Guidance game problems for linear integro-differential systems of Volterra type with control action under the integral sign

The paper is focused on pursuit-evasion game situations for controlled dynamic objects, the development of which is described by linear integro-differential and integral systems of Volterra type with control action under the integral sign. As a result, the control system has new essential features that are not obtained by traditional controlled differential systems. The author formulates a new definition of game position, the calculation of which requires the total memory of control action in a moment of aiming. Well-known extreme constructions by an academician N.N. Krasovsky modified by the author are used to solve these problems.