Narrow Results

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form

A differential game described by a nonlinear system of differential equations is considered in a finite-dimensional Euclidean space. The value set of the pursuer control is a finite set. The value set of the evader control is a compact set. The purpose of the pursuer is a translation of the system in a finite time to any given neighborhood of zero. The pursuer uses a piecewise open-loop strategy constructed only by using information on the state coordinates and the velocity in the partition points of a time interval. In the past work, sufficient conditions were obtained for existence of a neighborhood of zero from which the capture occurs. The statement of the capture theorem contains such a condition that some vectors set up a positive basis. In this research, we consider the case when these vectors set up a one-sided set. For this case, sufficient conditions are obtained for existence of a set of initial position, from which the capture occurs.

The purpose of this work is to study the pursuit-evasion problem and the “Life-line” game for two objects (called Pursuer and Evader) with simple harmonic motion dynamics of the same type in the Euclidean space. In this case, the objects move by controlled acceleration vectors. The controls of the objects are subject to geometrical constraints. In the pursuit problem, the strategy of parallel pursuit (in brief, the $\mathrm{\Pi }$-strategy) is suggested for the Pursuer, and by this strategy a capture condition is achieved. In the evasion problem, a constant control function is offered for the Evader, and an evasion condition is derived. Employing the $\mathrm{\Pi }$-strategy we generate an analytic formula for the attainability domain of the Evader (the set of all the meeting points of the objects), and we prove the Petrosjan type theorem describing that the attainability domain is monotonically decreasing with respect to the inclusion in time. In the “Life-line” problem, first, by virtue of the $\mathrm{\Pi }$-strategy solvability conditions to the advantage of the Pursuer are achieved and next, in constructing a reachable domain of the Evader by a control function, solvability conditions to the advantage of the Evader are identified. Differential games under harmonic motions are more complex owing to some troubles in determining optimal strategies and in building the meeting domain of objects. Accordingly, such types of games have not been fairly investigated than the simple motion games. From this point of view, studying the pursuit, evasion, and “Life-line” problems for oscillated motions arouses a special interest.

We study a differential game of one pursuer and one evader described by infinite systems of second order ordinary differential equations. Controls of players are subjected to geometric constraints. Differential game is considered in Hilbert spaces. We proved one theorem on evasion. Moreover, we constructed explicitly a control of the evader.