Scientific Bulletin. Physical and Mathematical Research


The theory of differential games is developed and resulted from modeling technical problems. Some of the problems in differential games theory can be described as controlling two moving objects, i.e. one of them is the follower that tries to catch the other object, and obviously the other object is the runner. The runner tries to run away from the follower. Most of the practical and theoretical IT problems, planning, technical and other challenges will be derived to the differential games theory for resolution. Thus researching this theory is one of most important topics currently. A lot of researchers contributed enormous work in studying the differential games. One can observe the definition of the differential games and related different problems in R. Ayzeks’s cases. In L.S. Pontryagin’s case, enough details were drafted to solve general follower problem in the linear differential games. Simple pursuit-runaway in a compact case was studied in one of the R.P. Ivanov’s main researches. In this problem it was proved that if the count of follower objects is n-1, the runner objects can always be at free without leaving the compact surface. If the followers count is n, then they can cross the runners’ lines and catch them. In below matter it has been looked as one follower catching the other runner at l-th try in the plain square field (n=2). As we mentioned above, based on R.P. Ivanov’s evidence the runner can always move without leaving the square and not being caught by the follower. We have proved that the follower can complete the game within limited timeframe based on l-catch manner in plain square field. We have researched building the structure of constant pursuit control system that can provide completing the game in limited time frame. On top this the time that was spent on completing the game has been derived from top. Simple differential games have drawn attentions of many researchers. Because there are a great deal of problems remaining unsolved even though they were detected in simple ways. Especially in this case it’s unknown if the followers and runners can complete the game without the knowledge of each other’s location and control manner or the runners can move without being caught all the time. In other words the information about the positions of runners and followers is very important in differential games. Having said that in this research we have obtained different results compared to the known results. For instance to construct the follower control system, the information of the runner player control is not required. In the contrary in many cases to construct the follower’s control at t moment it’s necessary to know the runner’s position at v(t). Beside estimated time to complete the game will be given from top. Results of this plain field problem will allow building intellectual computer games as well.

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