Numerical solution of differential games with electric potential distributions in some domain

Abstract

Various applications of the problem of electrostatics are considered. It is shown that the Poisson equation describes the electrostatic field at the points where free charges are located. Assuming that the value of the free charge is a function controlled by two opposite sides, it is shown that for any control of the evader, the pursuer, having some advantage, can construct his control so that the potential distribution remains within certain predetermined limits. The problem of continuous pursuit by finite difference methods has been transformed into a discrete game. Using the formula for the obtained solution of a discrete equation with boundary conditions, theorems are proved on the possibility of completing the pursuit in the sense of getting into a small neighborhood of the terminal set of a discrete game. The method of normalization of the potential distribution is indicated.