Some game problems in distributed controlled systems

Differential games of pursuit and evasion are considered in a system described by a partial differential equation containing an elliptic operator and additively occurring control parameters. Spaces are introduced using generalized eigenvalues and generalized eigenfunctions of this operator which depend on a non-negative parameter. Four versions of the formulation of the game problems are studied which differ in the constraints imposed on the control of the players. In the case of two of the versions, sufficient conditions are presented such that, when these are satisfied, evasion is possible from all initial states (the pursuit problem for these games has been studied earlier). In the case of the third version, two infinite non-intersecting sets are distinguished such that the completion of a pursuit is possible from the points of one of them and evasion is possible from the points of the second set. In the case of the fourth version, the possibility of completing a pursuit from any initial position in an arbitrary small neighbourhood of zero is demonstrated.