Differential Game: Solution Of Games By Differential Equation

Abstract

Differential equation game solutions have been demonstrated to translate into several solution concepts, such as the core, the sharply value, and in some cases, the nucleons. These centroids and nuclei are the stable critical points of the different differential equation systems, and they coincide with the ideas of classical solutions under a number of circumstances. A fresh demonstration of a value and optimal tactics for a two-person, zero-sum game is provided. Two distinct qualities of this proof seem to draw some interest: first, despite the algebraical nature of the theorem to be proved, a very straightforward proof can be obtained analytically; second, the proof is constructive in the sense that it can be used to actually compute the solutions of particular games. The process was rather easily mechanized for both analogy and digital approaches. In the latter instance, the sensitivity to equipment precision is likely far lower than in the relatively related task of solving linear equations or inverse matrix. Examples of differential games are given, where the state equation is a partial differential equation. They can be solved explicitly and show clearly how the values of the control functions enter in the solution. This enables us to set up a method of solving these games, which should also be applied to more complicated differential games, complementing known results about existence of solutions from the general theory.