On an irregular differential game
An example of a differential game is considered and its cost function found. If the cost function is known, the construction of the optimal strategies becomes relatively simple. In the regions where the cost function is differentiable, it satisfies a Hamilton-Jacobi type partial differential equation, i.e. the Isaacs-Bellman equation which, as a rule, is degenerate. Therefore we have, in position space, so-called singular sets on which the cost function is differentiable, and this creates considerable difficulties in determining this function. Only a small number of cases have been until how, for which the problem in question has been successfully solved. Although the equations of motion and the payoff function used in the present case are fairly simple, they fully illustrate the difficulties mentioned above. The results obtained can be of use in studying singular surfaces. The formulas obtained can be used to control the algorithms developed for computing the cost of the game. It should also be noted that the possibility, for the case in question, of applying differential inequalities to the cost function was tested.
Hamilton-Jacobi type partial differential equation
scientific article; zbMATH DE number 3985011