectories in the auxiliary problem of pathfinding by a mobile object. The boundary value problem was solved by the shooting method, the corresponding Cauchy problems were initially formed by setting the values of the of the shooting parameters on the grid of their possible values, and were integrated numerically by Runge-Kutta method with automatic step selection. Further, the values of shooting parameters were refined by the modified Newton's method. As a result of numerical modeling of 5 sensors placement, it turned out that it is most advantageous to place them in such a way that the first 6 best locally optimal trajectories of the evading object coincide in functional value.